embedding and unsolvability theorems for modular lattices

Algebra Universalis, 7 (1977) 47-84 Birkh~iuser Verlag, Basel

Embedding and unsolvability theorems for modular lattices

GEORGE HUTCHINSON

Abstract. Let R be a nontrivial ring with 1 and 8 a cardinal. Let L(R, 8) denote the lattice of submodules of a free unitary R-module on 8 generators. Let ~t be the variety of modular lattices. A lattice is R-representable if embeddable in the lattice of submodules of some R-module; ~(R) denotes the quasivariety of all R-representable lattices. Let to denote aleph-null, and let a (m, n) presentation have m generators and n relations, m, n =< to.

THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable 'word problem for any quasivariety ~, V c J i , such that L(R, to) is in V.

THEOREM. If L is a denumerable sublattice of L(R, 8), then it is embeddable in some sublattice K of L(R, 8*) having five generators, where 3" = 8 for infinite 8 and 3" --- 45(m + 1) if ~ is finite and L has a set of m generators.

THEOREM. The free 2e(R)-lattice on to generators is embeddable in th~ free 2e(R)-lattice on five generators.

THEOREM. If L has an (m, n) 2e(R)-presentation for denumerable m and finite n, then L is embeddable in some K having a (5, 1) 5e(R)-presentation.

1. Introduction

G i v e n a (2, n) s e m i g r o u p p r e s e n t a t i o n , there exists a (9, 13 + n) m o d u l a r la t t ice

p r e s e n t a t i o n such tha t a so lu t ion for the m o d u l a r la t t ice w o r d p r o b l e m yields a

so lu t ion for the s e m i g r o u p w o r d p r o b l e m [16: T h e o r e m 1, pp. 3 8 6 - 3 9 1 ] . Since Ju.

V. Mat iyasevi~ [20] has given a (2, 5) s emig roup p r e s e n t a t i o n wi th recurs ive ly

unso lvab le w o r d p r o b l e m , it fo l lows tha t the re is a (9, 18) m o d u l a r la t t ice

p r e s e n t a t i o n with unso lvab le w o r d p r o b l e m . R. F ree se p o i n t e d o u t to the au tho r

tha t the gene ra to r s m2 and m3 in [16] can be e l i m i n a t e d be c a use M 2 =

( M I v P 1 ) A ( M a v P 3 v P 2 ) and M 3 = ( M I v P I v P 2 ) A ( M 4 v P 3 ) in F(M4) . F u r t h e r

r e f inements of this na tu re led even tua l ly to the (5, 1) unso lvab le m o d u l a r la t t ice

p r e s e n t a t i o n given he re . In this case, we use the k n o w n unso lvab i l i ty of the w o r d

p r o b l e m for f i n i t e ly -p re sen t ed g roups .

I t r ema ins an o p e n p r o b l e m w h e t h e r any (4, n) m o d u l a r la t t ice p r e s e n t a t i o n

has an unso lvab le w o r d p r o b l e m . Clear ly , (3, n) m o d u l a r la t t ice w o r d p r o b l e m s

are so lvab le for any finite n (see [1: T h e o r e m 8, p. 63]). M a n y a t t e m p t s have b e e n

m a d e to solve free m o d u l a r la t t ice w o r d p r o b l e m s on 4 or m o r e gene ra to r s , bu t

Presented by G. Gr~tzer. Recdived April 7, 1975. Accepted for publication in final form April 1, 1976.

47

4 8 GEORGE HUTCHINSON ALGEBRA UNIV.

without success up to now. C. Herrmann and A. Huhn [9, 10] have recently showed that all denumerable free lattices have recursively solvable word problems for a number of varieties and quasivarieties contained in At, including ~e(R) if R is the ring of integers or is the ring of integers modulo m, m -> 2. We show by the methods of [9, 10] that all denumerable free ~s have solvable word problems if R is a finite ring with 1.

The configuration constructed to prove the unsolvability theorem can be adapted to prove the embedding theorem for denumerable sublattices of L(R, 8). It follows from this theorem that a lattice L having an (m, n) ~(R)-presentation, m, n-<to, can be embedded in some K having a (5, n) .Y(R)-presentation. Furthermore, if n = oJ and L has a recursively enumerable set of relations, then K can be chosen with a recursive set of relations. If n is finite, a more complex construction yields an embedding of L into K having a (5, 1) ~s The lattice embeddings of the above theorems don't necessarily preserve smallest and largest elements; they map into proper interval sublattices of the embedding codomains in general.

We will use the notation ~ ] ~} for presentations. If z is an algebraic type, ~ is a quasivariety if it is the set of r-algebra models of some set of identities and universal McKinsey (or "basic Horn") formulas, which are the universal closures of formulas:

(el = e2 & e3 = e~ &. �9 �9 & e2.-3 = e 2 n - 2 ) ~ e2.-1 = e2n,

for r-polynomials e,, e2, . � 9 e2n in some set of variables. It is known that a class of r-algebras is a quasivariety if and only if it is closed for isomorphic images, subalgebras, products including the trivial r-algebra, and ultraproducts. Also, a quasivariety admits direct limits of direct systems, and so an algebra belongs to a quasivariety W" if and only if all of its finitely generated subalgebras belong to og. (See [26] and [8] for discussion of quasivariety properties.) If X is a set, let W denote the free r-algebra of r-polynomials generated by X, and let gr be an arbitrary subset of W x W. Then V = ~ ~} is the ~-algebra generated by X subject to the relations ~ . That is, V is in ~" and there is a r -homomorphism h:W---> V such that h(dI) = h(d2) .for all (dl, d2) in ~ , and h is onto V. The defining universal property for ~ is the following: If Vo is in ~ and f : W--~ Vo is a r-homomorphism such that f(dl) = f(d2) for all (dl, d2) in ~ , then there is a unique r-homomorphism f*: V--~ Vo such that f*h =f . For a quasivariety ~ , ~{X I ~} exists for any such X and �9 (see [16 p. 386] for references). The homomorphism h above is called the "canonical projection." The word problem for ~ { X [ ~ } is recursively unsolvable if there is no recursive function computing the predicate h(xl) = h(x2) for all (x~, x2) in W x W.

Vol 7, 1 9 7 7 Embedding and unsolvability theorems for modular lattices 49

Throughout the paper, "denumerable" will mean "finite or denumerably infinite."

For a ring R with 1 and unitary left R-module M, F(M; R) or F(M) will denote the lattice of submodules of M. Given R and a cardinal 8, M(R, 8) will denote a free R-module freely generated by a set of 8 elements, so L(R, 8)= F(M(R, 8)).

We will sometimes refer to recursive functions between sets of polynomials of given algebraic types. In an appendix to this paper, we give definitions and results sufficient to justify the recursiveness assertions of the main text.

The author is grateful for helpful comments and additional references supplied by the referee.

2. An unsolvable word problem for relation algebras

In this section, a variety of algebras based on "additive relations" [3, 18, 21] will be introduced. We will construct a relation algebra presentation with two generators and one relation having a recursively unsolvable word problem. This construction is later used in the proof of the main unsolvability result.

Suppose R is a ring and M is a left R-module. The set of submodules of M x M is a lattice under inclusion. In addition, submodules of M • M may be regarded as "additive" relations on M, and provided with the relation operations of composition, converse and 1, the diagonal relation. The composite of additive relations is additive, the converse x#={(m2, mt) : (ml , m 2 ) E x } of an additive relation is additive, and the diagonal 1 = {(m, m): m ~ M} is additive. (We can also form composite additive relations x2xl:M~---~M3 from additive relations xl:M~--~ M2 and x2:M2---~ M3, and can form the additive relation converse X #" M2 ~ M~ of an additive relation x : M1 --~ M2, for any R-modules M~, M2, M3.)

Note that graphs of R-linear maps M~ --~ ME are precisely those additive relations which are everywhere-defined and single-valued, and similarly for endomorph- isms M --~ M.

DEFINITION. Let To and "roL denote the algebraic types (2, 1, 0) and (2, 1, 0, 2, 2), respectively. Let ~3 denote the variety of groups as To-algebras, with binary product xy, unary inverse x -1 and constant 1. Let FoL(M z) denote the set of submodules of M x M as a ~'oL-algebra, with binary composition xy, unary converse x #, nullary diagonal relation 1, and binary meet and join x A y and x v y, respectively. The first three ToL-operations are called the "multiplicative" operations, and the final two are "lattice" operations. We say that composition and product, converse and inverse, and diagonal 1 and group unit 1, respectively, are "corresponding" operations.

5 0 GEORGE HUTCHINSON ALGEBRA UNIV.

The variety S~o of ~'Ga-algebras is determined by the following identities:

(1) Commutativity, associativity and absorption laws for A and v . (2) Associativity for composition and 1 is an identity for composition. (3) (x#) # = x and (xy) # = y#x #.

That is, A in ~o is a lattice for meet and join and a monoid for composition and 1, and converse is a map A ~ A of period 2, antihomomorphic for composition. It is easily verified that FGL(M 2) is in S~o. Any quasivariety s~ of ZGL-algebras contained in S~o is called a "quasivariety of relation algebras."

T H E O R E M 1. Let ,~ be a quasivariety of relation algebras. Suppose that s~ contains FGL(M2), where M is the free left R-module generated by a denumerably infinite set, for some nontrivial ring R. Then there exists a presentation ~{Zl, z2[(bl , b2)} with two generators and one relation having a recursively unsolvable word problem.

Proof. By a well-known result obtained independently by W. W. Boone and P. S. Novikoff (see [22: Theorem 12.12, p. 298]), there exists a group presentation with finitely many generators and relations having a recursively unsolvable word

�9 problem. Using the embedding theorem of G. Higman, B. H. Neumann and H. Neumann [I2: Theorem IV, p. 251], we can construct a group presentation ~J{gl, g21A} with 2 generators and finitely many relations having an unsolvable word problem. As usual, we can suppose that

A={(p,(gl, g2), 1 ) : i = 1, 2 . . . . . m},

for suitable group polynomials pi(Xl, X2), i--- < m. We now outline the construction of the required presentation. In a lattice,

hence in an s~-algebra, the equation Ai%1 ei =~/~%1 ei is equivalent to e~ = e2 = . . . . e , . I f H is the set of elements of A =s~{Zl, z2 [ (b~, b2)} generated from Zx

and z2 by the multiplicative operations, and bl = b2 in A implies Z~Zx = ZxZ~ = z#2z2 = z2zg2 = 1 in A, then H is a group under the multiplicative operations. Let qi(xl, x2) be the TeL-polynomial obtained from the 1"G-polynomial pi(xl, x2) of A by replacing group operations by the corresponding multiplicative ~'Ga-operations. Let bl = / ~ 5 el and b2 = V ~ 5 e~, where el, e2 . . . . , e,,+5 equals:

1, Z~l Zl, z , z f , z~z~, z ~ z L qi(z,, z~) . . . . . qm(z,, z~).

We see that bl = b2 in A implies that H is a homomorphic image of G = ~J{gl, g2 [ A}. The hypothesis that F G L ( M 2) is in s~ is used to prove that H is isomorphic to G. It then follows that a solution of the word problem for s~{Zl, ZE[(bl , b2)} also solves the unsolvable word problem for ~J{gl, g2[ ~1}.

Vol 7, 1977 Embedding and unsolvability theorems/or modular lattices 51

To establish this contradiction, we consider Fig. 1 below. Here, W1 is the free l-a-algebra of group polynomials generated by gl and g2, and a : W~ ~ G is the canonical ~-G-homomorphism onto G =q~{gl, gzlA}. Also, W2 is the free rat.- algebra of ~-~L-polynomials generated by zl and z2, and [3:Wz.-.-> A is the canonical l"GL-homomorphism onto A = ~{Zl, z2 [ (bl, b2)}. Since G is denumerably infinite, we can suppose that M is the free R-module generated by the set G. Define ~b:G--~ FGL(M 2) by letting ~b(g) for g in G be the R-submodule of M x M generated by {(h, gh): h ~ G}. We can consider W~. A and Fc,,I.(M 2) to be

W1

G P v

Figure 1

~-~-algebras by replacing group operations by the corresponding multiplicative ~'aL-operations, and then ~b is a ~'G-homomorphism by direct computation. Since R is nontrivial and M is free, q~ is one-one. Using the universal property for free algebras, let tz: Wx---> W2 be the ~'a-homomorphism such that p.(g~)=z~ and /~(g2) = z2, and let v : WE ~ FGL(M:) to the ~-~L-homomorphism such that v(z~) = tha(gl) and v(z2) = t~ot(g2). Since v/~(gl) = ~ba(gl), V/-L(g2) = (~a(g2) and all mapsare za-homomorphisms, we have v/z = ~bct. Since v(bl) = v(b2) follows from v/~ = ~ba, there exists a ~'GL-homomorphism v*:A--> _F'GL(M 2) such that v*fl =v, from the defining universal property of A =s~{z~, z2l(bl , b2)}. By the relation algebra axioms, the ~'a-subalgebra /3tz[W~] of A has an associative product operation with. identity element 1. By induction on the length of the "ra-polynomial d, we can prove that [3~(d) # is an inverse for /3t~(d), using the equation fl(bl) =/3(b2) and the relation algebra axioms. So,/3/~[W1] is a group, and/3(b0 =/3(b2) implies that /3~(pi(gl, gz))=/3/~(1) for all i<=m. Therefore, there exists a l"a- homomorphism p,* : G ~ A such that /x*a =/3/z, using the universal property of ~{gl, g21 ~}- Since ~a = vtz = v*fllz = v*/~*a and a is onto, q~ = v*tz*. Since q~ is one-one, so is tz*. Since ~* is one-one and tz*a =/3/A the predicates a ( w 0 = a(w2) and /3/x(w~)=/3/x(w2) are equivalent for all (w~, w2) in W ~ x W ~ . By Proposition A in the appendix, ~ is recursive. So, if F(ex, e2) is a recursive

52 GEORGE HUTCHINSON ALGEBRA UNIV.

function on W2 x W2 solving the word problem for ~{zl , z2 [(bl, b:)}, ther~ F(l~(wl), ~(w2)) is a recursive function on W1 x W1 solving the word problem for ~g{gl, g21 A}. This contradiction proves Theorem 1.

3. Modelling additive relations in modular lattices

In [14], a theory of formal homomorphism graphs in a modular lattice was developed, similar to the von Neumann coordinatization techniques [27]. In [16], these methods were applied to construct modular lattice presentations with unsolvable word problems. We now generalize this technique to a theory of formal additive relation graphs in a modular lattice. Throughout this section, L will denote an arbitrary modular lattice.

DEFINITION. As in [14], 2 will denote the 2-element lattice {0, 1} with 0 c 1, and lattice homomorphisms A : 2 ~ L (corresponding to intervals of L) will have the associated notations A 1 and A ~ for A(1) and A(0), respectively, and A 1/A~ for A. Lattice homomorphisms 2 ~ L are called "objects" of L, and two objects A, B : 2 ~ L are called "r-disjoint" if A l/x B 1 = AO/x B o. A sequence of objects A1, A2 . . . . . A n : 2 ~ L for n_->2 is a " lef t" sequence if ( A l v A ~ v ' ' ' v A ~ - I ) A A ~ = A ~ 1 7 6 ' ' ' A A ~ for l<i<=n. A "mixed" sequence of objects is a sequence obtained by permuting the terms of a left sequence. A permutation or subsequence of a mixed sequence of objects is mixed, and distinct terms of a mixed sequence are r-disjoint in particular. We will use the distributivity lemmas [14: 3.1, 3.2, 3.3, p. 163] in the subsequent development.

Let U denote the lattice of Figure 2. Hereafter, let f - denote f ( e - ) for any

>,, .

Figure 2

Vol 7, 1977 Embedding and unsolvability theorems for modular lattices 53

lattice homomorphism f:U---> L, L a modular lattice. If A, B :2---> L are r- disjoint objects, let SR(A , B) denote the set of lattice homomorphisms f:U---~ L such that f (G) = A ~, f (H) = A ~ f(G') = B ~ and f(H') = B ~

PROPOSITION 1. The lattice U is the modular lattice generated by {G, H, G', H', e-} subject to the relations H c G, H' c G', G A G ' = H A H' and H v H' c e- c G v G'. For r-disjoint objects A, B : 2 ~ L, L a modular lattice, the function f ~ f-- is a one-one correspondence from S R ( A , B ) to the interval sublattice [ A ~ ~ A l v B 1] of L.

Proof. Let V be a real 8-dimensional vector space with basis {Vl, v2 . . . . . v8}, and let V(ul, u 2 , . . . , un) denote the vector subspace of V generated by vectors ul, u2 . . . . . un in V. Then U is isomorphic to the sublattice of V generated by H = V(vl), G = V(vb V2, V3, V4), H ' = V(vs), G'= V(vs, 1)6, I)7, Vs) and e - = V(vl, v2, v5, v6, V3-DT), by direct computation. So, U is a modular lattice generated by {G, H, G', H', e-}. Since the given relations hold in U, U is a homomorphic image of the modular lattice L generated by {G, H, G', H', e-} subject to the given relations. Now, a 25-element sublattice Lo of L isomorphic to U with e- deleted is generated by the two chains:

H A H ' c H c G A e - c ( G ' v e - ) A G c G and

H A H ' C H ' c G' ^ e - c ( G v e - ) A G ' c G',

computing from the relations of L using modularity. To verify that U and L are isomorphic, it suffices to show from modular lattice axioms and the given relations that any meet and join in L of e- and an element of Lo is the element of Lo U {e-} indicated by the diagram for U. We omit these computations.

Since the relations H c G, H ' c G' and G A G ' = H A H ' are the defining relations for a pair of r-disjoint objects, the second part follows. This proves Proposition 1.

DEFINITION. Suppose A, B:2--~ L are r-disjoint objects and f is in SR(A, B). Then there exists a unique homomorphism U ~ L in SR(B, A) , denoted f#, such that ( f#)-= f - . (There is an automorphism of U exchanging G with G', H with H', and leaving e- fixed.) For A ~ x c A 1 in L, let f[x] denote ( x v f - ) A B ~ in L. Note that B ~ ~.

Let S H ( A , B ) denote the set of f in S R ( A , B ) such that f [ A ~ ~ and f#[B~] = Ax; these relations are equivalent to f - A B ~ = B ~ and f - v B ~ = A ~ v B ~, respectively. Let SI(A, B) denote the set of f in SR(A , B) such that f [ A i] = B ~ and f#[B ~] = A ~ for i = 0, 1. (In [14], the definitions of S(A, B) and SI(A, B) are


essentially the same as the definitions of SH(A, B) and SI(A, B) here, respectively.)

Suppose A, B, C:2--* L is a mixed sequence of objects, f is in SR(A, B) and g is in SR(B, C). Then there is a unique map in SR(A, C), denoted g o f, such that (g o/)- = (Fv g-)A(A~v C~).

PROPOSITION 2. Suppose K is an ideal in the lattice of submodules of some module over a ring R, and let A, B : 2 ~ K be r-disjoint objects of K. Then there is a one-one lattice isomorphism o" from the lattice of submodules of A ~ / A ~ B~/B ~ (that is, additive relations A1/A~176 onto SR(A ,B) , given by negative graphs:

o'(U)- ={vl - v2 : vl ~ A x, v2~ B ~, (vl + A ~ v2 + B~ U}

and

O'-l(f) = {(Vl-h A 0, v2 + B~ v I ~ A 1, v2eB I, v , - v 2 e f - }

for U c A l /A~ x B1/B~

for f i n SR(A, B).

If A ~ x c A ~ in K and U[x] denotes the set of all v2 in B ~ such that (v~+ A ~ v2+B ~ is in U for some Vl in x, then U[x]=(xvcr(U)-)ABl=o'(U)[x] . I f U # c B ~ / B ~ ~ is the converse of U, then o'(U#)-=o'(U) -, so or(U#) = o'(U) # in SR(B ,A) . If A, B, C:2--> K is a mixed sequence and U2U1 is the relation composite of U~ c A 1 / A ~ B~B ~ and U2c B~/B ~ C~/C ~ then cr( U2 U1) = o'( U2) ~ o'( U1) in SR ( A, C). Furthermore, U is an everywhere-defined relation iff o'(U)#[B1] = A 1, U is a one-one relation iff o '(U)#[B~ ~ U is an onto relation iff o'(U)[A1] = B 1, and U is a single-valued relation iff o'(U)[A~ = B ~ It follows that ~r(U) is in SH(A, B) iff U is the graph of an R-linear transformation, and o-(U) is in SI(A, B) iff U is the graph of an isomorphism A~/A~ B~/B ~ and U # is the graph of the reciprocal isomorphism in the latter case.

We omit the straightforward calculations proving Proposition 2. If U is an R-linear map AI/A~ B~/B ~ note that the image U[x/A ~ equals o-(U)[x]/B ~ for A ~ 1, and the inverse image U-I[y/B ~ equals o'(U)#[y]/A ~ for B ~ 1.

PROPOSITION 3, I rA , B :2--> L are r-disjoint objects and f is in SR(A, B), then B 1 v F = B l v f # [ B 1] and B 1 A f - = f [ A ~ The mapping x~--~x] takes [A ~ A 1] into [B ~ B 1] and is a projectivity isomorphism in L from [f#[B~ f#[BI]] onto [f[A~ f[A~]]. The reciprocal of this projectivity isomorphism is the map


y ~ f#[y]. If f is in SI(A, B), then x ~ f[x] is a projectivity isomorphism from [A ~ A 1] onto [B ~ B1]. IrA, B, C : 2 ~ L is a mixed sequence, f is in SR(A, B), g is in SR(B, C) and A~ c x c A 1, then (go f)[x]= g[f[x]].

Proof. The first part follows from the hypotheses and modularity. Again by modularity and the hypotheses, f[x]=f[Xo] for Xo=f#[B~ in [f#[B~ f#[BI]]. The second part then holds because [GAe- , (G've- )AG] and [G'Ae- , ( G v e - ) A G ' ] are both transposes of [e-, ( G v e - ) A ( G ' v e - ) ] in U. (Check Fig. 2, and see [1: Theorem 13, p. 13].) Assumethe hypotheses of the final part. Applying [14: 3.3] to B, C, A, it follows that B l v g - , C 1 and A 1 distribute, and so A ~ A C I C C ~ g- implies that:

(x v f - v g-)A Cl ~ (A l v B l v g-)A Cl c B l v g -,

But x v F , B ~ and g- also distribute by [14: 3.3], so:

(g o f ) [ x ] = (x v (g o f ) - ) ^ c ' = (x v f - v g - ) ^ c ' =

(x v f - v g-) A (B1 v g-)A C 1 =

(((x v f-) A B 1) v g-) A C 1 = g[f[x]],

using modularity. This proves Proposition 3.

PROPOSITION 4. If A, B :2 ~ L are r-disjoint objects and f is in SR(A, B), then (f#)#=fi If A, B, C :2---~ L is a mixed sequence, f is in SR(A, B) and g is in SR(B, C), then (g o f ) # = f # o g#.

Proof. By direct computation from the definitions, using Proposition 1.

PROPOSITION 5. If A, B :2---~ L are r-disjoint objects and f is in SI(A, B), then f# is in SI(B, A). If A, B, C : 2 ~ L is a mixed sequence, f is in SI(A, B) and g is in SI(B, C), then g o f is in SI(A, C).

Proof. Use Propositions 3 and 4.

PROPOSITION 6. If A, B :2---~ L are r-disjoint objects and f, g are in SR(A, B) such that f f c g-, f~[B 1] = g#[B 1] and f[A ~ = g[A~ then f = g. If A, B, C:2--~ L is a mixed sequence, f is in SR(A ,B) and g is in SR(B, C), then (g ~ D of # = g if f is in SI(A, B), and g# o (g o f) = f if g is in SI(B, C).


Proof. Assuming the hypotheses of the first part, note that B t v f - = B ~ v f # [ B 1 ] = B I v g # [ B ~ ] = B ~ v g - and B ~ A f - = f [ A ~ 1 7 6 - by Proposition 3. Therefore, f - = g - by [1: Lemma 1, p. 36], and so f = g by Proposition 1.

Assume the hypotheses of the second part, and suppose that f is in SIVA, B). then g - c f - v A ~ v C 1, and so

g- c ( f - v [ ( A ' v C 1) A ( f - v g-) I ) A ( B ' A C 1) = ((g of)o f # ) - ,

using modularity. We then have (g o f) o f# = g by applying the first part above and using Propositions 3 and 4. The remainder of the second part can be proved. similarly, completing Proposition 6.

PROPOSITION 7. (Associativity) Suppose A, B, C, D :2 -* L is a mixed sequence, f is in S R ( A , B ) , g is in SR(B, C) and h is in S R ( C , D ) . Then h o (g o JO = (h o g) o f = k in SR (A, D), where k - = (f- v g- v h-) A (A t v D 1).

Proof. Assume the hypotheses, and define k in SR(A, D) with the given k- by Proposition 1. Clearly ((h o g) o D- c k-.

We first prove the lemma: If'w1, w 2 c A t v B 1 v C 1 and either f - v g - c w I or f - v g - c w2, then wl, w2 and D ~ distribute. Assuming the.lemma hypotheses, this follows because:

( A ~ 1 7 6 1 7 6 1 7 6 1)

c (wlvw2)AD 1C ( A t v B t v C I ) A D 1)

= ( A ~ 1 7 6 1 7 6 ~ ,

using [1: Theorem 12, p. 37] and [14: 3.2]. Using the lemma with wi = A 1 and W 2 = f-- V g - v h#[D 1], we obtain:

k#[D1]= A 1 A ( f - v g - v h - v D 1)

= A 1 A (.f- V g - v h#[D1] v D 1)

= (A1A (f-V g - v h#[D1]))v(A1A D l)

= A t A ((g of)- V h#[D ~]) = (g of)e[h#[D ~]]

= (ho (g o f))#[D1],

also applying modularity, Propositions 3 and 4 and A1A D I c A ~ f- .

Vol 7, 1977

Now use the

k[A ~ = D 1 ̂

= D 1 A

= O l A

= D 1 A

Embedding and unsolvability theorems for modular lattices

lemma with w l = f - v g - and Wz = C ~, so that:

k-

(h - v ((f-- v g-) ^ ( C 1 v D 1)))

( h - v ( ( f - v g-)A C 1 ) v ( ( F v g-)A D1))

(h- v (go f)[AO]) = (ho (g o f))[A~

57

using modularity, Proposition 3 and ( f - v g - ) ^ D 1 c D ~ But then h o (g of) = k by Proposition 6. Since f # o ( g # o h # ) = k # by the same argument, we have (h o g)o ; f= k by taking converses and using Proposition 4. This completes the proof of Proposition 7.

4. An unsolvable modular lattice word problem

We begin by establishing a framework useful for both the modular lattice unsolvability and embedding theorems. Let "rL denote the algebraic type (2, 2) for lattices, with operations meet x ^ y and join x v y. A "quasivariety of modular lattices" is a quasivariety of ~z-algebras contained in ~ , the variety of all modular lattices.

A subset U of L in ~t such that card ( U ) ~ 2 is " independent" over y in L if V {x : x ~ U1} ̂ V {x: x e u2} = y for any finite disjoint nonempty subsets Ut and U2 of U. By modularity, V { x : x ~ U1}AV{x :x~ Ua} = V { x : x e Ulf'l Uz} if Ut and U2 are finite but not disjoint subsets of an independent set U. A subset of cardinality 2 or more of a set independent over y is also independent over y. A finite U = {xt, x2 . . . . . xn} for card ( U ) = n---2 is independent over y if[ we have:

( x i v x z v �9 ' ' v x i - 1 ) A x i = y ,

for 1 < i N n, by [1: Corollary 2, p. 74]. In general, W(X) will denote the free ~'L-algebra of all lattice polynomials

generated by a set X of variables. Let Z = {zx, z2, z3, z4, zs}, generating lattice polynomials W(Z).

P R OP OS I TION 8. Suppose L is a modular lattice containing elements M, for iN4 and Pj for j<-5 such that (1) {M1, M2, M3, M4} is independent over Mo = M1 ^ ME ̂ M3 A M4, (2) there exist hi in SI(MJMo, M2/Mo), h3 in SI(M31Mo, MJMo) and h4, h5 in SH(M4/Mo, MJMo) such that h[ = Pj for j = 1, 3, 4, 5, and (3) P2 c M2 v M3 and P2 ^ M: = Mo. Then there is a sublattice K of L having five generators and containing M, for i <-_ 4 and Pj for ] <- 5.


Specifically, let f: W(Z)---~ L be the unique 1"L-homomorphism satisfying the equations below.

(i) f ( z l ) = M1, f(z2) = P3, f(z3) = M = v M3,

f(7`4) = P1 v P4, f(7,5) = P2 v Ps,

Define mi for i <= 4 and pj for j <= 5 in W(Z) by the equations below.

(ii) m 1 = z1 , m 2 = Z 3 A ( Z 1 V Z 4 ) ;

I"1"l 3 = Z 3 A (Z2 V ((Z1 V Z4) A (7,1 V 7,5))),

1714 = (7' 3 V 7,2) A ((7 ,1V 7,4) A (7,1V Z5)) ,

Pl = 7,4A(Z1V T,3), P2 = Z3AZs , P3 = Z2,

P4 = 2:4 A (2:1V ZS), P5 --'-- 7̀ 5 A (7`1V Z4).

Then f(mi) = Mi for i <= 4 and f(Pi) = PJ for j <- 5.

Proof. Assume the hypothesis. Then f (ml)= M1 directly, and f(mz)= M2 by modularity and the hypotheses, since M2 = (ME v M3) A (ME V M1 V M4).

Now M3/Mo, M2/Mo, M4v M1/Mo is a mixed sequence in L by (1), and so by (3) and [14: 3.3] we have:

P2A (M2v M4v MI) c (P2A M2) v Mo = Mo.

Therefore, ( f (Zl )Vf(z4))A( f (z l )v f (zs) )=M4vMI, by modularity and (2). The formulas f ( r n3 )= M3 and f ( m 4 ) = M4 then follow by modularity, (1) and (2).

By using [14: 3.3] as above, we see that P4A(M1vM2vM3)=Mo, and f(PO = PI follows by modularity. Clearly PsA(M2vM3)= Mo by (1) and (2), and f(P2) =P2 follows from modularity and (3). We have f(p3) =P3 directly, and f(pa) = P4 and f(Ps) = P5 follow from the hypotheses and the equation for M4 v M1 given above. This completes the proof of Proposition 8.

If L = L(R, 8) for some ring R and cardinal 8, then we can construct R-linear transformations from elements of L satisfying the Proposition 8 hypotheses, by Proposition 2. In Fig. 3, we give the appropriate diagram, including an additive relation o--1(h2) corresponding tq h2 in SR(M:/Mo, M3/Mo) such that h~" = P2.

DEFINITION. In a modular lattice L, a "relative n-frame" for n >= 2 consists of a mixed sequence A1, A z , . . . , A , : 2 ~ L plus hj in SI(Ai, Ai+l) for j =< n - 1.


o-1(h4) M4/Mo �9 M1/Mo

M3/Mo 0.. ~ (h2) M2/Mo

Figure 3

If L has a smallest element O, then the relative n-frame above is called an "absolute" n-frame if A ~ = O for i _-< n. The terminology is suggested by the yon Neumann coordinatization theorem [27: Theorem 14.1, p. 208]; a yon Neumann "normalized frame of order n" [27: p. 118] in a complemented (0, I) modular lattice is essentially the same as an absolute n4rame such that AlvA~v - - - v

1__ A , - I . Part of the utility of relative n-frames may be seen from the next result.

PROPOSITION 9. Suppose A b A2, A3, A4:2--* L and hj in SI(Aj, Ai+l) for ]<-3 form a relative 4-frarne in a modular lattice L. Then SR(A4, A~) can be made into a relation algebra, where the "rGL-operations are given for f, g in SR(A4, A~) by the equations below.

(iii) fg = (f o h3) o (h2 ~ h~ o g),

/a#) = ( h f o h f ) o (h~ o f# o he) o (h~ o hf ) ,

1 = hfo h~o he,

( f^ g)- = f - ^ g-, ( fv g)- = f - v g-.

(Here, f~#) denotes the converse operation on SR(Aa, At), and f# denotes the converse in SR(A1, A4) of f in SR(A4, A~).) If L = F(N; R) for some ring R and R-module N, then U~---> o-(Uo--l(1)) determines a 1"eL-isomorphism FGt.(M2) -'>

SR(A4, AO with reciprocal f~-->o'-l(f)o'-l(1) - t , where M = A ~ o 1/A1 and o--~(1): AI/AO ~ ~ o A 1/A t is the R-linear isomorphism o'-l(hl)-l~-l(h2)-lcr-l(ha) -1.

Proof. Assume the hypotheses. Since SR(A4, At) is lattice isomorphic to the interval sublattice [ A ~ ~ A~vA~] of L by Proposition 1 and the lattice equations of (iii), SR(A4, A1) is a modular lattice. From Propositions 1 and 7, it is clear that all operations are well-defined and closed for SR(A4, A1). To prove associativity of composition, repeated use of Proposition 7 is made, as in [16: p.


390]. The identities f l = i f = f , f(#~(#~=f and (fg)C#~= g~#~)a#~ are provable by Propositions 4, 5, 6, and 7; we omit these computations. So, SR(A4, AO is in S~o.

If L = F(N; R), we can verify by direct computations from (iii) and Proposi- tion 2 that U,--> o'(Uo'-1(1)) and f~--~ O'-l(f)o'-l(1)-i determine reciprocal r~L- isomorphisms between F~L(M") and SR(A4, A1). This completes Proposition 9.

We now develop techniques for constructing a relative n-frame under suitable conditions.

PROPOSITION 10. Suppose f : W---> L is a rL-homomorphism for W a rL- algebra and L a modular lattice, and let a 1, a ~ b t, b ~ and e ~ be elements of W such that A = f( a 1)/f( a o) and B = f( b 1)/f(b~ are r-disjoint objects of L. Define e 1, c -1, c ~ d l, d o in W by the equations below.

(iv) e I = (a~ b~ e~ ^ ( a Iv bI),

c l = ( b l v e l ) A a l ' c~ = el ^ a l,

d l = ( a l v e l ) A b 1, d ~ 1.

Then f( a ~ c f( c ~ c f( cl) c f( a t) and f( b ~ c f( d~ c f( d I) c f( b t), and there exist g in SR(A , B) and h in SI(C, D) such that g - = h - = f(el), where C=f (c l ) / f ( c ~ and D =f(dl) / f (d~ I f there exists go in SR(A , B) such that go=f(e~ then g = go. I f there exists ho in SI(A, B) such that ho-'- f(e~ then C = A, D = B and h = ho. I f there exists ho in SI(A, Bo) such that h o = f ( e ~ and B ~ c B ~ then D = B .

The proof, by direct calculation and Proposition 1, is omitted.

DEFINITION. If dl, d2, d3, d4, ds, are elements of a l-c-algebra W, let IV (db d2, d3, d4, ds; d6, d7, ds, d9, dlo) indicate that dl, d2, d3, d4, d5 are to be substituted fo r e ~ a 1, a ~ b 1, b ~ respectively, in the equations (iv), and that d6, d7, ds, d9, dlo then represent elements of W corresponding to e I, c l, c ~ d l, d ~ respectively, as given by the equations (iv) with the indicated substitutions. In some cases we need only the definition of e 1, which is indicated by IV(dl , d2, d3, d4, ds; d6).

In the next result, we use Proposition 10 repeatedly to construct a relative n-frame from arbitrary elements of a modular lattice, in such a way that if the given elements already formed an absolute n-frame, then the constructed relative n-frame equals the given absolute n-frame. (The referee points out that similar constructions occur in [7, 11, 13].)


The statement of the result is quite complex, so we will consider the case of

4-frames before turning to the general statement. Suppose W is a ~-L-algebra containing elements mi for i_--<4 and p~ for j<_-3, and g : W ~ L is a ~'L- homomorph i sm into a modular lattice L. Define mi for 0 <_- i -<_ 4 in W as follows:

For l<-_i<j<k<-4, m#k=m~vmjvmk.

i n 0 ~--- m123 A Fr1124A m 1 3 4 A m234~

1~11 -~" /'/I,123 A /"F/124 A m134, !i!2~--- /'F/123 A/T/124A m234,

!113 ~--- /1'/123 A DI134A m234, 1114 -~/T/124 A/T/134A m234.

By ordinary modular lattice calculations, {g(ral), g(m2), g(m3), g(m4)} is indepen-

dent over g(mo) in L. Now define el, ea . . . . . e25 in W by:

IV(pl, m~, too, m~_, too; e~, e2, e3, e4, es),

IV(p2, e4, es, m3, too; e6, e7, e8, e9, elo),

IV(p3, e9, e~o, m4, too; e~l, e12, e13, e14, ets),

_/'V(e6, eT, es, el2, e~3; et6, e17, els, e19, e20),

IV(et, e2, e3, e~7, e~s; e21, e22, e23, e24, e25).

In Figs. 4 and 5, we show partial sublattices of L corresponding to the first three

and last three of the five terms above. (The points g(en) have been labelled "n.")

g(ml) ~ g(rn2) ~ g(m3) g(m4)

2 9

14

g(mn) Figure 4

62 GEORGE HUTCHINSON ALGEBRA UNIV. g(ml) g(rn 2) g(m 3) g(m4)

2 ~ 21 14

g(mo) Figure 5

By Proposition 10, there exist ki in SI(g(ei+l)/g(ei+2), g(ei+3)/g(e~+4)) such that ki = g(e,) for i = 1, 6, 11, 16, 21. The hypothescs of the last part of Proposition 10 are satisfied for the fourth and fifth terms above, so g(elg)/g(e2o)= g(e~2)/g(e13) and g(e24)/g(e25) = g(e~7)/g(e~s) as shown in Fig. 5. It then follows that B~, B2, B3, B4 and k2t, k16, k11 form a relative 4-frame in L, for Bl = g(e22)/g(e23), B2 =

g(el7)/g(ela), B3 = g(e12)/g(e13) and B4 = g(e14)/g(e15). Furthermore, if g(mi) and g(pj) already form an absolute 4-frame in L, then the constructed 4-frame is equal to it. More precisely, suppose A1, A2, A3, A4:2--* L and hj in SI(Aj, Aj+O for ]_-<3 form an absolute 4-frame in L such that A~ = g(m~) for i_-<4 and hT= g(pi) for j_-<3. Then {g(ml), g(m2), g(m3), g(m4)} is independent over O in L, so g(mo) = O and g(m~)= g(m~) for 1_- < i_-<4. So, A~ =Bi for i_-<4 and hi = kl = k21,

h2 = k6 = k16 and ha = kH, by Proposition 10. We now state and prove this result for all n-frames, n_-> 2.

PROPOSITION 11. Suppose W is a "eL-algebra and g: W--->L is a "rr.- homomorphism for a modular lattice L. Let n >= 2, and suppose mi for 1 <- i <= n and pj for 1 <= ] < n belong to W. Define mi for 0 <= i <= n in W by the equations below.

(v), For l<-i<=n, m * = V { m j : ] < - n , ] # i } .

mo=m*Am2*/x ' ' ' Am*,

* ' ' < j# i} . For l<-i<=n, m i = A { m j . i = n ,

Vol 7, 1977 Embedding and unsolvabitity theorems for modular lattices 63

Now define e~ for 1 N i <= 1 0 n - 15 in W as shown below.

(vi) First: I V ( p b ml, too, mz, too; el, e2, e3, e4, es).

For ] = 2, 3 , . . . , n - 1, successively:

IV(pi, esj-6, es~-s, m/+l, llllo; esi-4, e5i-3, esi-2, esi-1, esi).

For ]= n, n + l . . . . . 2 n - 3 , successively, with i = 2 n - 2 - ] :

IV(e5~_4, esl-3, e5i-2, e5i-8, e5i-7; esi-4, esi-3, esi-2, esi-1, esj).

Then there is a relative n- frame B~, B2 , . � 9 B , : 2 ~ L and k~ in SI(B, , Bi+I) for 1 N i N n - 1 in L, given as follows:

(vii) For 1 N i <-- n - 1 and ] = 2 n - 2 - i, B~ = g(esi-3)/g(esi-2)

and kT= g(esi-4).

B . = g(es.-6)/g(es.-s) .

I f there is an absolute n- frame Ai :2 --~ L such that A~ = g(mi) for 1 N i N n and

h i in S I ( A i, A~+x) such that h [ = g ( p i ) for l < - ] < n , then A i = B ~ for i n n and

hi = ki for ] < n.

Proof. Assume the hypotheses. By computation, {g(ml) : i N n} is independent over g(mo) in L. Applying Proposition 10 and using induction on j for the first n - 1 terms of (vi)n, we have:

There exists 1~ in SI(g(esi-a)/g(esi-2), g(esj-1)/g(esi)) such that ~ = g(esi-4), for l < - - ] N n - 1 .

g(mo) c g(e3) c g(e2) c g(m~).

For 2 N ] N n - I :

g(mo) c g(esi-5) c g(esj-2) c g(e5i-3) c g(esi-6) c g(mi).

g(mo) c g(es,-5) c g(es,-6) ~ g(m,).

We now use Proposition 10, the relations above, and the last n - 2 terms of (vi)n to prove by induction on ] that:

For n < = j N 2 n - 3 and i = 2 n - 2 - ] : There exists/~ in SI(Bi, Di+l) such that f f = g(esj-4), where Di+l = g(es;-1)/g(esj), and D~+I = Bi+l. g(mo) c B ~ B~ c g(m,).


Defining k~ = f2n-2-i, we have kl in SI(B~, B~+~) for 1 ~ i ~ n - 1 . By the indepen- dence of {g(mi) : i -< n} and the inclusion relations above, it is easily checked that B1, B2 . . . . , B, is a mixed sequence in L. Therefore, B~ for i_- < n and ki in SI(Bj, Bj+I) for ] < n form a relative n-frame in L.

Now suppose that there is an absolute n-frame A~ such that A~ = g(m~) for i_-_n and hj in SI(Aj, Aj+I) such that h~=g(pj ) for ] < n . Since A ~+x ̂ (A ~ v �9 �9 �9 v A ~) = O for 1 _-< i < n, clearly {g(mi) : i _--- n} is independent over O in L. So, g(mo) = 0 and g(m~) = g(m~) for i - n. From (vi), and Proposition 10, we have h T = g(esj-4), Ai = g(esj-3)/g(e51-2) and Ai+I = g(esj-1)/g(esi) for 1 ---]-< n - 1 . So, B , = A , and k,_x=h,_~. Continuing, we have for n<-_j<=2n-3 and i = 2 n - 2 - ] that h? = g(esj-4), Ai = g(esi-3)/g(esi-2) and A~+~ = g(esi-1)/g(e~i). But then B~ = A~ for i _-< n and k~ = h~ for j < n, completing the proof of Proposition 11.

Combining Theorem 1 and Propositions 8, 9 and 11, we obtain the modular lattice unsolvability theorem.

T H E O R E M 2. Suppose ~ is a quasivariety of modular lattices, ~ A t , such that L(R, to) is in ~ for some nontrivial R. Then there exists a (5, 1) presentation ~ I (ut, U2)} having a recursively unsolvable word problem.

Proof. Assume the hypotheses, and let M = M(R, oJ). In [16: Theorem 1], the semigroup of R-linear transformations for a certain R-module M was modelled by negative graphs in F(M4; R), where M4=M(~M(~M(gM. A binary algebraic function 12 for lattices was chosen by coordinatization techniques [27, 14], so that semigroup composition was modelled by 12. We now modify and extend this procedure, using algebraic functions related to the equations (iii) of Proposition 9 to map the relation algebra FoL(M 2) into F(M4).

For our purposes, M is chosen to be the free R-module on the denumerably infinite set G as in Theorem 1. Since F ( M 4) is isomorphic to L(R, o~), it belongs to ~ by hypothesis. For i = 1 , 2, 3, 4, let e~:M--->M 4 be the ith injection (el(v) = (v, 0, 0, 0), etc.), and let Mi = ei[M] in F(M4). For i=<4, let Ai = M.,/O, an object of F(M4), and let hj in SI(Aj, Aj+I) be given by:

h~=Pi={e~(v)-ei§ for j<-3.

Then A~ for i = < 4 and hj for ] =< 3 form an absolute 4-frame in F(M' ) , since {M1, M2, M3, M,} is independent over O in F(M4). By Proposition 9, SR(A4, At) is a relation algebra for the l"~L-operations of (iii), and a ~-GL-isomorphism FGL((MJO)2)--* SR(A4, A~) is given by U~--> r Since M and M J O are isomorphic R-modules via a modification of c~, a z~L-isomorphism K* : F~L(M 2) ---> SR(A4, A~) can be defined. A lattice homomorphism


W7 ~ W2 e W3

G P" 0" L

, , ,. [ ' ( M 4) rsL(MZ) x

Figure 6

t~:/~c;t.(M2)--~_F'(M 4) can be obtained by taking negative graphs: (K*(U))-. Specifically, we have:

K ( u ) =

K(U) ={C4(1)1)--C1(/)2): (l)l, 1)2)~ U}.

(Compare ~ with F in [16: pp. 387-388].) Consider the diagram of Fig. 6, extending Fig. 1. Let W3 denote W(Z), and

define rnl for i _-< 4 and pj for j _-< 5 in W3 by the equations (ii) of Proposition 8. Let : W3 ~ F(M 4) be the unique TL-homomorphism satisfying equations (i) for the

given M, i<_-4, and Ps, j-<_3, in F(M4), plus P4= K~ba(gl) and P5 = K~ba(g2). By Proposition 8, ~'(mi)= M~ for i_--__4 and ~(pj)= Pi for j--_ 5.

We now define m~ for 0_-<i=<4 in W(Z) by equations (v)4 of Proposition 11, and el, e2 . . . . . e2~ in W(Z) by the terms (vi)4 of Proposition 11 applying the equations (iv). By Proposition 11, the equations (vii)4 hold in F(M 4) for g replaced by ~, B~ replaced by A~ for i_- <4 and kj replaced by h i for j <_-3.

Dependent upon our subsequent definition of Ul and u2 in W3, let L = ~ [ (ul, u2)} and let 3/: W3--~ L be the canonical projection. Defining B~:2---> L for i <-_ 4 and ki in SI(Bs, Bi+l ) for j <- 3 by the equations (vii)4 with 3' replacing g, we obtain a relative 4-frame in L by Proposition 11.

We now make W3, L and F(M 4) into ~'oL-algebras, retaining the lattice operations and introducing multiplicative operations by algebraic functions as given below.

(viii) For e, e' in W3,

ee'= (p(e)vp'(e'))A(e14ve22), for

p(e)=(eve11)/x(elave22) and

p'(e') = (e 'v e21 v el6) A (el4 v e12).


e # = (P2,3 V p"(e) v p 1.2) A (el, v e22),

P2.3 = (e~t v e16)A (ei4 v e17),

p"(e) = (e~ v e v e11)A (e~7 v el2)

Pl.2 = (e 16 V e2t) ̂ (e~2 v e22).

1 = (e~ v e~6 v ez~)/x (el~ v ez2).

for

and

To define the multiplicative operations in L, replace e~ by 3"(e~) in (viii), and similarly replace e~ by ff(ei) in (viii) for F(M4). Then 3, and ff are roc- homomorphisms. Comparing the equations (viii) with the equations (iii), we see that f~-->f-:SR(A4, A1)--* F(M 4) is a one-one ZGL-homomorphism, and so K : F6L(M 2) ~ _F'(M 4) is a one-one ~'or-homomorphism.

We now define e26 and e27 in W3 by the equations below.

IV(p4, el4, ezs, e22, e2s: e26),

IV(ps, e14, e15, e22, ezs; e27).

By Proposition 10, there exist maps in SR(B4, BI) with negative graphs 3'(e26) and y(e27) in L, and also ~(e26)= P4 = Kqbo~(gl) and ~(e27)= P5 = K~ba(g2) in F(M4).

Let 0 : W2 ~ W3 be the unique ~-oL-homomorphism such that O(zi) = e26 and 0(7.2) = e27; recall that Wz is the rGL-algebra of "rGL-polynomials on {zz, z2}. Since ~O(zi) = Kd~a(gi) = Ku*/3tx(gi) = Kv*/3(zi) for i = 1, 2, we have if0 = Kv*/3 because if0 and ~v*/3 are r6L-homomorphisms.

Letting A=s~o{Zl, z21(bi, b2)} as in Theorem 1, we now complete the definition of L=~{Zl(u~, u2)} by setting ui = O(bi) for i = 1, 2. Since /3(b0 = fl(b,), we have ~'(u~) = ~O(b~) = ~:v*/3(bl) = Kv*/3(b2) = ~O(b2) = ~(u_,). and so there is a unique TL-homomorphism (*:L---> F(M 4) such that if*3" = if, by the universal property for ~{Zl(Ul , u2)}.

By Proposition 9, SR(B4, BI) is an S~o-algebra under the operations (iii), with kj replacing hj for j _-< 3. It is easily checked that f ~ f - is a roc- isomorphism from SR(B4, B1) onto the interval sublattice [Xo, Xi] of L, where x o = B ~ ~ and x~ = Bt4vBl. So, [Xo, xi] is a rcc-subalgebra of L isomorphic to SR(B4, B~). Since Xo C 3"O(zi)= 3"(e~+25)cxl for i = 1,2, it follows that 3"0[W2] is in s~o. Now 3'0(bl) = 3"(u0 = 7(u2)= 3'0(b2), so there is a unique ZoL-homomorphism O*:A--> L such that 0*/3 = 3,0, by the universal property for Sgo{zl, zzl(bl, b2)}. This completes the definition of the diagram of Fig. 6.

Since Kv*/3 = g0 = ~ '70 = ~*0"/3 and /3 is onto, we have Kv*= ~*0", and so ~*0*tL*= Kv*/.~* = K4>. Since Kck is one-one, so is 0*g.*. Fur thermore, 0*/x*a = 0*/3t~ = 3"0Ix, so a(wi)= a(w2) and 3"Ot~(wO = 3'0/~(w2) are equivalent predicates for (Wl, w2) in W~ x W1. Clearly p. and 0 are recursive functions by Proposition A


in the appendix. Therefore, if F(dl, d2) is a recursive function solving the word problem for OF{Z [ (ul, u2)}, then F(Olx(wl), 0/x(w2)) is a recursive function solving the unsolvable word problem for ~d{gl, g2 1 A}. This contradiction proves Theorem 2.

Let R be a ring with 1. A lattice L is "representable by R-modules" if it is embeddable in the lattice of submodules of a unitary left R-module. The class ~s of lattices representable by R-modules is a quasivariety of modular lattices. (See [15, 17] for discussion of this result. It is easily verified that ~ (R) admits isomorphic images, sublattices and the trivial lattice. Since embeddings L~ F(M~; R) for i in I can be combined into an embedding IL~ILi -o F(I-L~IM~; R) in an obvious way, ~(R) admits products. The model theory technique of B. M. Schein [24: Main Theorem, p. 15] may be used to show that ~s admits ultraproducts.) Note that ~s is the class of lattices embeddable in some lattice of subgroups of an abelian group, where Z is the ring of integers.

If R is a ring with characteristic n a prime or a product of distinct primes, then ~s =.Y(Zn), where Zn is the ring of integers modulo n. If R is a torsion-free ring then .~(R)=.Le(Q(PR)), where Q(Pa) is the unitary subring of the rational field Q generated by the inverses of all primes p such that 1 + 1 + �9 �9 �9 + 1 (p times) is an in~ertible element of R. (See [17:Theorem 5, p. 88] for these and related results.)

It follows from Theorem 2 that the (5.1) presentation ~(R){Z [ (ul, u2)} has a recursively unsolvable word problem for nontrivial R. However, C. Herrmann and A. Huhn [9, 10] have shown that all denumerable free lattices have solvable word problems with respect to certain of the quasivarieties ~(R) and certain related varieties. More precisely, they show in [10: Kor. 9, Kor. 11] that all denumerable OF-free lattices have solvable word problems if ~ is one of the following quasivarieties or varieties of "rE-algebras:

(1) OF is 5e(z,~) or is the variety HSs of homomorphic images of lattices in ~s for any m---2.

(2) OFis ~(Z), the quasivariety of lattices representable by abelian groups, or OF is H~e(Z).

(3) OF is the variety HSP(~K) of lattices generated by the class ~ = ~s U I_1 {~e(zp) - p prime}.

(4) OF is the variety HSP(~) of lattices generated by the class % of all complemented modular lattices.

(5) OF is the variety HSP(~) of lattices generated by the class ~s = % U ~e(Z).

(Of course, S(~) is the class of isomorphic images of sublattices of lattices in ~ , and P(~) is the class of isomorphic images of products of lattices in ~ , including the trivial lattice, as usual. See [8: Theorem 2, p. 152].)


We insert a preparatory result.

PROPOSITION 12. Let R be a ring with 1 and L a lattice representable by R-modules. If 8 = to+card(R)+card(L) , then L is embeddable in L(R, 8).

Proof. Assume the hypotheses, so that there is an R-module M and a lattice embedding f : L -~ F(M). For distinct x, y in L such that x c y, choose u(x, y) in M such that u(x, y) is in f(y) but not in f(x). Let Xt ={u(x, y ) : x c y in L, x # y}, and let N~ be the submodule of M generated by X1. Since card (X1)<_-8 obvi- ously, we have card (N1)N 8 by [17: Proposition 6, p. 75]. Now define Xi and N~ recursively, for i>--t. If x, y are in L and u is in N~ A(f(x)vf(y)) , choose v = v(u, x, y, i) and w = w(u, x, y, i) in f(x) and f(y), respectively, such that u = v + w. For i >_- 1, let Xi§ equal:

Ni LJ {v(u, x, y, i), w(u, x, y, i): x, y ~ L, u ~ Ni A (f(x)v f(y))},

and let N~+~ he the submodule of M generated by X~+~. By induction and [17: Proposition 6], card (X~)N 8 and card (N~)_-< 8 for i ->l . Let N = U~---1 N~, so N is a submodule of M and card (N) - 8. Define g :L ~ F(N) by g(x) = f(x) A N. Clearly g preserves meets, and g is one-one by the choice of X~. For any x, y in L we have NiA( f ( x ) v f ( y ) ) c (N~+lAf(X))v(Ni+lAf(y)) for all i = 1 by the construction of X~§ So, g preserves joins, and is a lattice embedding. Since F(N) is isomorphic to an interval sublattice of L(R, 8) (see [17: Proposition 5, p. 75]), Proposition 12 follows.

The next theorem is a slight generalization and simplification of one result of Herrmann and Huhn, proved by their methods. It may be useful as an introduction to [9, 10].

T H E O R E M 3. If R is a finite ring with 1, then every denumerable ~s lattice has a recursively solvable word problem.

Proof. Assume the hypotheses, and let W be the free ~-L-algebra of lattice polynomials on the variables Y={y~:i->_ 1}. By [8: Theorem 2, p. 163], h(el)= h(ez) for the canonical projection h: W---~ L onto the free ~(R)-lattice with to generators iff the identity e l = e2 holds in every lattice in ~s So, by [5: Theorem 1.5, p. 67; see also p. 74], the word problem for L is recursively solvable iff A and A ~ = ( W x W ) - A are recursively enumerable subsets of W x W, where:

A --- {(el, e2)~ W x W: el = e2 is satisfied in all L in ~e(R)}.


We now sketch the first-order theory if(R) constructed by M. Makkai and G. McNulty [19] to axiomatize representability by R-modules. The variables of if(R) include the variables Y that generate W. The function symbols of i f (R) consist of binary meet, join and addition, one unary function r for each r in R, and a constant.0. The predicate symbols of if(R) are equality, unary predicates L and M, and a binary predicate E. (The L and M predicates represent the " la t t i ce ' and "module" parts of the model, respectively, and E(x, y) represents membership of a module element x in a submodule y.) The nonlogical (first-order) axioms of if(R) are just sufficient to prove that models X of if(R) satisfy the properties of the next paragraph:

The domain X M of the predicate M is closed for addition and for each r, r in R, and contains the constant 0. Under the restriction of addition and scalar multiplications r to X M, X M is an R-module with zero element 0. The domain X L of L is closed for meet and join, and E(x, y) in X implies x is in X ~ and y is in X L. The function which assigns {x: E(x, y)} to each y in X c determines a zL-embedding X t" --~ I ' ( xM; R). (So, X L is in ~(R).)

Since R is finite, if(R) can be constructed with finitely many nonlogical axioms. Therefore, the set of theorems of if(R) is recursively enumerable by [25: p. 126]. For el, e2 in W, let ~(e t , e2) denote the formula:

(y~, y2 . . . . ym)(L(yl) & L(y2) & - - " &L!ym) ~ e~ = e2),

where m is the smallest positive integer such that i ~ m if there is any occurrence of yi in el or e2, for i>=1. By the G6del completeness theorem [25: p. 43], one can easily check that (el, e2) is in A iff ---(el, e2) is a theorem of if(R). Since we can recursively decide whether a theorem of if(R) has the form -~(e~, e2) and can recursively compute (et, e2) from _-"(e~, e2), it follows that A is recursively enumerable.

To prove A c is recursively enumerable, it suffices to show that A c corresponds to the set of finitely refutable equations for models of if(R). That is, (el, e2) in A c implies ~(el , e2) fails in some finite model of if(R), corresponding to a lattice of submodules of a finite R-module. (If so, A c corresponds to the semicomputable predicate Rl(v)=(3u)R2(u, v), where R2(u, v) is the predicate: u is the G6del number of some standard effective description of a finite model X for if(R), and v is the G6del number of (e~, e2) in W x W such that ~(e~, e2) fails in X. Clearly, R2(u, v) is computable by determining whether the finitely many nonlogical axioms of if(R) are satisfied in the finite model X and ~(e t , e2) fails in X. (See [5: Theorem 1.2, p. 66].) This finite refutability is proved by adapting a method used by D. Sachs [23] for partition lattices.


Suppose (ei, e2) is in A c, so el --- e2 fails in some lattice in Y(R). Clearly, el = e2 fails in some m-generated lattice in Y(R), so el = e2 fails in L(R, to) by Proposi- tion 12. Let K be the sublattice of L(R, to) of all finitely generated submodules of M(R, to). (Since R" is the free n-generated R-module and R is finite, finitely generated R-submodules of M(R, to) are finite, and so K is an ideal of L(R, to).) Now L(R, to) is isomorphic to the lattice of ideals of K, mapping an ideal of finite submodules of M(R, to) to its union. Therefore, el = e2 fails in the lattice of ideals of K, and so fails in K by [23], and so fails in some m-generated sublattice of K, and so fails in some interval sublattice [0, x] of K. But x is a finite R-module, and so a finite model of f t (R) for which _=(el, e2) fails can be constructed from x and [0, x]. Therefore, A c is recursively enumerable, and so the word problem for the free Y(R)-lattiee on to generators is recursively solvable. It follows that all denumerable ~s lattices have solvable word problems, completing Theorem 3.

Unfortunately, the indicated decision algorithms in [9, 10] and Theorem 3 are ineffective for practical use.

In view of the above results, it is unlikely that the methods of Theorem 2 could be used to prove that some free finitely-generated modular lattice has an unsolvable word problem. Similarly, it seems that the methods of Theorem 2 could not be used to obtain a (4, n) modular lattice presen.tation having an unsolvable word problem for any finite n. This conjecture is motivated by the result [9: Theorem 19, p. 116] of C. Herrmann: All (4, n) word problems are recursively solvable for the variety of lattices generated by all complemented modular lattices. Recent studies of quadruples of subspaces of finite dimensional vector spaces by I. M. Gelfand and V. A. Ponomarev and by S. Brenner (see [2]) also suggest that lattices generated by four subspaces of a vector space are considerably more restricted than those generated by five or more subspaces.

5. Embedding theorems for modular lattices

For a given variety or quasivariety W of algebras, it may happen that the free n-generated W-algebra has denumerably many elements in "general position" with respect to each other. This can be stated as an embedding theorem of the form: The W-free algebra on to generators can be embedded in the W-free algebra on n generators. For example, P. M. Whitman [28] proved that the free lattice on to generators is embeddable in the free lattice on three generators. Such an embedding theorem may appear in a stronger form, that each denumerably generated W-algebra is embeddable in some n-generated W-algebra. R. A. Dean


f6] extended Whitman's result in this way, proving that each denumerable lattice is embeddable in some lattice with three generators. The Higman-Neumann- Neumann result [12: Theorem IV, p. 251] previously mentioned also has this form, stating that each denumerable group is embeddable in some group with two generators. (Their result also implies that the free group on to generators is embeddable in the free group on two generators.) In the following, we consider embedding theorems of this type for the lattice quasivarieties ~(R) , with the critical value n = 5. An open problem suggested by these results is the following: Is each denumerable modular lattice embeddable in some modular lattice with five generators?

We begin with the basic embedding construction.

T H E O R E M 4. Let R be a ring with 1 and 8 a cardinal. I f L is a denumerable sublattice of L(R, 8), then L is lattice embeddable in a sublattice K of L(R , 8*) generated by five elements, where 8* = 8 if 8 is infinite, and 8* = 4 n ( m + 1) if 8 = n and L has a generating set of cardinality m.

Outline of proof. Suppose 8 is infinite, and let M denote the free R-module generated by a set X, card (X)= 8. By partitioning X into a family of to subsets each of cardinality 8, we can construct a set { N i : i ~ Z } of submodules of M, independent over O in F(M). Note that F ( M ) = L ( R , 8), and, furthermore, L(R, 8) is lattice isomorphic to each of its interval sublattices [O, Ni], i~ Z.

A "translating" R-linear automorphism S : M - ~ M such that S[Ni] = N~-I for all i in Z is easily constructed. Now, L is isomorphic to some denumerable sublattice {T~ : i >_- 1} of [O, No]. We observe that {T~ : i >_- 1} can be recovered from S, No and T=V{S-~[T~]:i>--I} because of the formulas T ~ = S i [ T ] ^ N o , i>-1. (For iN 1, S ~ and S -~ denote iterated composition of S and S -1, respectively.)

To prove the first part, we use the configuration of Proposition 8 to put the above in lattice terms (see Fig. 3). The elements M1, ME, M3, M4 of Proposition 8 are isomorphic copies of M, and P3, P4, P1 are negative graphs of suitable relative 4-frame isomorphisms. The negative graph of the translating automorphism S, expressed as an isomorphism M4-~ M1, is Ps. Finally, P2 is constructed so that T and No can be recovered from it. The second part of the theorem, for 8 = n, can be proved by adapting the above technique.

It is convenient to prove Theorem 4 in a more technical form, for reference purposes.

DEFINITION. Let mi for i_-_N4 and pj for ]>=5 be defined in W ( Z ) by equations (ii) of Proposition 8. Let 'no and si, ti for i ->_ 1 be given in W ( Z ) by the


equat ions below.

(ix) no = (((m3vp2)Amz)vpl)Aml.

Sl = (P3 V Ps) A (m3 v rex).

For i >_-- 1, si+~ = (((st v p4) A (m3 v m4)) v Ps) A (m3 v ml) .

For i>-1, t~ = ( ( m3 /x p2) v S~) A no.

Let Y be a set of co distinct variables {y~ : i _-__ 1}; and let Y,, = {y~ : i <= m} for m >- 1. Let p : W(Y)---> W(Z) be the unique ~-t.-homomorphism such that p(y~)= t~ for i > - 1. For m ~ 1, let p,, : W(Ym)---> W(Z) be the unique ~ 'L-homomorphism such that p,,(y~) = t~ for i - m. Also, let Y., = Y and p,, = p.

P R O P O S I T I O N 13. Let R be a ring with 1 and let m, 8 be cardinals, m <= to. Let No = M(R, 8) and N = M(R, 8"), where 8 * = m +8 if m or 8 is infinite and 8 * = 4 n ( m + l ) - n if 8 = n and re<to. So, F(No)=L(R,(5) and F(No(DN)= L(R, ~ + 8*). Let ~ : F(No) ---) F(No(DN) be the embedding induced by the insertion No---> No(gN. If @: W(Ym)-->F(No) is a zL-homomorphism, then there exists a ~'L-homomorphism t~+: W(Z)--->F(No(DN) such that ~b+(no)=~.(No) and ,.~b= 4,§

Proof. Assume m is infinite, and let M = No(DN. Choose a family {X,,n :i =< 4, n r Z} of pairwise disjoint subsets of F(M), each of cardinali ty (5, such that Xt,o

freely genera tes ~(No) and X - X I , o f reely genera tes O (D N for X = U {X~.~ : i --<4, n ~ Z}. Note that X freely genera tes M. Le t M,., for i<=4 and n a Z deno te the R - s u b m o d u l e of M genera ted by X,.~, so Ml,o = ~(No). Le t M~ for i<=4 deno te the R- submodu le of M genera ted by I.J {x~,n :n ~ Z}. Clearly {M1, M2, M3, M4} is i ndependen t over O in F(M).

By free R - m o d u l e propert ies , we can construct R- l inea r i somorphisms

h~ : M1 "-~ M2, h3 : M3 --~ M4, ha : M4 ~ M t and h5 : M4 --~ M1 such that hl[Ml,.] =

ME.n, h3[M3.n] = M4.~, h4[M4,n] = M~,n and hs[M4.n] = M~.~_~ for all n in Z. By Proposi t ion 2, o-(hl) is in SI(M~/O, M2/O), tr(h3) is in $1(M3/0, M 4 / O ) and o-(h,), tr(hs) are in SI(M4/O, MI/O). Let Pi = o'(hj)- in F(M) for ] = 1, 3, 4, 5.

Le t h2,o: M2.o---> M3,o be an R- l inea r isomorphism, and define P2.0 = tr(h2.0)- in F(M) for tr(h2,0) in SI(M2.0/O, M3,o/O). For i - i , h31h4~(h4h~)'[Mt,o] = h31h-~[Ml,i]=M3,1. We define P2.,=h3~h4~(h,h~X)'[~t~(y,)] for i_>-1; since ~ b ( y ~ ) ~ ( N o ) = M L o , we have P2 . i c M3,i. Finally, let P2=P2,0vP * for P * = V{P2.~:i >- 1}, which is p roper since F(M) is complete .

For i - 4, let M* = V {M~,n v M~_, : n >- 1}, so M~.o v M* = M~ and M~.o A M* = O. Now M3/O, M2,0/O, M*/O is a mixed sequence in F(M), so P2.0AM2 = P2.0^(M2.0vM*)~(Pe,o^M2.0)vO= O, by [14: 3.3]. Similarly, M3,0/O, M2/O,


M*/O is a mixed sequence, and so P2,o, M2 and P~ distribute by [14: 3.3] because P * c M * . So, P2^M2=(Pz,o^Mz)v(P*AM~)=O. Therefore, the hypotheses of Proposition 8 are satisfied. Let to+: W ( Z ) ~ F(M) be the unique ~'L-homomorphism given by the equations (i), to§ replacing f. By Proposition 8, to+(m~)=M~ for i - 4 and to+(pj)=Pj for j<=5.

Note that M2.o = (M3 v P2)/x M2 because P* c M3 and M3,oV P2,o = M3,oV M2,o.

So, to+(no) = (Mz,oVo'(hl)-)AM1 = h~-l[M2.o] = ~(No). Furthermore, P* = P* v (P2,0/x M3) = P2 A M3 by modularity. (By applying [14: 3.3], P2,o,X M3 = O just as Pz,o^ Mz = O.)

For i > - 1, let k~ :M3---> M1 be the R-linear isomorphisrfi (hsh2~)ih4h3. From the equations (ix), we prove that to§ cr(k~)- for i>_-1, by induction on i. Clearly cr(kO=o'(hsh3)=o'(hs)oo'(h3) by Proposition 2, and so to+(sl) = (P3 v Ps) ^ (M3 v Mr) = o-(kl)-. Assume the induction hypothesis, for i > 1. Since k~ = hsh2tkH, o'(k~)=o'(hs)o (o'(h4) # ~ o'(ki-1)) by Proposition 2. But then to*(s~) = o'(k~)- by (ix), completing the induction.

We observe that k~[P~j]ck~[M_~.i]=M~.~=~ for i,]>-l. So, /q[P*]= k~[P,_.~]vN~.i for NL~=V{k~[P~.~]:j>=I, i# j }~M*, for i_-__l. Therefore, for i>-1:

Lto(y~) = k~[P,_.,] = (k,[P2.~] v N,.,)A M,.o = k,[P*]A M,.o

= (P* v cr(ki)-) A ~(No) = ((M3 ̂ Pz) v to+(si)) A to+(no)

= to+(ti)= to+p(y~).

But then Lto = 4,+0 since Y generates W(Y). This completes the proof for infinite n'~.

For ~ = n and rn<~o, define M~,i for i<_-4 and O<=]<=m in L(R, 4 n ( m + l ) ) such that Ml,o = ~(No), and each Mi.j is freely generated by n elements. This is done so that the union of the generating sets for the remaining M~ 4 is a set of 4n(m + 1 ) - n free generators for O ~ N. Modify the definitions of MI, M2, M3, M4 and P1, P3, P4 in the obvious way. Redefine P5 by choosing a cyclical translation hs, such that hs[M4,j] = M1.~-1 for 1 _-__]_-< m and hs[M4,o] = Ml,m, and make P* a join of Pz.~ for 1 N i_<- m. The argument then carries through as in the infinite case. This completes the proof of Proposition 13.

Now to[W(Y)] is an arbitrary denumerable sublattice of L(R, 8), and ~ induces an embedding tO[ W(Y)] --> to+[ W(Z)], since ~to = to§ But to+[ W(Z)] is generated by five elements. Similarly, to[W(Ym)] is an arbitrary sublattice of L(R, n) with rn generators, and ~ induces an embedding to[W(Y,,)]--* to+[W(Z)], if 8 = n and m < o). So, Theorem 4 follows from Proposition 13.


Suppose that a "system of ring equations" is a finite set of expressions, each of form xl + x2 = x3 or x~x2 = x3, where each term xj is either a variable vi or 0 or 1. A system of ring equations is "satisfiable" in a ring R with 1 if some assignment of ring elements to variables vi makes every system equation true in R, with 0 and 1 interpreted as zero and unit in R. M. Makkai and G. McNulty [19], adapting and extending the model theory technique of B. M. Schein [24], proved the following theorem: If R1 and R2 are rings with 1 such that every system of ring equations satisfiable in R2 is also satisfiable in R1, then 5~(R~)c ~(R2). They note that every ring R has a denumerable unitary subring Ro such that ~e(R)= Le(Ro), since there are only denumerably many systems of ring equations.

This result may be used in conjunction with the next proposition.

PROPOSITION 14. Suppose R is a ring with 1 and Ro is a denumerable unitary subring of R such that 5~(R) =Se(Ro). Then a lattice L is in ~(R) if and only if every finitely generated sublanice of L is embeddable in L(Ro, to). Suppose that T is a quasivariety of ~'L-algebras containing L(Ro, to). Then ~(R) c T, and every denumerable lattice L in ~s is embeddable in a lattice K in T which is a homomorphic image of the free M-lattice with five generators.

Proof. Assume the hypothesis. By Proposition 12, a finitely generated sublattice of L is in ~(Ro) if and only if it is embeddable in L(Ro, to). Since L is in ~(R) if and only if every finitely generated sublattice of L is in 5e(R), the first part follows.

Since L(Ro, to) is in T, every finitely generated lattice in ~s is in T, and so Se(R) ~ T.

By Theorem 4 and Proposition 12, every denumerable lattice L in Se(R) is embeddable in a sublattice K of L(Ro, to) having five generators. Clearly K is a homomorphic image of the free V-lattice with five generators, and K is in T. This proves Proposition 14.

Recall the function p: W ( Y ) ~ W(Z) defined previously.

PROPOSITION 15. If R is a ring with 1 and L in ~e(R) is definable by denumerably many generators subject to a set of relations of cardinality 6, then L can be embedded in a lattice K in ~(R) definable by live generators and at most 8 relations.

Specifically, if g tc W(Y)x W(Y), let p(gZ)c W(Z)x W(Z) be given by:

p(gr = {(p(d,), p(d2)} : (d,, d2) ~ gt}.

Then there is a unique lattice embedding O* :L1 ~ L2 for LI =~(R){YI ~} and


L2 = ~ ( R ) { Z I p(x[y)} such that p*a =/30 if a : W ( Y ) ~ L1 and/3 : W ( Z ) --~ L2 are the canonical projections. I f ~ is recursively enumerable, then O( ~ ) is recursively enumerable and there exists a recursive set O(gt)*c W ( Z ) x W ( Z ) equivalent to p ( ~ ) with respect to Sg(R). I f rF is recursive, then p(gt) is recursive.

Proof. Assume the hypotheses, and consider the diagram of Fig. 7. Since

W(Y) Q W(Z)

I o 1

L(Rb) t ~- L(R.5)

Figure 7

L1 =~(R){YI ~} is representable by R-mochales, it is embeddable in L(R , 3) for sufficiently large infinite 8 by Proposition 12. Therefore, we can construct lattice embeddings 0 and ~ and a q'L-homomorphism u = (0o~) § such that up = ,.~ba as in Fig. 7, by Proposition 13. For ( d l , d2) in ~, we have up(dO = ~Oa(dl) = ~ba(d2) = up(d2) since ct(dl) = a(d2), so there exists u* : L2--* L(R, 3) such that 0"/3 = o, by the universal property of ~e(R){ZIo(~)}. Finally, there exists a lattice homomorphism P*: L1--~ L2 such that O*c~ =/30 by the universal property for ~(R){Y I ~}, since /3p(d0 =/3p(d2) for (p(dl), p(d2)) in p(~) . As usual, ~ba = up = 0*/30 = o*O*o~ implies L0 = o'O* since c~ is onto, and so p* is an embedding since ~ and ~ are one-one.

The recursiveness results follow from Propositions A, B and C. This completes the proof of Proposition 15.

Corollary to Proposition 15, we see that the free ~(R)-lattice with to generators is embeddable in the free ~e(R)-lattice with five generators.

By more elaborate use of the techniques now available, we can improve Proposition 15 if finitely many relations are given. Our goal is the following result.

T H E O R E M 5. Let R be a ring with 1. I f L has an ~e(R)-presentation with denumerably many generators and finitely many relations, then L is embeddable in some K having an ~(R)-presentation with five generators and one relation.


After a preparatory result, a further use of relative n-frames is made. A single lattice equation is constructed that is equivalent to n equations within one of the n-frame intervals,

PROPOSITION 16. If A~, A2 . . . . . . A , :2---> L is a mixed sequence of objects in a modular lattice L, n >-_ 2, then for 1 <- i <- n it follows that:

A ~ ^ V{A~:] <--n,]# i } = A ~ V{A~ n , j # i}.

Pro@ By induction on n; an easy generalization of the proof of [14: 3.2, p. 163].

PROPOSITION 17. Suppose B~ for i <- n and k~ in SI(Bj, BI+I) for ] < n form a relative n-frame in a modular lattice L. For x in [B~ B I], let k(i)(x) be given by k<~ = x and k")(x) = ki[k(*-l)(x)] = (k"- l ) (x)v k[)^B~+l for 1 <- i <- n - 1. S u p - pose al and bi are in [B ~ BI] for l <-i<-_n. Then a~=b~ for i n n if and only if:

V {k"-t~(a~) : i ~ n} = V {k"-l~(b~) : i _-< hi.

Pro@ Assume the hypotheses, let a~ = k(~-l)(a~) and b~ = kCi-U(b~) for iN n, and let a= V{k"-~>(a~):i<-n}=a~va2v . . . v a , and b= V{k(~-l)(bi):i<--n} = b i v b 2 v . " vbn. Clearly, ai=b~ for i n n implies a = b .

Suppose a = b and I<-_iNn. Now k "-l~ is a projectivity isomorphism [B ~ BI]---, [B ~ B~] by Proposition 3 and induction. So:

ai c a A B ~ caiv(V{ai:]_<_ n, i ~ i}AB~)c

a ,v(B~A V{B~:j=< n , ] ~ i } ) c a i v B i ~

using the relations above, modularity and Proposition 16. So, a, = a ^ B~, and so a~ = bl since b~ = b/x B~ similarly, for each i_-- < n. But then a~ = bi because k ~-1) is one-one, for i _ <- n. This completes the proof of Proposition 17.

To prove Theorem 5, we adapt the proof of Proposition 15, using Proposition 11 to set up a relative n-frame in the embedding codomain and then using Proposition 17 to construct a single lattice equation which induces n given equations in the first n-frame interval.

DEFINITION. Suppose n _-> 2. Let mi for i_- < 4 and pj for ] =< 5 be defined in W(Z) by the equations (ii), and then let no and si, ti for i >- 1 be defined in W ( Z ) by the equations (ix). Define mi for 0 - i < - n by the equations (v), with t,


replacing m~ for i <- n. Then define el, e2 . . . . . elo.-15 in W(Z) by the terms (vi)., with t.§ replacing pj for 1-<_j<-n-1. Let h . : W ( Y ) - - > W ( Z ) be the unique zc-homomorphism such that:

'~.n (Yi) = (e10n-17 V t2n- l+i ) A elo.-18,

for i >= 1. Also, define a ~-L-algebraic function f , : W(Z)" --~ W(Z) as given below.

(x). For dbd2 . . . . . d. in W(Z), f . (d l , d2 . . . . , d.)

= go(dl)v gl(de)v �9 �9 �9 v g . - l (d . ) , where:

go(d) = d,

gi(d)=(gi_l(d)vesi_a)Aesj_8 for l < - i < - n - 2

and ] = 2 n - 2 - i ,

g. - l (d) = (g . -2(d)vesn-9)Aes . -6 , for d in w(z).

(Comparing Proposition 17 and (vii)., we see that g~ corresponds to k(~).)

PROPOSITION 18. Suppose R is a ring with 1 and L1 = ~(R){Y I ~} for some finite ~ ={( a , , b,}:i<-_n}, n>-2. Let L2=Sg(R){ZI(d, e)}, where d=f~(hn(a~), h~(a2), . . . ,h,(a,)) and e=f~(h,(b~), hn(b2),. . . ,h,(b,)). Then there exists a lattice embedding h* : LI ~ L2 such that h*a =/3h,, where a : W(Y) ~ L~ and /3 : W(Z) ~ L2 are canonical projections.

Proof. Assume the hypotheses. By Proposition 12, there exists a lattice embedding ~bl:Li--* L(R, 8) for sufficiently large infinite 8. Let No and N be R-modules isomorphic to M(R, 8) and M = N o ~ N, as in Proposition 13.

Partition a free generating set of No into n subsets each of cardinality 6, and let VI, V2 . . . . . Vn be the submodules of No generated by the respective subsets. Clearly {V~, V 2 , . . . , V,} is independent over O in F(No), and so VJO, V2/O, . . . , V,]O is a mixed sequence of objects of F(No). For ] < n, we can construct an R-l inear isomorphism Vj--~ Vi+ ~ by free R-module properties, so there exists uj in SI(Vj/O, Vi§ by Proposition 2. Therefore , VJO for iN n and uj for ] < n form an absolute n-frame in F(No).

Consider the diagram of Fig. 8. Let rI :W(Y)--> W(Y) be the unique ~'L- homomorphism such that ~(yi )= y2,-l+i for iN 1. Let Lo = ~s "O(~)}, where n(g~) = {(n(ai), "0(bi)): i <_N n}, and let ~/: W(Y) --> Lo be the canonical projection.

By the universal property for L1, there exists a lattice homomorphism ri* : L1 --~ Lo such that ~*a = "Y'O. Since V1 is isomorphic to M(R, 8), we can suppose the


w(Y) n ~ i i , W(Y) ( ) .

e y

LI q" L 0 P" v

r (vo h ' F(No) ~ '~ Figure 8

embedding ~bl of L1 into L(R, 3) has codomain F(V1) as in Fig. 8. Let ~I:F(V~)---> F(No) be the lattice embedding corresponding to the inclusion of the interval sublattice [O, V~] into F(No). Let q~: W(Y)---> F(No) be the unique rL-homomorphism such that ~b(yi) = Vi if i =< n, ~b(y~) = (u,-n)- if n + 1 -< i -< 2 n - 1 , and ~b(y,)=~l~blct(y~_2n§ if i > 2 n - 1 . It is apparent that ~brl(y~)= ~.xqS~a(y~) for i >= 1, so &-q = ~l~b~ct. It then follows from the universal property of Lo that there exists ~b*:Lo--->F(No) such that ~b*3,=~b. Since a is onto and

Let ~:F(No)---> F(M) be the lattice embedding induced by the insertion No---> N o ~ N. By Proposition 13, there exists o = (~b*3,) + such that o(no) = ~(No) and oO = ~b*~,= ~b. Since O(yi) = t, for i>-1, we conclude that o(t~)= ~(V,) for i<-n, o(ti)=~((ui_,)-) for n+l<-_i<-_2n-1, and O(ti)=~lC~lct(yi-2,+l) for i > 2 n - 1 .

Clearly, A, = ~(%)/0 for i <- n and h i in SI(Aj, Aj+I) such that h~-= ~(uT) for j < n form an absolute n-frame in F(M). Applying Proposition 11, we see that the equations (vii)n hold if g is replaced by o, Bi is replaced by A~ for i_- < n, and kj is replaced by h i f o r / ' < n. In particular, u(elo,_17)=A ~ 0 and o(elo,_~s)=A~= ~(Vt). So o(eton-17)~ ~ ( x ) c o(elo~-x8) for all x in F(V1).

Let ~:W(Y)---> W(Z) be the unique a-~-homomorphism such that /~(y~)= t~

for i<-2n-1 and I~(y,)=(exo,-17vt,)Ae~o~-s for i > 2 n - - 1 . Then o(ti)= ~lqS~ot(yl-2,+l) = op,(yl) for i>2n - 1 by the inclusions above, and so o/~(y,)= up(y,) for i >- 1, and so o~.= up = ~b*3,. Since )t. = / ~ by examination, we see that v)~. = up.- o = ~q5"7- q = ~ b ~ a . Therefore , vA.(a~) = vh.(b~) since a(a~) = a(bi) for each i_- < n, and so v(d)= o(e) for d = f . ( ) t . (a t ) . . . . . ,k.(a.)) and e = f . ( ) t . (bt) . . . . . A.(b.)), since f . is a zL-algebraic function. But then o*:L2-'-~ F(M) exists such that o*/3 = o by the universal property for L2.

By Proposition 11, there exists a relative n-frame B~ for i <_- n and k~ for j < n in Lz given by the equations (vii). with /3 replacing g. We observe that Bt =

/3(e~o.-ts)//3(e~o,-17) and /3~)(y~)= (/3(elOn_lT)V/3(t2n_1+i))A/3(elon_18) for i >- 1.


So, /3 /zn[W(Y)]c [B ~ BI], and/3p`,0(al) and/3p`,0(b,) are in [B ~ BI] for i n n in particular. It follows from (vii)n and (x), that r k('(Ci~,0(x)) for x in W ( Y ) , 0 <- i<n . So, /3(d)=/3(e) in L2 implies that V{k(~-~)([3p`,0(a,)):iNn}= V {k(~-l)(l~p`,0(bl)) : i <_- n}, and therefore/3p`n(a3 = ~p`n(b3 for 1 < i - n by Propos- ition 17. Since the relations ,0(gt) define Lo, there exists a lattice homomorphism p`*:Lo--~ La such that P`*7 =/3p` by the universal property of Lo.

We have now completed the definition of Fig. 8, and we have the com- mutativities ,0*a = 7"0, ~'1(~1 ~ t~*,0 ~*, P`*7 --'~ /~]s and also L~b* = o'p`* since ~b*7 = up, = o*/3p` = o*p`*y and 7 is onto. So p`*,0*a =/3/z~ = [3h, and o'p`*,0* = ~1(~.

Since ~bl, ,1 and L are embeddings, h*=/z* ,0* is a lattice embedding L1--~ L2 such that h*a =/3h,, which proves Proposition 18.

Since L1 has an arbitrary (to, n) ~(R)-presentat ion for n _->2 and L 2 has a (5, 1) .Y(R)-presentation, Theorem 5 follows from Proposition 18 for m = to and n_->2. If m is finite and n>___2, Theorem 5 follows from Proposition 18 plus the observation that ~(R){Ymlg '} is embeddable in , ~ (R){ Y Ig t} if g t c W ( Y ~ ) x W(Yr,) . Finally, Proposition 15 suffices if n <___ 1, completing the proof of Theorem 5.

Appendix. Recursive algebraic functions and sets

We give here a standard approach to the recursiveness results needed in Theorems 1 and 2 and Proposition 15. For reference purposes, the results are given with more generality than is needed in the main text.

DEFINITION. Let W be the free monoid of words on the four letter alphabet {u, v, w, a}, provided with the juxtaposition operation and the empty word as unit. A recursive subset of W is called a W-set, and a recursive function from a W-set into a W-set is called a W-function. Note that any finite subset of W is a W-set, and any function from a finite W-set into a W-set is a W-function.

An algebraic type z is "recursive" if the set S(z) of r -operat ions is denumerable and has an enumerat ion Uo, u~, u 2 , . . , such that the arity of ui is a recursive function of i. Note that z is recursive if S(z) is finite.

If Y is a denumerable set of variables having an enumerat ion Vo, v~, v2 . . . . , then the free z-algebra P(Y , z) of r-polynomials is represented as a W-set in a standard way. The variable vi of Y, where i < card (Y), is represented by the word va i of W, with a repeated i times. Similarly, the r -operat ion u, i < card (S(z)), is represented by the word ua ~ of W. Like the definition appearing in [8: p. 39],


P(Y, r) is characterized as follows.

(1) A variable word va ~ is in P ( Y , r) if / < c a r d (Y). (2) Suppose i<card(S( r ) ) . If u, has arity n > 0 and eo, e l , . . . , e , - I are in

P ( Y , r), then uaieoel �9 �9 �9 e,,_~ is in P ( Y , .r). If ul has arity zero, then ua ~ is in P ( Y , r).

(3) P(Y , r) is the smallest subset of W satisfying (1) and (2).

Note that P ( Y , r) is nonempty if Y is nonempty or if there is a z-operation of arity zero.

Algebraic functions P(Y , -r) ~ --~ P(Y, ~') are also representable by words in W, where the word wa ~ represents a variable w~ in an algebraic function expression. For example, if Uo is a binary operation, the binary algebraic function f on P(Y, r) given by f(wo, w l ) = Uo(Wo, Uo(V2, wl)) is representable by the word uwuvaawa of W. The free r-algebra A ( Y , r) of algebraic function expressions is the smallest subset of W satisfying (1) and (2) above with A ( Y , r) replacing P(Y, 7), plus the condition (1') wa ~ is in A ( Y , I") for all i ->__0. We say that wa ~ "occurs" in e in W if e=eiwa~e2 for words el, ez of W such that e27~ae3 for any e3 in W. The "minimal arlty" of e in A(Y, r) is zero if e is in P(Y , r) and is l + m a x { i : w a i occurs in e} otherwise. (Note that an e in A ( Y , r) either belongs to P(Y, r) or has an occurrence of some wa~.) If e is in A ( Y , r) with minimal arity no greater than n and eo, el . . . . . en-i are in P(Y, r), then there is a unique element of P(Y, ~-), denoted e(eo, el . . . . . e~_l), obtained by substituting e~ for each occurrence of wa ~ in e, for i < n. Of course, this operation equals eval/mtion of the corresponding ~--algebraic function at an n-tuple in P(Y, T) ~.

For appropriate recursive algebraic types o- and r, we want to construct certain W-functions of the form f :P (X , 0 - ) ~ P(Y , T) for denumerable X and Y. These functions are given by specifying a recursive g:X--~ P(Y , r), and also making P(Y, ~') into a o--algebra by specifying a ~--algebraic function or ~--operation to correspond to each o--operation. Then f is defined to be the unique cr- homomorphism P(X, ~r) ~ p(Y, r) extending g, using the free o--algebra property of P(X, o').

DEFINITION. Let X and Y be denumerable sets of variables and o- and r recursive algebraic types. An "X-specification" for P(Y, r) is a W-function g : V ( X ) ~ P(Y, r), where V ( X ) = {va ~ : i < card (X)} is the W-set corresponding to X. A "o--specification" for P ( Y , r) is a W-function h: U(o- )~ A ( Y , r), where U(cr) = {ua ~ : i < card (S(o-))}, and the arity of each o--operation ui is not less than the minimal arity of the r-algebraic function expression h(uai) . Given a o-- specification h for P ( Y , r), there is a corresponding o--algebra structure for


P(Y, ~): If u~ has o'-arity n, then the o'-operation u~ :P(Y, r) ~---, P(Y, r) is given by:

u~(eo, e~ . . . . . e~-l) --- h(ua i) (eo, el . . . . , e~_~).

Of course, u~ = h(ua ~) in P(Y, .;) if u~ has o'-arity zero.

PROPOSITION A. Let ~y and �9 be recursive algebraic types and let X and Y be denumerable sets of variables, all having appropriate recursive enumerations. For any X-specification g: V(X) ---> P(Y, 1-) and o'-specification h : U(cr) --> A ( Y , I"), there exists a unique o~-homomorphism P(X, o')---;P(Y, ~,) extending g, where P(Y, r) has the o'-algebra structure corresponding to h. This cr-homomorphism, denoted fg, h, is a W-function (recursive), and is uniquely characterized as follows: (1) For i< card (X), fg.h(va ~) = g(va*). (2) For i< card (S(o')), ui with ~-arity n, and eo, e l , . . . , e~-i in P(X, ~r), fg, h(uaieoel . . . e~-l)= h(uai)(f,.h(eo), fg.h(el) . . . . . fg, h(e,-1)). (If ui has o'-arity zero, fg, h(ua i) = h(uai).)

ProPosition A can be proved by standard inductions on word length in P(X, ~r), G6del numbering techniques (see [5: Chap. 4]), and recursive function properties. We omit these calculations.

Let lel denote the word length of e in W, on the alphabet {u, v, w, a}. A W-function f : Wo--* W1 is "decrease bounded" if there exists a recursive function k : to ~ to such that lel--< k(lf(e)[) for e in Wo. (That is, lel is bounded by k(ldl) if e is any preimage under f of a given d in W1.) Note that a W-function f : Wo--* W1 is decrease bounded if Wo is finite or if f is nondecreasing for word length (that is, lel<-If(e)[ for all e in Wo.)

A o'-specification h : U ( c r ) ~ A ( Y , ~') is called "complete" if for each i < card (S(o-)), there is an occurrence of wa j in h(ua ~) for each ] less than the o--arity of ui. (That is, all arguments of each o'-operation induced on P(Y, 1-) by h have occurrences in the corresponding T-algebraic expression. In this case, the minimal arity of h(ua ~) equals the o'-arity of ui, for all / < c a r d (S(o')).)

PROPOSITION B. Let f : P(X, tr) --> P(Y, r) be a W-function for recursive algebraic types tr and ~" and denumerable sets of variables X and Y. I f A is a recursively enumerable subset of P(X, ~)", n >-1, then f("~[A] is a recursively enumerable subset of P(Y, ~')~, where:

ff")[a] = {(f(eo), f(el) . . . . . f(e,_,)) :<eo, e l , . . . , e,_,) s A}.

Furthermore, f(")[A] is recursive if zl is recursive and f is decrease bounded. I f g: V(X)---> P(Y, ~') is a decrease bounded X-specification and h: U(o')--> A ( Y , r)


is a complete and decrease bounded (r-specification such that h(ua j) # w for all j < card (S(tr)), then fg, h is decrease bounded.

Proof. Assume the hypotheses of the first part. For n -> 1, mapping (eo, el . . . . . e , - l ) to (f(eo), f(el) . . . . f (e,- l)) yields a recursive function P(X, o')" ~ P(Y, r) n. So, a recursive enumeration of A can be modified to obtain a recursive enumeration of )a"~[A], proving that ff")[z~] is a recursively enumerable if A is recursively enumerable.

Suppose A is recursive and f is decrease bounded, so there exists a recursive k : to --> to such that lel =< k(If(e)l) for all e in P(X, o'). Since {u, v, w, a} is finite, there are only finitely many e in P(X, o') such that tel =< m, for any integer m. Assume d, is in P(Y, r), and note that f(e~) = di implies le, I--< k(ld, I), for 0 - i < n. Since A is recursive, it follows that (do, dl . . . . . d , - i ) is in la")[A] iff the recursive predicate given next is satisfied: Among the finitely many n-tuples (eo, ei . . . . . e , - t ) in P(X, o-)" such that le, I--< k(Id~l) for 0--<i< n, there exists one belonging to J such that f(e~)=d~ for O<=i<n. Therefore , /(")[~] is a recursive subset of P( Y. r)".

Assume the hypotheses for g and h. Without loss of generality, we can choose a nonzero, strictly monotonically increasing recursive function k : to --+ to such that ]g(e)l_-<p implies le]<--k(p)for e in V(X) and ]h(e)l<-p implies lel<=k(p)for e in U((r). Recursively define k*(p) by k*(O)= 1 and k*(p)= k(p2)+pk*(p - 1) for p>O. By induction on [el, we can show that Ifg, h(e)l<-_p implies lel_- < k*(p) for all p => O. (If e = uaidodl . . . d , - i , where j < card (S(cr)), u i has cr-arity n > O, and do, d, . . . . . d ,_, are in P(X,o-), then we can verify from the hypotheses and ]g.,,(e)l-<_p that n<=p, ]d~l<=k*(p-1) for O<=i<=n-1 and Ih(ua*)l<=p 2, from which luaq <- k(p2), ] d o d l ' . ' d,-l l<-pk*(p - 1) and l e t - <_ k*(p) follow.) But then fg, h is decrease bounded via k*, completing the proof of Proposition B.

DEFINITION. If ~" is a quasivariety of algebras of type z, a "replicating" unary polynomial p(x) for ~ is an element of P({x}, z) distinct from x but having an occurrence of x such that the identity p(x)= x is satisfied in every member of ~V. For example, lx is a replicating polynomial for every z~-quasivariety of groups, and x v x is a replicating polynomial for every z~_-quasivariety of lattices.

PROPOSITION C. Suppose r is a recursive algebraic type and ~ is a quasivariety of r-algebras having a replicating polynomial p(x). I f A is a recursively enumerable subset of P(Y, r) z for denumerable Y, then there exists a recursive A * c p ( y , r) 2 which is equivalent to za for ~V. That is, if h : P ( Y , r)---> A for A = ~{YI a} and h*:P(Y , r) ~ A* for A* :-: ~{YI A*} are canonical projections, then there exists a r-isomorphism f : A ~ A * such that fh = h *.


Proof. Assume the hypotheses . If A is finite, A * = za suffices. So, assume zl is

infinite and has the recursive enumera t ion (di, ei), i - 1. We adapt the m e t h o d of

W. Craig [4]. Let p i (x ) be given in P({x}, r) by" p l ( x ) = p (x ) and p i ( x ) = p(p~- l (x ) )

if i > 1. Le t za* = {(p~(d~), e~): i ->__ 1}. Clearly pi(x) = x is an identi ty for ~ for i => 1.

So, by the universal proper t ies of A and A*, we can show that a r -

h o m o m o r p h i s m f : A ~ A * exists such that fh = h*, and f is a r - i somorph i sm

because there exists g : A*---> A reciprocal to f. It remains to show that A* is

recursive. Suppose d and e are in P ( Y , r ) , and Id[=n . Now, p ( x ) has an

occurrence of x but doesn ' t equal x, so Ip'(x)[>i for all i->__ 1. So, (d, e) is not in

{(p~(dl), e i ) : i > n} because i > n implies Ip~(d~)l > n. Therefore , we can recursively

decide whe ther (d, e) is in A* by de termining whether (d, e) belongs to the finite

and computab le set {(p~(di), e~):i <= n}. This proves Proposi t ion C.

REFERENCES

[1] G. BIRKHOFF, "Lattice Theory". Third ed., Amer. Math. Soc. Colloquium Publications XXV, Providence, R.I., 1967.

[2] S. BRENNER, On four subspaces of a vector space. J. of Algebra 29 (1974), 587-599. [3] H. -B: BRIrqKMANr~ and D. PUPPE, "Abelsche und exakte Kategorien, Korrespondenzen".

Lecture Notes in Mathematics 96, Springer-Verlag, Berlin, Heidelberg and New York, 1969. [4] W. CRAIG, On axiomatizability within a system. J. of Symbolic Logic 18 (1953), 30-32. [5] M. DAVIS, "Computability and Unsolvability". McGraw-Hill, New York, Toronto and London,

1958. [6] R. A. DEAN, Component subsets of the free lattice on n generators. Proc. Amer. Math. Soc. 7

(1956), 220-226. [7] R. FREESE, Planar sublattices of FM(4). Algebra Universalis 6 (1976), 69-72. [8] G. GRATZER, "Universal Algebra". Van Nostrand, Princeton, N. J., 1968. [9] C. HERRMArqN, On the equational theory of submodule lattices. Proc. University of Houston

Lattice Theory Conference, 105-118, Houston, 1973. [10] .- and A. HUHN, Zum Wortproblem liar freie Untermodulverbi:inde. Archiv fiir Mathematik 26

(1975), 449-453. [11] 185"194. ' Zum Begriff der Charakteristik moduIarer Verbiinde. Math. Zeitschrift 144 (1975),

[12] G. HIGMAN, B. H. NEUMANN and H. NEUMANN, Embedding theorems for groups. J. London Math. Soc. 24 (1949), 247-254.

[13] A. HUHN, Schwach distributive Verbi~nde I. Acta Sci. Math. 33 (1972), 297-305. [14] G. HUTCHINSON, Modular lattices and abelian categories. J. of Algebra 19 (1971), 156-184. [15] , The representation of lattices by modules. Bull. Amer. Math. Soc. 79 (1973), 172-176. [16] , Recursively unsolvable word problems of modular lattices and diagram-chasing. J. of

Algebra 26 (1973), 385-399. [17] , On classes of lattices representable by modules. Proc. University of Houston Lattice

Theory Conference, 69-94, Houston, 1973. [18] S. MACLANE, An algebra o[ additive relations. Proc. Nat. Acad. Sci. USA 47 (1961), 1043-1051. [19] M. MAKKAI and G. MCNULTY, Universal Horn axiom systems for lattices of submodules.

Manuscript, 1973. [20] JU. V. MATIYASEVIr Simple examples of unsolvable canonical calculi (Russian). Trudy. Mat.

Inst. Steklov. 93 (1973), 56-88, MR36#4996.

84 GEORGE HUTCHINSON

[21] D. PUPPE, Korrespondenzen in Abelschen Kategorien. Math. Ann. 148 (1962), 1-30. [22] J. ROTMAN, "The Theory of Groups, An Introduction". Second ed., Allyn and Bacon, Boston,

1973. [23] D. SACHS, Identities in finite partition lattices. Proc. Amer. Math. Soc. 12 (1961), 944-945. [24] B. M. SCHEIN, Relation algebras and [unction semigroups. Semigroup Forum I (1970), 1-62. [25] J. SHOENFIELD, "Mathematical Logic". Addison-Wesley Pub. Co., Reading, Mass., 1967. [26] A. SHAFAAT, On implicationaUy defined classes of algebras. J. London Math. Soc. 44 (1969),

137-140. [27] J. YON NEUMANN, "Continuous Geometry". Princeton University Press, Princeton, N.J., 1960. [28] P. M. WHITMAN, Free lattices II. Annals of Math. 43 (1942), 104-115.

National Institutes of Health Bethesda, Maryland U.S.A.

embedding and unsolvability theorems for modular lattices

Documents