universal algebra and its links with...

Universal Algebra and its Links with Logic,Algebra, Combinatorics, and Computer ScienceProc. "25. Arbeitstagung liber Allgemeine Algebra", Darmstadt 1983P. Burmeister et al. (eds.)Copyright Heldermann Verlag 19841-13

UNIVERSAL ALGEBRA AND LATTICE THEORY:A STORY AND THREE RESEARCH PROBLEMS

G. Gratzer

I would like to tell a story and present three researoh

problems on universal algebra and lattioe theory, with speoial

emphasis on the interaotion of these two fields. Although the

topics chosen reflect my special interests, I hope they give the

reader the flavour of this area of research.

The story: A free m-latt1ge. Lattice theory is two

faceted: on the one hand, a lattice is a universal algebra

<L; A, V) with two binary operations satisfying eight identities

(see, e.g. [8], §I.1); on the other hand, a lattice is a special

kind of poset <L; S) in which any two elements have a least

upper bound and greatest lower bound.

The universal algebraic approach gives lattice theory

such concepts as congruence relation, free lattice, free lattice

on a poset P (notation: F(P», free prod~ct, equational class,

etc.

The poset approach gives for example the concept of

completeness. The two approaches are sometimes difficult to

reconcile. For instance, there is no free complete lattice of 3generators since there are complete lattices of arbitrarily large

size completely generated by 3 elements.

In a series of papers [10], [11], [12], [13], [14],

[15], D. Kelly and the author continued the development of the

theory of m-lattices which was started by P. Crawley and R. A.

Dean [2]: a poset L is an m-lattice if all subsets X with

o < Ix I < m have a. least upper bound and greatest lower bound

(see also [9]). The theory of m-lattices is an important

contribution of universal algebra to lattice theory. (For a

survey of some aspects of this theory, see [16].)

2 G. Gratzer

if it is one-to-one , thenIt is an isomorphism iff

case , 58 is isomorphic toimage of B.

The development of this theory is quite lattice

theoretic in nature. However, occasionally, universal algebra

comes to the rescue.

Let H be the poset of Figure 1. The free m-lattice

D(m) (with an additional a and 1) on H is shown of Figure

4. It is made up of the lattice A of Figure 2, the "mirror

image" of A: the lattice B, for each 1 a dyadic rational, a

copy C1 0 f the I a t tic e C( m) 0 f Fig uI'e 2 ( C( tto) i sin the

"middle" of C(m); the upper part of c(m) is not shown), and

for each real t, 0 < t < 1, t non-dyadic, a copy Ci of the

two-element chain. Figure 5 shows in more detail how the

elements of A and B interact with the Ct'

The result D(No) = F(9) is due to I. Rival and R.

Wille (27J.

The proof that D(m) - {y, Y'} is the free nt-lattice on

H uses a result of P. Crawley and R. A. Dean [2]; it is

necessary to verify condition (W m): if P = I\x ~ VY = q, then

(X u Y) n [p, q] ". is. ( X, Y .& D ( m), 0 < I X I, !Y I < m}. The

proof is very long and requires a detailed analysis of where p

and q are.

In the paper [13], D. Kelly and the author found a

universal algebraic proof avoiding most of the tedious

computations. This proof uses the m = NO case (the result of

Rival and Wille) and some universal algebraic trivialities.

Let ! be a variety (equational class) of algebras of

some finitary or infinitary type. For ~ = (A; F> E ~ and

9 £ A, we define the partial K-algebra ~ = (H; F> on 9 as

follows: if f E F, a O' a 1 , ... E 9 and f(a o' a" ••• ) = a E H

in ~, then (and only then) f( aO' a" ••• ) is defined on H

and equals a. ~ is called a relatiye. algebra of ~.

If e = (B; F> is a partial algebra of the type of K,then F(~) denotes the free (-algebra generated by I. The

canonical map of 5'8 into F (~) is not necessarily one ... to-one;

it is an embedding of ~ into F( 58).

58 is a partial ~-algebra; in this

the relative algebra of ~ on the

Now let uo and ~1 be partial ~-algebras,

Universal algebra and lattice theory

AO n A1 = A2 suoh that ~2 is the same as a relative algebra of

~O and of ~1· We shall say that ~O and ~1 can be etronglyamalgamated oyer ~2' if there is an algebra ~3 ~ ~ of whioh

both ~O and ~1 are relative algebras and AO n A1 = A2 in~3. The following easy lemma should give the flavour of the new

proof.

3

1..e..m.mA.let ~t be the

and F(~t) canalgebra [At)

isomorphic to

Let U be a partial (-algebra, let At $ A, andoorresponding relative algebra of ~. If ~

be strongly amalgamated over Ut , then the subof F(~) generated by At is naturally

F ( Ut ) •

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

First Problem: Let U be an idempotent algebra such

that Pi (tt) < a::I for all i. Does there exist an n suoh that

<P2(~)' ••• , Pn(U» has the Minimal Extension Property?

The concept of the Pn-sequence was introduoed by

E. Marozewski. Take an algebra ~,and consider the sequence<fo ' f 1 , f 2 , ••• >, where f n is the size of the free algebra

over ~ with n generators. The Basic Problem is the

characterization of such sequenoes. Now look at Figure 6,

depicting the free algebra with 2 generators. The shaded areas

are two copies of F(l), they overlap in F(O). Thus

f 2 ~ 2f, - f O• There is a similar inequality for f n for every

n (the, so oalled, inolusion exolusion prinoiple). So it is

more convenient to study Pn' the number of essentially n-arypolynomials ("essentially" means that they depend on all n

variables). The size of the unshaded area in Figure 6 is P2.So let PO(U) be the number of oonstant unary poly

nomials, let Pl(~) be the number of non-oonstant (essentially

unary) polynomials exoluding p(x) = x; for n > 1, P (ti) isn

the number of essentially n-ary polynomials.

The Basio Problem restated is: which sequenoes <Pn>

oan be represented as <Pn(~» for some algebra U. It was

proved in [22] that if Po - 0, <Pn> is always so representable.

In the oase Po = 0, there are many representation

4 G. Gratzer

results. The tollowing, also from (22], is typical: If > 0

for all n ~ 1, or if n divides for n even and > 0for n odd, or if n divides for all n, then is so

repre.sent.able.All three oonditions impose some restriction on each

Pn' but the~ of PiI ,. j.

does not influenoe the size of if

, ..... J

In the idempotent case = P, = 0, that is, there areno nullary operatioas and f(x, ••• , x) = x for all operations

f) the situation is very different ([21] aad [24]):Let U be aa idempotent algebra which is not equivalent

to a semilattiae, a diagooal algebra «Bo ; t>, where1 1f«a 1 , ... , a>,

1 0<a 1 , , ••• , defines the o-ary operation f), an idempotentreduat of a Boolean group (let <G; +> be a Boolean group, that

is, 2x = 0; the idempotent reduat is <0; x + y + z». Thenthere exists an n such that

(~O < eu) <

So in the idempotent case < is strictly iriereasingtrom some point on (with three exceptions). Of course, given a

eU», we oan trivially ooastruct a <PiCe» whioh agrees with(U» up to a, and from n on <ptce» increases faster:

(m> > (U) for all i > n.For the free idempoteat semigroup satisfying xyz = xzy,

obviously Pn = n for all n ~ 2. The following result ofJ. P~onka [26] proves an astonishing converse: Let U be an

idempotent algebra satisfying (U) = 2, (U) = 3, and P4(U> =4. Then p eU) ~ n for all n a 2.

nIn other words, <0, 0, 2, 3, 4) has a "minimal"

extension <0, 0, 2, 3, 4, 5, •••• n •••• ); every other extension

is a the minimal extension.

Definition: <0, 0, P > has the MinimalExtension Property Iff there ' a~ 'lctemBotent algebra.'satisfying (1) Pi = Pi (U) for 2 Sis nj (Ii) for any

Idempotent algebra 8. it Pi = Pi(8) for 2 SiS D, thenPj(t{) S (8) tor all jan.


It was proved in my paper with R. Padmanabhan [17] that

(0, 0, 1, 3, 5> has the Minimal Extension Property and theminimal extension is given by the algebra <Ai 0>, where (Ai +>is an abelian group of exponent 3, and x 0 y = 2x + 2y.

The First Problem asks, whether any idempotent algebra ti

gives rise to a sequenoe <0, 0, Pi(ti), ••• , Pn(ti» with the

Minimal Extension Property if n is large enough.

For a more detailed introduotion to this problem, see(4]. For more reoent referenoes, oonsult Algebra Universalis,

espeoially, A. Kisielewioz [24].

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Second Froblem: Find all lattioes K with ~he 'property

that whenever K oan be embedded in the ideal lattioe I(L) ofa lattioe L, then K oan also be embedded into L.

This problem illustrates how developments in universal

algebra open up new fields in lattioe theory. Chapter 7 of (7]is devoted to the following question in universal algebra: whioh

first ordered properties are preserved under algebraiooonstruotions (e.g., formations of subalgebras, homorphio images,

direot produots).

The formation of the ideal lattioe I(L) of a lattioe

L is an algebraio oonstruotion that is speoial to lattioe

theory. So it is natural to raise the question what properties

are pr~served under the formation of ideal lattices.I discussed this question briefly in 1961 (see (5]).

Sinoe there seem to be too many such properties, I proposed to

investigate the special case: the lattioe has a sublattice

isomorphio to a given lattioe K. This is how we arrive at theSecond Problem.

Lattices K satisfying this property are called

transferable. There is also a stronger oonoept. Let <p be an

embedding of K into I(L)i for every a E K, a<p is an ideal ofL. If K is transferable, then there is an embedding ~ of K

into L. Now it seems natural to require, for a E K, that a~

be in the ideal a<p, but not in any b<P where b < a. If there

is always such a map ~, K is oalled aharply transferable.

5

6 G. Gratzer

of "The story" with m = ~O' that Is,v v implies that {x, y, u, v} n

It is an instructive exercise to check that N5 is

sharply transferable. (In fact, K. A. Baker and A. Hales

observed that all finite projeative lattices are transferable;

Algebra Universal Is 4 (1974), 250-258.) Sharp transferability is

easier to handle than transferability since we know roughly where

aw should be. Let us start the discussion with somedefinitions:

1. Let <PI ~> be a partially ordered set. For

Xt I S P, define X < Y to hold if and only if for every x € X

there exists y E I suoh that x ~ y.2. Let <S; v> be a join-semilattice, p ~ S, and

J S S. We say that <p, J) is a minimal pair if and only if the

following three conditions hold:

(i) p ~ J;

(ii) p ~ V;(Iii) if J'.s S, P ~ VJ', a1?d J' < J ,then J.= J'

3. A semilattioe (5; v> is said to satisfy (T) if

and only if there exists a linear order relation R on S suohthat if <p, J> is a minimal pair, then p R x holds for all

X E J.

4. A lattice <L; At v> is said to satisfy ( if

and only if <L; v> satisfies (T), and to satisfy ) if and

only if the dual of <L; A> satisfies (T).

Iheorem. A finite lattice is sharply transferable if

and only if it satisfies the three oonditions (Tv), (T A), and

on.Condition (N) is (Nm)

the condition: x A y S U

X A y, U v v] ~ ~.

Later, in 19], I generalized this result with Platt to

arbitrary lattices.

The first result for transferability is in [5] (see [6]

for a proof): A transferable lattice cannot have doubly

reduoible elements.

The proof relies on the following result: Every lattice

can be embedded in the ideal lattice of a lattice without doubly

irreducible elements.


This led to the following

Conjecture. The class of transferable lattices is the

intersection of all classes K of lattices with the propertythat every lattice can be embedded in the ideal lattice of some

lattice in ~.

Such classes were investigated in [18] and [20].

Sufficiently many such classes were constructed to conclude:

Theorem. For finite lattices, transferability and sharptransferability are equivalent.

It is quite likely that proving the Conjecture would

lead to a solution of the Second Problem.

* • • • • • * * * * * * * * * * * • * * * * * * * * * * * * * * •

Third fcoblem: Find a nontrivial class ! of groupoids

(algebras with one binary operation +) such that for every

~ E K, a, b, c, d E A, c .i. d (9(a, b» iff c = a + y and

d = b + Y for some YEA.

This problem comes from a universal algebraic problemx1a lattice theory.

For an algebra ~,a, b E A, let 9(a, b) denote thesmallest congruence relation under which a.i. b. Mal'cev's Lemma

(see, e.g., (7], Theorem 10.3) describes 9(a, b) as follows:

c .i. d (9(a, b) iff there exists a sequence Zo = c,

zl' ••• , zn = d of elements of A such that for each i < n,there exists a polynomial P.(x, Xl' ••• , x) with

1 mPi (a, Yl'"'' Ym) = zi+f(i)' Pi (b, Y1' ... , Ym) = zi+l-f(i) forsome Yl"'" Ym E A, where f is a choice function, f(i) = 0

or f(1) = 1-

The choice of n, of PO"'" Pn- 1 ' and of f depends

on a, b, c, and d. Would it not be nice, if they could bechosen 1ndependently of a, b, c, d? In most classical algebraic

systems this is not the case. Take a Boolean group:

c _ d (9(a, b» Iff c + d = a + b or c + d = 0; 1n th1s case,

n = 1 t p(x, y) ;: x + y, but f depends on a, b, 0, and d.

7

8 G. Gratzer

For an interesting example, lattice theory came to the

rescue. I proved with E. T. Schmidt [23J that in a distributive

is called a congruence scheme E.

to have a uniform congruencePo' ... , Pn-1 ' and f

An equational class ~ is saidscheme,E, if for all ~ to K, a,

lattice L, a, b, c, d to L, a S b, c S d, c.:. d

(a v Y1) " Y2 = c, (b v Y1) " Y2 = d for some

there are examples where PO' ••• , Pn-1' fare

whole equational class.

b, c, d to A, c

can always be described with the same PO' ••• ,

This concept was introduced by R. Magari [25J.

('l(a, b)) iff

Yl' Y2 to L. Thusthe same for a

_ d ('l (a, b.))

Pn-1 ' and f.See also [3].

In [1J, I considered with J. Berman the question how

congruence schemes may look like. If there are constants, the

problem seems very difficult. Even n = 1, f(O) = 0, p(x, y) =o + «0 + x) + y) cannot be the congruence scheme for an algebra

with more than one element. Now if there are no constant

operations, then the result is very nice:

Theorem. PO' ••• , Pn-1' f is the uniform congruence

scheme for a nontrivial variety iff all Pi are at least binary.

The condition, all Pi are binary, simply means that no

Pi has only x as variable.

The equational class constructed in the proof is of

unspecified type. Of course, the type contains all the

operations needed to build up the Pi' but it contains a number

of additional operations. The Third Problem asks, in the

simplest possible case, what happens if the type is fixed.

The congruence scheme in the Third Problem is p(x, y) =x + Y (and f(O) = 0). One can raise the same problem with any

other reasonable polynomial (y + (x + y), Y + «x + y) + x),

etc.) or pairs of polynomials, e.g., PO(x, y) = x + y, p,(x, y) =

y + «x + y) + (x + y» f(O) = 0, f(1) = 1 that is,

c _ d ('l(a, b))

iff

c = a + Y1' b + Y1 = Y2 + «a + Y2) + (a + Y2)'

d = Y2 + «b + Y2) + (b + Y2))·


References

[1] J. Berman and G. Gratzer: Uniform representations ofcongruence schemes. Pacific J. Math. 76 (1978), 301-311.

[2] P. Crawley and R. A. Dean: Free lattices with infiniteoperations. Trans. Amer. Math. Soc. 92 (1959), 35-47.

[3] E. Fried, G. Gratzer, and R. W. Quackenbush: Uniformcongruence schemes. Algebra Universal is 10 (1980),176-188.

[4] G. Gratzer: Composition of functions.In Proceedings of the Conference on Universal Algebra,Queen's University, Kingston, Onto (1969), 1-106.

[5] G. Gratzer: Universal Algebra.In Current Trends in Lattice Theory.D. Van Nostrand, (1970), 173-215.

[6] G. Gratzer: A property of transferable lattices.Proc. Amer. Math. Soc. 43 (1974), 269-271.

[7] G. Gratzer: Universal Algebra. Second Edition.Springer Verlag, New York; Heidelberg, Berlin, 1979.

[8] G. Gratzer: General Lattice Theory.Pure and Applied Mathematics Series, Academic Press,New York, N. Y.; Mathematische Reihe, Band 52, BirkhauserVerlag, Basel; Akademie Verlag, Berlin, 1978. (Russiantranslation: MIR PUblishers, Moscow, 1982.)

[9] G. Gratzer, A. Hajnal, and D. Kelly: Chain conditions infree products of lattices with infinitary operations.Pacific J. Math. 83 (1979), 107-115.

[10] G. Gratzer and D. Kelly: A normal form theorem forlattices completely generated by a subset. Proc. Amer.Math. Soc. 67 (1977), 215-218.

[11] G. Gratzer and D. Kelly: Freem-products of lattices. 1and II. Colloq. Math., to appear.

[12] G. Gratzer and D. Kelly: The free m-lattice on the posetH. ORDER, to appear.

[13] G. Gratzer and D. Kelly: A technique to generate m-aryfree lattices from finitary ones.

[14] G. Gratzer and D. Kelly: An embedding theorem for freem-lattlces on slender posets.

[15] G. Gratzer and D. Kelly: A description of free m-latticeson slender posets.

9

lOG.Gratzer

[16] G. Gr~tzer and D. Kelly: The construction of some freem-lattices on posets. Proceedings of the Lyon Conferenceon Ordered Sets.

[17] G. Gratzer and R. Padmanabhan:and non-associative groupoids.(1971), 75-80.

On idempotent, oommutative,Proc. Amer. Math. Sao. 28

[18] G. Gritzer and C. R. Platt: Two embedding theorems forlattices. Proc. Amer. Math. Soc. 69 (1978), 21-24.

[19] G. Gritzer and C. R. Platt: A characterization of sharplytransferable lattices. Canad. J. Math. 32 (1980), 145-154.

[20] G. Gritzer, C. R. Platt, and B. Sands: Embedding latticeso lattice of ideals. Paoific J. Math. 85 (1979),

-75.

[21 G. Gritzer and J. Plonka: On the number of polynomials ofan idempotent algebra. I and II. Pacific J. Math. 32(1970), 697-709 and 47 (1913), 99-113.

[22] G. Oritzer, J. Plonka, and A. Sekanina:polynomials of a universal algebra. I.(1910),9-11.

On the number ofColloq. Math. 22

[23] G. Gr~tzer and E. T. Schmidt: Ideals and congruencerelations in lattices. Acta Math. Acad. Sci. Hungar. 9( 1958 ), 131-175.

[24) A. Kisielewicz: The sequences of idempotent algebras

are strictly increasing. Algebra Universal Is 13 (1981),

233-250.

[25] R. Magari: The classification of idealizable varieties(Congruenze ideali IV), J. of Alg., 26 (1973), 152-165.

[26] J. P~onka: On algebras with at most n distinctentially n-ary operations. Algebra Universal is

1971).80-85.

[27] I. Rivai and R. Wille: Lattices freely generated byially ordered sets: Which can be "drawn"? J. Reine •

. Math. 310 (1979),56-80.

University of ManitobaWinnipeg, ManitobaR3T 2N2 Canada

Universal algebra and lattice theory 11

The poset HFigure I

L1+1 °1+1 bi+1

OJ

LI i limit

°1

Lo Q"'Oo

The lottice A

Figure 3

Figure 2

<0.1>

A

G. Gratzer

The m-lofticeD(m)

Figure 4

Details of D(m)

Figure 5

t real nondyadic

<1,0>

i dyadic

<s.r> 8

r< t<s

r. s dyadic

t real noodyadic


Figure 6

13

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