GJSFR Classification – F (FOR) 010101,010105,010206

When Is An Algebra Of Endomorphisms An

Incidence Algebra?

1Viji M., R.S.Chakravarti2

Abstract-Spiegel and O’Donnell give a characterization of

algebras of n×n matrices which are isomorphic to incidence

algebras of partially ordered sets with n elements. We

generalize this result to get a characterization of algebras of

endomorphisms of a vector space which are isomorphic to

incidence algebras of lower finite partially ordered sets.

AMS Subject Classification: 16S50.

Keywords-incidence algebra, partially ordered set, lower

finite, endomorphism.

I. INTRODUCTION

A partially ordered set X is said to be locally finite if, the

subset Xyz = {x X : is finite for each y, z

X such that is said to be lower finite if the subset

is finite for each z X and is said to

be upper finite if the subset is finite

for each z X. If a partially ordered set X is lower or upper

finite then it is clearly locally finite.

The Incidence algebra I(X,R) of a locally finite partially

ordered set X over the commutative ring R with identity is

with operations defined by

for all f, g I(X,R), r R and x, y, z X. The identity

element of I(X,R) is

The JacobsonRadical, denoted by J(T) of a ring with identity

is the intersection of all its maximal right ideals. This is

always a two sided ideal and it is the largest ideal J of the

ring T such that 1 − t is invertible for all t J. It is proved

([1], Theorem 4.2.5) that the Jacobson Radical of an

incidence algebra consists of all the functions f X. So we

have,

About1Viji M.,Dept. of Mathematics, St.Thomas’ College, Thrissur-680001,

Kerala E-mail:vijigeethanjaly@cusat.ac.in 2R.S.Chakravarti, Dept. of Mathematics, Cochin University of Science and

Technology, Cochin-682022, Kerala.

E-mail:rsc@cusat.ac.in

Proposition 1.- ([1], Cor.4.2.6) Let X be a locally finite

partially ordered set

and R a commutative ring with identity. Then

The following result gives a relation between multiplication

in an incidence algebra and matrix multiplication.

Proposition 2-([1], Proposition 1.2.4) Let X be a locally

finite partially ordered set and R a commutative ring with

identity. Then I(X,R) is isomorphic to a subring of M|X|(R),

the R−module of all maps from X×X to R with pointwise

addition and scalar multiplication.

Then a natural question that arises is that, which subalgebras

of M|X|(R) are incidence algebras? For incidence algebras of

finite posets over a field, we have the following

characterization,

Theorem 1-([1], Theorem4.2.10) Let K be a field and S a

subalgebra of Mn(K). Then there is a partially ordered set X

of order n such that I(X,K) if and only if

i. S contains n pairwise orthogonal idempotents, and

ii. is commutative.

II. A CHARACTERIZATION of I(X,K) WHERE X IS a

LOWER FINITE PARTIALLY ORDERED SET AND K IS a

FIELD

Theorem 2- Let V be a K−vector space with dimension |X|,

for a suitable set X. Let S be a subalgebra of EndKV . Then

there exists a lower finite partial ordering in X such that S

I(X,K) if and only if,

i. 1 S

ii. S/J(S) is commutative

iii. For each x X, there is an Ex S of rank 1, such

that

iv. Xy = is finite for each y X

Proof-First we prove that the conditions given are

sufficient. Let S be A subalgebra of EndKV satisfying

conditions (1), (2), (3) and (4). From condition (3) it is clear

that we may find a basis for V such that

Define Exy EndK(V ) by Exy(vz)

Define an order in X by if and only if Exy S

for all x X, is reflexive

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antisymmetric.

Now from condition (4), it is clear that is a lower

finite partially ordered set, since

We prove that Observe that, for each

we have by writing

So

It is clear that will preserve addition and scalar

multiplication and will map identity of S to identity in

I(X,K). Now we will prove that will preserve

multiplication.

for two elements P,Q S.

Then,

So that

So preserves multiplication, and hence is a

homomorphism. If then pxy = qxy for each pair

x, y X and this implies P = Q. So is injective. Let

Define T such that (this

is possible since X is lower finite). Clearly, (T) = f.

Hence is surjective and is an isomorphism from S to

I(X,K).

Now we have to prove that conditions (1), (2), (3) and (4)

are necessary. Suppose that I(X,K) S where S is a

subalgebra of EndK(V ) of a K−Vector

space V , with dimension |X|, where X is a lower finite

partially ordered set. Let us consider the map : I(X,K) !

EndKV where is the map that is defined in the proof of

sufficiency part.Thus where Tf is such that

will be isomorphic to

I(X,K) and will satisfy the conditions (1), (3), (4) clearly and

(2) follows from Proposition 1 and the fact that

Will also have the properties specified in conditions (1), (2),

(3) and (4). Hence the theorem.

III. REFERENCES

1) E.Spiegel and C.J.O‘Donnell ; Incidence Algebras,

Monographs and Textbooks in Pure and Applied

Mathematics, Vol.206, Marcel Dekker, New York,

1997.

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