GJSFR Classification – F (FOR) 010101,010105,010206 When Is An Algebra Of Endomorphisms An Incidence Algebra? 1 Viji M., R.S.Chakravarti 2 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, About 1 Viji M.,Dept. of Mathematics, St.Thomas’ College, Thrissur-680001, Kerala E-mail:[email protected] 2 R.S.Chakravarti, Dept. of Mathematics, Cochin University of Science and Technology, Cochin-682022, Kerala. E-mail:[email protected] 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 M n (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 End K V . 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 E xy End K (V ) by E xy (vz) Define an order in X by if and only if E xy S for all x X, is reflexive Page |68 Vol.10 Issue 4(Ver1.0),September 2010 Global Journal of Science Frontier Research