ALGEBRAS ASSOCIATED WITH POSETS

Abstract

In this paper we introduce a class of algebras whose bases over a field K are pogroupoids.We discuss several properties of these algebras as they relate to the structure of their associated pogroupoids and through these to the associated posets also.In particular the Jacobi form is 0 precisely when the pogroupoid is a semigroup, precisely when the poset is (C2 + l)-free.Thus, it also follows that a pp-algebra KS over a field K is a Lie algebra with respect to the commutator product iff its associated poset S(<) is (C2+l)-free.The ideals generated by commutators have some easily identifiable properties in terms of the incomparability graph of the poset associated with the pogroupoid base of the algebra.We conjecture that a fundamental theorem on the relationship between isomorphic algebras and isomorphic pogroupoids holds as well. IntroductionAlgebras have been associated with a variety of different objects including groups, semigroups and posets themselves via such algebras as incidence algebras and Stanley-Reisner algebras [3], [10].In [4], J. Neggers defined a pogroupoid and he obtained a functorial connection between posets and pogroupoids and associated structure mappings.The present authors [5] demonstrated that a pogroupoid S(-) is a modular iff its associated poset S(<) is (C2 + I)-free, a condition which corresponds naturally to the notion of sublattice (in the sense of Kelly-Rival [1], [2]) isomorphic to N&, and that this is equivalent to the associativity of the pogroupoid.The present authors [6] showed that the modular pogroupoid (semigroup) S(-) is selfdistributive if and only if its associated poset S(<) is (1 © 2)-free, and the present authors with Y. H. Kim [8] discussed a dimension (parallel dimension) of a pogroupoid associated with posets and related to their pogroupoid algebras.This dimension is also an invariant of the incomparability graph (Harris diagram) of the poset under graph isomorphism (incomparability