Structures characterizing partially ordered sets, and their automorphism groups

Abstract

By finding invariant embeddings of a partially ordered set X into the semigroups it is shown that the semigroup of order ideals of X, where the semigroup operation is set union, and the semigroup and semiring of order preserving maps from X into the positive cone D + of a partially ordered integral domain D, all characterize X to within poset isomorphism. The automorphism group of the semigroup of order ideals is isomorphic to the automorphism group of X. The same holds for the semiring of order preserving maps from X into the non-negative integers, and this semiring is the closed linear span of its idempotents.