ACTA issues

Finite posets whose monoids of order-preserving maps are abundant

 Abstract. The ${\cal L}^*$ relation on a semigroup $S$ is defined as follows. For $a,b\in S$, $a{\cal L}^*b$ iff $a$ and $b$ generate the same left ideal in some semigroup $T$ of which $S$ is a subsemigroup. The relation ${\cal R}^*$ is defined dually. As introduced by Fountain, the semigroup $S$ is {\it abundant} providing each ${\cal L}^*$-class and each ${\cal R}^*$-class contain an idempotent. Since regularity of a semigroup may be defined analogously in terms of its ${\cal L}$ and ${\cal R}$ relations, abundancy is a generalisation of the classical concept of regularity. We determine all finite partially ordered sets $P$ for which $\mathop{\rm End}(P)$, the monoid of order-preserving selfmaps of $P$, is abundant. Moreover, in answer to a question of V. Gould, we show that abundancy and regularity are equivalent for endomorphism monoids of semilattices. AMS Subject Classification (1991): 06A06, 06A12, 20M17 Keyword(s): Poset, semilattice, endomorphism, abundant semigroup, regular semigroup Received July 15, 1998, and in revised form November 20, 2000. (Registered under 2771/2009.)