IDEMPOTENT POINTLIKE SETS

Abstract

In this paper we answer a question posed by John Rhodes: "What are the aperiodic-idempotent-pointlike subsemigroups of S?" Answer: Precisely those aperiodic-pointlike subsemigroups that are idempotents, i.e. EPlA(S)={X|X≤E=E2∈PlA(S)}. In the proof we define, for a given variety V (closed under n-tuple expansion) and a given relation R:S-V∈V computing the V-pointlike subsets of S, an "improved" relation R(n):S-V(n) that computes the V-idempotent-pointlike subsemigroups of S. Consequently, for any W with decidable membership problem W ⓜ A is decidable.