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. EPl A (S)={X|X≤E=E 2 ∈Pl A (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.