Algorithmic Problems for Amalgams of Finite Semigroups

Abstract

We prove that there exists an amalgam of two finite 4-nilpotent semigroups such that the corresponding amalgamated product has an undecidable word problem. We also show that the problem of embeddability of finite semigroup amalgams in any semigroups and the problem of embeddability of finite semigroup amalgams into finite semigroups are undecidable. We use several versions of Minsky algorithms and Slobodskoj's result about undecidability of the universal theory of finite groups.

Groups and semigroups: connections and contrasts

“…Recently, Luda Markus-Epstein [92] has used an extension of the methods of Stallings foldings to study algorithmic problems for finitely generated subgroups of amalgamated free products of finite groups. Her construction is related to a construction used by Cherubini, Meakin and Piochi [27] to prove that the word problem for amalgamated free products of finite inverse semigroups in the category of inverse semigroups is decidable: this is in contrast to a result of Sapir [123] that shows that the word problem for amalgamated free products of finite semigroups (in the category of semigroups) is undecidable in general. It seems plausible that further investigation of these techniques might prove fruitful in the study of algorithmic problems for subgroups of finitely presented groups and for other algorithmic problems about groups and semigroups.…”

Decidability of the representation extension property for finite semigroups

“…II, page 138). More recently, Sapir [7] has shown that it is in fact undecidable whether an amalgam of (finite) semigroups embeds in any (finite) semigroup. A semigroup S is called an amalgamation base for semigroups if every amalgam of semigroups containing S as a subsemigroup embeds in some semigroup.…”

Decidability Versus Undecidability of the Word Problem in Amalgams of Inverse Semigroups

“…The word problem is decidable for amalgamated free products of groups and is undecidable for amalgamated free products of semigroups (even when the two semigroups are finite [29]) but it is not known under which conditions on the inverse semigroups the word problem for amalgamated free products is decidable in the category of inverse semigroups (Problem 5 of [21]). In the sequel, we will briefly illustrate some sufficient conditions on amalgams of inverse semigroups for the word problem being decidable in the amalgamated free products [7,8] and a negative recent result [28].…”