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.

Amalgams of inverse semigroups and reversible two-counter machines

“…In the paper [3], Birget, Margolis, and Meakin proved that the amalgamated free product of two finitely presented semigroups with solvable word problems and a nice common subsemigroup may have undecidable word problem. This result has been further strengthened by Sapir [12] by showing that an amalgamated free product of finite semigroups may have undecidable word problem. These results are in contrast to the situation for amalgamated free products of groups where the word problem is decidable (see [7]) assuming general nice conditions on the amalgam.…”

“…It is well known that every amalgam of groups embeds in a group (and hence in the amalgamated free product of the group amalgam) [7]. However there are many examples showing that in the category of semigroups not every amalgam of two semigroups is embeddable into a semigroup, see [6,12]. On the other hand, T.E.…”

Minsky Machines and Algorithmic Problems

“…Theorem 7.1 (Sapir, [32]). (a) There exists an amalgam of two finite semigroups such that the word problem is undecidable in the corresponding free product with amalgamation and the amalgam embeds in the amalgamated free product.…”

Amalgams of inverse semigroups and $C^*$-algebras

“…We remark that the structure of amalgamated free products of semigroups or of inverse semigroups is far from understood in general. For example, it is known that the word problem for an amalgamated free product S 1 * U S 2 of semigroups (in the category of semigroups) may be undecidable even if S 1 , S 2 and U are finite semigroups [33]. On the other hand, the word problem for an amalgamated free product S 1 * U S 2 of finite inverse semigroups in the category of inverse semigroups is decidable [7].…”