Amalgams of finite inverse semigroups

Abstract

We show that the word problem is decidable for an amalgamated free product of finite inverse\ud semigroups (in the category of inverse semigroups). This is in contrast to a recent result of M. Sapir\ud that shows that the word problem for amalgamated free products of finite semigroups (in the category\ud of semigroups) is in general undecidable

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

“…This construction is provided in [8] and is mainly along the lines of the one given in [2] for the amalgamated free products of a class of amalgams which satisfies some quite technical conditions are recalled in the following definition. …”

“…Then if the closure of each lobe is a finite graph, B * is a finite cactoid automaton whose lobes are finite DV -quotients of Schützenberger lobes relative to X i |R i for some i ∈ {1, 2}. We refer to Lemma 4 of [8] for the proof. Of course since in that paper S 1 and S 2 were assumed finite, the closures of the lobes were always finite automata and the construction always terminates.…”

“…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]. The paper is organized as follows.…”

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.…”

“…idempotents of F IM (X)). The automata were used by several authors to study free products of inverse semigroups [67], and various classes of amalgamated free products and HNN extensions in the category of inverse semigroups (e.g [14], [65], [26], [27]..) Related work on the structure of amalgams and HNN extensions of inverse semigroups has been done by Haataja, Margolis and Meakin [56], Bennett [15], Yamamura [146], [147], and Gilbert [47].…”