On the Word Problem for Compressible Monoids

Abstract

A monoid is compressible if it admits a presentation in which all words in the defining relations begin and end with some specified word. Associated to any compressible monoid M is a compressed left monoid L(M ). We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of compressible monoids. We prove that if C is a superAFL, then M has word problem in C if and only if L(M ) has word problem in C. In particular, this applies when C is the class of context-free languages. As a corollary, we deduce that if L(M ) has context-free word problem, then the rational subset membership problem for M is decidable. We also consider subspecial monoids, i.e. one-relation monoids which can be compressed to special monoids. In this case we extend the above results, and show that a subspecial monoid has context-free word problem if and only if all its maximal subgroups are virtually free. This gives a generalisation of the Muller-Schupp theorem to subspecial monoids.

Non-finitely Generated Maximal Subgroups of Context-free Monoids

“…Here the group of units of a monoid M is the maximal subgroup containing the identity element 1 of M . The author has also showed analogous results for subspecial and compressible monoids [39]. Thus it is also natural to ask to what extent the word problem (in the above sense) of a monoid relates to the word problems of its maximal subgroups.…”

The Word Problem for One-relation Monoids: A Survey

“…This algebraic connection has been exploited by Kobayashi [Kob00] (see Section 6 and Gray & Steinberg [GS19]. The author of the present survey has also investigated the language-theoretic aspects of compression [NB20b].…”

“…One is "weak", and is based on finding a selfoverlap free word as a prefix and suffix of both words in the defining relation. This method has been used and analysed in-depth by Lallement [Lal74], a paper by Adian & Oganesian [AO78], Kobayashi [Kob00], Gray & Steinberg [GS19], and Nyberg-Brodda [NB20b]. There is also another, more general, form of compression, which bears some similarity to the first, and is "stronger" in the sense it only requires that the two words in the defining relation have some shared prefix and some shared suffix; this is featured only in a paper by Adian & Oganesian [AO87] and the surveys by Lallement [Lal88,Lal93,Lal95].…”

“…We shall present the theory of weak compression essentially as it appears in the work of Adian & Oganesian [AO78], while blending in some notation from Kobayashi [Kob00] (see §6 for more details on Kobayashi's work). We remark that compression can easily be defined for arbitrary monoids, not necessarily one-relation, but for brevity we only deal with the one-relation case below; see [NB20b] for full details.…”

Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

“…The terminology monadic ancestor property was introduced by the author in [96], and also appears in [95,97], but was treated implicitly already in [22,76], see especially [76,Lemma 3.4]. The idea of defining classes of languages via ancestry in rewriting systems is not new, and can be traced back at least to e.g.…”