Non-finitely Generated Maximal Subgroups of Context-free Monoids

Nyberg-Brodda, Carl-Fredrik

doi:10.48550/arxiv.2107.12861

Cited by 2 publications

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References 33 publications

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“…Indeed, the language is sufficiently natural to have been studied independently two decades prior to Duncan & Gilman's article, see [BJW82,Corollary 3.8]. The language (1.3) has been studied by several authors [HOT08, HHOT12, Bro18, BCP19, NB20a, NB20b,NB21b]. The approach via studying this language has also led to new insights in the group case; for example, a new proof of Herbst's result (see [Her91,HT93]) that a group has one-counter word problem if and only if it is virtually free [HOT08], which avoids the deep results involved in proving the Muller-Schupp theorem.…”

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On the Word Problem for Free Products of Semigroups and Monoids

Nyberg-Brodda¹

2021

Preprint

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We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which generalise e.g. the context-free or the indexed languages. When C is a super-AFL closed under reversal, we prove that the semigroup (monoid) free product of two semigroups (resp. monoids) with word problem in C also has word problem in C. This recovers and generalises a recent result by Brough, Cain & Pfeiffer that the class of context-free semigroups (monoids) is closed under taking free products. As a group-theoretic corollary, we deduce that the word problem of the (group) free product of two groups with word problem in C is also in C. As a particular case, we find that the free product of two groups with indexed word problem has indexed word problem.

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On the Word Problem for Free Products of Semigroups and Monoids

Nyberg-Brodda¹

2021

Preprint

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“…(2) There exists a finitely generated special monoid with context-free word problem, whose group of units is not finitely generated (indeed none of whose maximal subgroups is finitely generated). We refer the reader to Nyberg-Brodda [65] for this example, which answered in the negative a question of Brough, Cain and Pfeiffer (Question 10.4 in the pre-print version of [13]). (3) There exists a finitely generated special monoid with context-free group of units, but such that the word problem for the monoid is not context-free.…”

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confidence: 99%

On the word problem for special monoids

Nyberg-Brodda

2022

Semigroup Forum

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A monoid is called special if it admits a presentation in which all defining relations are of the form $$w = 1$$ w = 1 . Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the word problem, in the sense of Duncan and Gilman, for special monoids in terms of their group of units. We prove that a special monoid has context-free word problem if and only if its group of units is virtually free, giving a full generalisation of the Muller-Schupp theorem. This fully answers, for the class of special monoids, a question posed by Duncan and Gilman (Math Proc Camb Philos Soc 136:513–524, 2004). We describe the congruence classes of words in a special monoid, and prove that these have the same language-theoretic properties as the word problem. This answers a question first posed by Zhang (Math Proc Camb Philos Soc 112:495–505, 1992). As a corollary, we prove that it is decidable (in polynomial time) whether a special one-relation monoid has context-free word problem. This completely answers another question from 1992, also posed by Zhang.

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Non-finitely Generated Maximal Subgroups of Context-free Monoids

Cited by 2 publications

References 33 publications

On the Word Problem for Free Products of Semigroups and Monoids

On the Word Problem for Free Products of Semigroups and Monoids

On the word problem for special monoids

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