On the Word Problem for Free Products of Semigroups and Monoids

Nyberg-Brodda, Carl-Fredrik

doi:10.48550/arxiv.2112.10665

Cited by 3 publications

(11 citation statements)

References 15 publications

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“…Similarly general statements involving reversal-closed super-AFLs appear as the main results in previous work by the author[95,96,97].…”

supporting

confidence: 79%

“…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.…”

Section: Rewriting Systemsmentioning

confidence: 99%

“…In this section, we will generalise (in a fairly uncomplicated manner) the bipartisan ancestors introduced in [97] to polypartisan ancestors, and prove that this construction preserves certain language-theoretic properties of the languages it is applied to. We will use this construction to obtain the multiplication table for a monoid free product from the table for a semigroup free product.…”

Section: Polypartisan Ancestorsmentioning

confidence: 99%

“…, R k ). A useful intuition for polypartisan ancestors is as follows: one may imagine the k-shuffled language Polypartisan ancestors generalise in an easy way the bipartisan ancestors introduced by the author in [97]. It is clear that R (k) (L) is a k-shuffled language.…”

Section: Polypartisan Ancestorsmentioning

confidence: 99%

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

Nyberg-Brodda¹

2022

Preprint

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This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied e.g. by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids, or groups with a C-multiplication table, where C is any reversal-closed super-AFL. In particular, we deduce that the free product of two groups with ET0L resp. indexed multiplication tables again has an ET0L resp. indexed multiplication table.

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“…Similarly general statements involving reversal-closed super-AFLs appear as the main results in previous work by the author[95,96,97].…”

supporting

confidence: 79%

Section: Rewriting Systemsmentioning

confidence: 99%

Section: Polypartisan Ancestorsmentioning

confidence: 99%

Section: Polypartisan Ancestorsmentioning

confidence: 99%

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

Nyberg-Brodda¹

2022

Preprint

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“…We refer the reader to Nyberg-Brodda [61] for a full (rather straightforward) proof. Thus we have a combinatorial characterisation of super-AFLs in terms of rewriting systems.…”

Section: Proposition 11 Let C Be a Full Afl Then C Is A Super-afl If ...mentioning

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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Multiplication Tables and Word-Hyperbolicity in Free Products of Semigroups, Monoids and Groups

Nyberg-Brodda

2023

J. Aust. Math. Soc.

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This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc.136(3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a $\mathbf {C}$ -multiplication table, where $\mathbf {C}$ is any reversal-closed super- $\operatorname {\mathrm {AFL}}$ . In particular, we deduce that the free product of two groups with $\mathbf {ET0L}$ with respect to indexed multiplication tables again has an $\mathbf {ET0L}$ with respect to an indexed multiplication table.

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On the Word Problem for Free Products of Semigroups and Monoids

Cited by 3 publications

References 15 publications

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

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

On the word problem for special monoids

Multiplication Tables and Word-Hyperbolicity in Free Products of Semigroups, Monoids and Groups

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