Free products with amalgamation of monoids

Dekov, Deko V.

doi:10.1016/s0022-4049(96)00145-4

Cited by 4 publications

(8 citation statements)

References 3 publications

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“…By Proposition 3.11, we have that g = 1. Thus by part (7) of Lemma 3.1, the functions g → g| x are injective for all letters x. It follows by part (6) of Lemma 3.1 that the functions g → g| x are injective for all strings x.…”

Section: 2mentioning

confidence: 86%

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A correspondence between a class of monoids and self-similar group actions II

Lawson

Wallis

2015

Int. J. Algebra Comput.

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Abstract. The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa-Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych, In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be signficant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

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Section: 2mentioning

confidence: 86%

“…Thus by uniqueness g = 1, and so by part (7) of Lemma 3.1 the function from G to itself given by g → g| x is injective. Thus (1) holds.…”

Section: 2mentioning

confidence: 92%

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A correspondence between a class of monoids and self-similar group actions II

Lawson

Wallis

2015

Int. J. Algebra Comput.

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“…We now make use of the following classical theorem of Bourbaki ( [22]), in the left hand dual of the form given by Dekov in [33].…”

Section: Monoid Hnn-extensionsmentioning

confidence: 99%

Semigroup and Category-Theoretic Approaches to Partial Symmetry

Wallis¹

2017

Preprint

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“…However, the special word problem W = W (M 1 * M 2 , α) is such a language; (1) and (3) obviously hold for W and (2) is a consequence of the normal form theorem for free products (see [4] or [7]). …”

Section: The Word Problem Of a Free Productmentioning

confidence: 99%

Automata with Counters that Recognize Word Problems of Free Products

Corson

Ross

2015

Int. J. Found. Comput. Sci.

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An M -automaton is a finite automaton with a blind counter that mimics a monoid M . The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M -automata play a central role in understanding the family L(M ) of all languages accepted by M -automata. If G 1 and G 2 are finitely generated groups whose word problems are languages in L(M ), in general, the word problem of the free product G 1 * G 2 is not necessarily in L(M ). However, we show that if M is enlarged to the free product M * P 2 , where P 2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M 1 * M 2 lies in L(M * P 2 ) whenever M 1 and M 2 are finitely generated monoids with special word problems in L(M * P 2 ). We also observe that there is a monoid without zero, denoted by CF 2 , that can be used in place of P 2 for this purpose. The monoid CF 2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P 2 . The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P 2 in the above result and under what circumstances P 1 (or the bicyclic monoid) is enough to do the job of P 2 .

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Free products with amalgamation of monoids

Cited by 4 publications

References 3 publications

A correspondence between a class of monoids and self-similar group actions II

A correspondence between a class of monoids and self-similar group actions II

Semigroup and Category-Theoretic Approaches to Partial Symmetry

Automata with Counters that Recognize Word Problems of Free Products

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