Languages associated with saturated formations of groups

Ballester‐Bolinches, A.; Pin, Jean-Éric; Soler–Escrivà, Xaro

doi:10.1515/forum-2012-0161

Cited by 17 publications

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References 28 publications

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“…The second step of our programme [6] has been precisely to study this operation, and the corresponding operation on languages, when H is a formation. The definition of a Hall variety can be readily extended to formations and as we said earlier, formations are a more flexible tool than varieties in finite group theory.…”

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confidence: 99%

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Ballester‐Bolinches

Soler–Escrivà

2012

Forum Mathematicum

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Abstract. We present an extension of Eilenberg's variety theorem, a well-known result connecting algebra to formal languages. We prove that there is a bijective correspondence between formations of finite monoids and certain classes of languages, the formations of languages. Our result permits to treat classes of finite monoids which are not necessarily closed under taking submonoids, contrary to the original theory. We also prove a similar result for ordered monoids. This paper is the first step of a programme aiming at exploring the connections between the formations of finite groups and regular languages. The starting point is Eilenberg's variety theorem [10], a celebrated result of the 1970's which underscores the importance of varieties of finite monoids (also called pseudovarieties) in the study of formal languages. Since varieties of finite groups are special cases of varieties of finite monoids, varieties seems to be a natural structure to study languages recognized by finite groups. However, in finite group theory, varieties are challenged by another notion. Although varieties are incontestably a central notion, many results are better formulated in the setting of formations . This raised the question whether Eilenberg's variety theorem could be extended to a "formation theorem".The aim of this paper is to give a positive answer to this question. To our surprise, the resulting theorem holds not only for group formations but also for fo rmations of finite monoids. We also prove a similar result for formations of ordered finite monoids, extending in this way a theorem of [17] .Before stating these results more precisely, let us say a word on the aforementioned research programme and give a brief overview of the existing literature.

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Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Ballester‐Bolinches

Soler–Escrivà

2012

Forum Mathematicum

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“…More precisely, they spotted a bijective correspondence between formations of finite monoids and the so-called formations of languages. Using this "formation theorem" the authors not only recovered the previously mentioned results on nilpotent groups, soluble groups and supersoluble groups, but, relying on the local definition of a saturated formation [5], they exhibited new examples, like the class of groups having a Sylow tower [3].…”

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confidence: 83%

Languages and formations generated by D4 and Q8

Soler–Escrivà

2019

Theoretical Computer Science

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We describe the two classes of languages recognized by the groups D 4 and Q 8 , respectively. Then we show that the formations of languages generated by these two classes are the same. We also prove that these two formations are closed under inverses of morphisms, which yields a language theoretic proof of the fact that the group formations generated by D 4 and Q 8 , respectively, are two equal varieties.Most monoids and groups considered in this paper are finite. In particular, we use the term variety of groups for variety of finite groups. Similarly, all languages considered in this paper are regular languages and hence their syntactic monoid is finite.

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“…Languages corresponding to saturated formations of finite groups were studied in [6,39]. Naturally rises the following question: How to describe the languages corresponding to s-closed (one-generated) totally composition formations of finite groups?…”

Section: Definition 8 [5]mentioning

confidence: 99%

Algebraic lattices of solvably saturated formations and their applications

Tsarev

Kukharev

2020

Bol. Soc. Mat. Mex.

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In each group G, we select a system of subgroups sðGÞ and say that s is a subgroup functor if G 2 sðGÞ for every group G, and for every epimorphism u : A ! B and any H 2 sðAÞ and T 2 sðBÞ, we have H u 2 sðBÞ and T u À1 2 sðAÞ. We consider only subgroup functors s such that for any group G all subgroups of sðGÞ are subnormal in G. For any set of groups X, the symbol s s ðXÞ denotes the set of groups H such that H 2 sðGÞ for some group G 2 X. A formation F is s-closed if s s ðFÞ ¼ F. The Frattini subgroup UðGÞ of a group G is the intersection of all maximal subgroups of G. A formation F is said to be solvably saturated if it contains each group G with G=UðNÞ 2 F for some solvable normal subgroup N of G. Composition formations are precisely solvably saturated formations. It is shown that the lattice of all s-closed totally composition formations is algebraic. Keywords Finite group Á Subgroup functor Á Formation of groups Á Satellite of formation Á Totally composition formation Á Algebraic lattice of formations Á Formal language Á Hypergroup Mathematics Subject Classification Primary 20F17 Á Secondary 20D10 Á 43A62 Á 20M35 This work has been supported by the Russian Science Foundation under Grant 18-71-10007.

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Languages associated with saturated formations of groups

Cited by 17 publications

References 28 publications

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Formations of finite monoids and formal languages: Eilenberg's variety theorem revisited

Languages and formations generated by D4 and Q8

Algebraic lattices of solvably saturated formations and their applications

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