Classes of monoids with applications: formations of languages and multiply local formations of finite groups
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Classes of monoids with applications: formations of languages and multiply local formations of finite groups Aleksandr Tsarev1 · Andrei Kukharev2,3 Received: 10 July 2020 / Accepted: 25 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract The Frattini subgroup 𝛷(G) of a group G is the intersection of all maximal subgroups of G. A formation 𝔉 of groups is said to be saturated if G∕𝛷(G) ∈ 𝔉 always implies G ∈ 𝔉 . A formation of finite groups is saturated iff it is local. A local satellite of 𝔉 is a function with domain the set of all primes whose images are formations of finite groups. If the values of this function are themselves local formations, then this circumstance leads to the definition of multiply local formation. The languages corresponding to n-multiply local and totally local formations of finite groups are described. Keywords Formation · Saturated formation · n-Multiply local formation · Totally local formation · Regular language · Formation of languages · Monoid · Algebird · Twitter · Data mining Mathematics Subject Classification 20F17 · 20D10
1 Introduction The Eilenberg [1] theorem establishes that there exists a bijection between the set of all varieties of regular languages and the set of all varieties of finite monoids. Formations extend the notion of a variety, and an Eilenberg-like theorem holds for formations, i.e., there is a one-to-one correspondence between formations of finite monoids and the formations of languages [2, Formation Theorem]. Thus we can study classes of regular languages which do not form varieties of languages.
* Aleksandr Tsarev [email protected] Andrei Kukharev [email protected] 1
Department of Mathematics, Jeju National University, Jeju 690‑756, South Korea
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Siberian Federal University, 79 Svobodny Avenue, Krasnoyarsk, Russia 660041
3
Department of Mathematics and IT, P.M. Masherov Vitebsk State University, 33 Moscow Avenue, 210038 Vitebsk, Belarus
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A. Tsarev, A. Kukharev
A group language is a language whose syntactic monoid is a group, or, equivalently, is recognized by a finite deterministic automaton in which each letter defines a permutation of the set of states. Pin and Soler-Escrivà [3] described the two classes of languages recognized by the groups D4 and Q8 , and they proved that the formations of languages generated by these two classes are the same. The theory of saturated (local) formations introduced by Gaschütz [4] became an integral part of group theory. “After completion of the classification of simple groups, the main problem in the theory of finite groups remains the problem of mastering mechanisms of their interaction in arbitrary groups. The most important handicap here is p-groups. These small bricks are encountered almost everywhere and, in addition, the possibilities of their interaction are displayed infinitely, like a horde of insects. The theory of formations is an attempt to be engaged in the theory of groups, that is to say, modulo p-groups. For all that, separa
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