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Classes of monoids with applications: formations of languages and multiply local formations of finite groups

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Abstract

The Frattini subgroup \(\varPhi (G)\) of a group G is the intersection of all maximal subgroups of G. A formation \({\mathfrak {F}}\) of groups is said to be saturated if \(G/\varPhi (G) \in {\mathfrak {F}}\) always implies \(G \in {\mathfrak {F}}\). A formation of finite groups is saturated iff it is local. A local satellite of \({\mathfrak {F}}\) 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.

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Acknowledgements

The authors thank the referee for careful reading of the paper.

Funding

This work was supported by the Russian Science Foundation under Grant 18-71-10007.

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Authors and Affiliations

  1. Department of Mathematics, Jeju National University, Jeju, 690-756, South Korea

    Aleksandr Tsarev

  2. Siberian Federal University, 79 Svobodny Avenue, Krasnoyarsk, Russia, 660041

    Andrei Kukharev

  3. Department of Mathematics and IT, P.M. Masherov Vitebsk State University, 33 Moscow Avenue, 210038, Vitebsk, Belarus

    Andrei Kukharev

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  1. Aleksandr Tsarev
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  2. Andrei Kukharev
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Correspondence to Aleksandr Tsarev.

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Tsarev, A., Kukharev, A. Classes of monoids with applications: formations of languages and multiply local formations of finite groups. Rend. Circ. Mat. Palermo, II. Ser 70, 1257–1268 (2021). https://doi.org/10.1007/s12215-020-00556-9

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