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Semilatice Decompositions of Semigroups. Hereditariness and Periodicity—An Overview

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Algebraic Structures and Applications (SPAS 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 317))

Abstract

A semigroup is an algebraic structure consisting of a set with an associative binary operation defined on it. We can say that most of the work within theory is done on semigroups with a finiteness condition, i.e. a semigroups possessing any property which is valid for all finite semigroups—like, for example, completely \(\pi \)-regularity, periodicity are. There are many different techniques for describing various kinds of semigroups. Among the methods with general applications is a semilattice decomposition of semigroups. Here, we are interested, in particular, in the decomposability of a certain type of semigroups with finiteness conditions into a semilattice of archimedean semigroups. Having in mind that the definition of finiteness condition may be given, also, in terms of elements of the semigroup, its subsemigroups, in terms of ideals or congruences of certain types, we choose to characterize them mostly by making connections between their elements and/or their special subsets. We are, also, going to list some of the applications of presented classes of semigroups and their semilattice decompositions in certain types of ring constructions. This overview, which is, by no mean, comprehensive one, is mainly based on the results presented in the book [27], and articles [8, 28, 29].

“Semigroups aren’t a barren, sterile flower on the tree of algebra, they are a natural algebraic approach to some of the most fundamental concepts of algebra (and mathematics in general), this is why they have been in existence for more then half a century, and this is why they are here to stay.”

Boris M. Schein, [41]

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Acknowledgements

Melanija Mitrović is financially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, Grant 174026, and by the Faculty of Mechanical Engineering, University of Niš, Serbia, Grant “Research and development of new generation machine systems in the function of the technological development of Serbia”. Melanija Mitrović is grateful to Mathematics and Applied Mathematics Research Environment MAM, Division of Applied Mathematics at the School of Education, Culture and Communication at Mälardalen University for cordial hospitality and excellent environment for research and cooperation during her visit.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Mechanical Engineering, University of Niš, 18000, Niš, Serbia

    Melanija Mitrović

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Box 883, 72123, Västerås, Sweden

    Sergei Silvestrov

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  1. Melanija Mitrović
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  2. Sergei Silvestrov
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Correspondence to Melanija Mitrović .

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Editors and Affiliations

  1. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Sergei Silvestrov

  2. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Anatoliy Malyarenko

  3. Division of Applied Mathematics, School of Education, Culture and Communication, Mälardalen University, Västerås, Sweden

    Milica Rančić

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Mitrović, M., Silvestrov, S. (2020). Semilatice Decompositions of Semigroups. Hereditariness and Periodicity—An Overview. In: Silvestrov, S., Malyarenko, A., Rančić, M. (eds) Algebraic Structures and Applications. SPAS 2017. Springer Proceedings in Mathematics & Statistics, vol 317. Springer, Cham. https://doi.org/10.1007/978-3-030-41850-2_29

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