Idempotents, Group Membership and their Applications

0497269 - ÚI 2019 RIV SK eng J - Článek v odborném periodiku
Porubský, Štefan
Idempotents, Group Membership and their Applications.
Mathematica Slovaca. Roč. 68, č. 6 (2018), s. 1231-1312. ISSN 0139-9918. E-ISSN 1337-2211
Grant ostatní: GA ČR(CZ) GA17-02804S
Program: GA
Institucionální podpora: RVO:67985807
Klíčová slova: multiplicative semigroup * finite semigroups * power semigroups * idempotent elements * finite commutative rings * principal ideal domain * Euler-Fermat theorem * Wilson theorem * matrices over fields * maximal groups contained in a semigroup * periodic sequence * multiplicative semigroup of ℤm * semigroup of circulant Boolean matrices
Obor OECD: Pure mathematics
Impakt faktor: 0.490, rok: 2018

Š. Schwarz in his paper [SCHWARZ, Š.: Zur Theorie der Halbgruppen, Sborník prác Prírodovedeckej fakulty Slovenskej univerzity v Bratislave, Vol. VI, Bratislava, 1943, 64 pp.] proved the existence of maximal subgroups in periodic semigroups and a decade later he brought into play the maximal subsemigroups and thus he embodied the idempotents in the structural description of semigroups [SCHWARZ, Š.: Contribution to the theory of torsion semigroups, Czechoslovak Math. J. 3(1) (1953), 7–21]. Later in his papers he showed that a proper description of these structural elements can be used to (re)prove many useful and important results in algebra and number theory. The present paper gives a survey of selected results scattered throughout the literature where an semigroup approach based on tools like idempotent, maximal subgroup or maximal subsemigroup either led to a new insight into the substance of the known results or helped to discover new approach to solve problems. Special attention will be given to some disregarded historical connections between semigroup and ring theory.
Trvalý link: http://hdl.handle.net/11104/0289830