Basis reduction for cryptogroups and orthogroups
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Basis reduction for cryptogroups and orthogroups Ana Casimiro1 · Eduardo Skapinakis2 Received: 8 June 2020 / Accepted: 26 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The goal of this note is to provide equivalent bases of identities for subvarieties of completely regular semigroups. Keywords Equational characterizations · Completely regular semigroups · Cryptogroups · Orthogroups The search for equivalent definitions of famous classes of algebras has been attract‑ ing the attention of mathematicians for the last one hundred years [9–24, 26–28, 30–33]. Especially famous cases are Tarski’s single law for abelian groups [30] and Higman and Neumann’s single law for groups [10]. Here we carry the same study for semigroups as in [1–6, 8, 29]. We study equivalent definitions for completely regular semigroups, cryptogroups, and orthogroups. Unlike what happens with groups, it is a folklore result in equa‑ tional algebra that no class of completely regular semigroups in this note is single based [2, 4, 5]. It is worth observing that the difficulty of the results, rather than from proofs, comes more from the task of finding elegant and simpler versions of the original bases of identities. We will concentrate in finding equivalent set of identities which define classes of completely regular semigroups, cryptogroups and orthogroups, giving an alternative characterization of them, with less number of identities. A principal reference for the definitions and properties of these classes of semigroups is the book [25]. Communicated by Edmond W. H. Lee. * Ana Casimiro [email protected] Eduardo Skapinakis [email protected] 1
Departamento Matemática, Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829‑516 Caparica, Portugal
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Departamento Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829‑516 Caparica, Portugal
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A. Casimiro, E. Skapinakis
Throughout the paper (S, ⋅ , � ) is a semigroup with a unary operation, E(S) its set of idempotents, x◦ ∶= xx� and ◦x ∶= x� x , and associativity will be assumed except for completely regular semigroups and left regular semigroups, where the associativity axiom is replaced by another one. The classes induced by the corre‑ sponding right notions, that is, right regular (normal) cryptogroup or orthogroup, are treated similarly (see [25, Corollary IV.2.12]). Table 1 summarizes new characterizations for different subvarieties of completely regular semigroups, orthogroups and cryptogroups. The second column has the defi‑ nitions, the third the characterization provided by [25] and the last one has ours. The new results described in the fourth column of Table 1 are proved in the fol‑ lowing paragraphs. Completely regular semigroups That identities (1)–(4) imply identity (5) is trivial. Conversely, we only need to prove that identities (2), (3) and (5) imply identity (4). Note that x◦ = (x� )◦ (*), inde
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