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On spectra of solvability and attainability for varieties of semigroups. (English) Zbl 0679.20053

Let r be a variety of semigroups, \({\mathcal X}\) a subvariety of r, and S a semigroup from r. The \({\mathcal X}\)-verbal of S is the set of congruence classes determined by the congruence \(P({\mathcal X},S)\) (i.e. \(S/\rho\) belongs to \({\mathcal X})\). The \(\alpha\)-th \({\mathcal X}\)-verbal of a semigroup S is defined for any ordinal \(\alpha\) by transfinite induction, and it is denoted by \({\mathcal X}^{\alpha}S\), that is, if \(\alpha\) is a non-limit ordinal then \({\mathcal X}^{\alpha}S={\mathcal X}({\mathcal X}^{\alpha -1}S)\); if \(\alpha\) is a limit ordinal then \({\mathcal X}^{\alpha}S=\cap_{\beta <\alpha}{\mathcal X}^{\beta}S\) where \(\cap\) denotes the intersection in the lattice. Then we have a descending chain \[ S={\mathcal X}^ 0S\geq {\mathcal X}^ 1S\geq...\geq {\mathcal X}^{\alpha}S\geq.... \] This chain is called the \({\mathcal X}\)-verbal chain of the semigroup S. The variety \({\mathcal X}\) is called \(\gamma\)-attainable on the semigroup S if \({\mathcal X}^{\gamma}S={\mathcal X}^{\gamma +1}S\). The smallest \(\gamma\) with this property is called the step of attainability of \({\mathcal X}\) on S, and denoted by step(\({\mathcal X},S)\). If there is an ordinal \(\gamma\) such that \({\mathcal X}\) is \(\gamma\)-attainable on all semigroups of the variety r, then \({\mathcal X}\) is called \(\gamma\)-attainable in r. The attainability introduced by the reviewer [J. Algebra 3, 261-276 (1966; Zbl 0146.027)] is nothing but 1-attainability here. \(E_ S\) denotes the family of \(\rho\)-classes such that each \(\rho\)-class is a subsemigroup of S. If for some ordinal \(\gamma\), \({\mathcal X}^{\gamma}S=E_ S\), then the semigroup S is called \(\gamma-{\mathcal X}\)-solvable, and in this case the ordinal step\(({\mathcal X},S)\) is called the step of \({\mathcal X}\)-solvability of S. The class of all ordinals that occur as steps of \({\mathcal X}\)-solvability of some semigroups from \({\mathcal K}\) is called the S-spectrum of \({\mathcal X}\) in \({\mathcal K}\). The S-spectrum is called complete if it coincides with the class of all ordinals. The main result of this paper is the following: Let \({\mathcal X}\) be an arbitrary non-trivial combinatorial variety of semigroups such that \({\mathcal X}\subseteq [x^ 2=x]\). Then S-spectrum \({\mathcal X}\) is in the class \({\mathcal N}\) of all commutative nil semigroups. From this result it follows that S-spectra of non-trivial varieties of semigroups in the class of all semigroups are completely determined. It is very interesting for the reviewer that the following classes of commutative semigroups are investigated: the class of commutative idempotent-free semigroups with power joined generators, and the class of commutative nil semigroups with power-joined generators. At the end of this paper it is shown that the variety of left [right] zero semigroups is not \(\omega\)-attainable in the class of all semigroups where \(\omega\) is the first infinite ordinal.
Reviewer: T.Tamura

MSC:

20M07 Varieties and pseudovarieties of semigroups
08A30 Subalgebras, congruence relations
20M15 Mappings of semigroups

Citations:

Zbl 0146.027
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References:

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