Identities for existence varieties of regular semigroups. (English) Zbl 0666.20028

Hall, T. E.

Bull. Aust. Math. Soc. 40, No. 1, 59-77 (1989).

A class \({\mathcal V}\) of regular semigroups is called an existence variety (or e-variety) if it is closed under the operations of taking all homomorphic images, regular subsemigroups and direct products. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities. For any regular semigroup (S,\(\circ)\), there is by the axiom of choice a unary operation \('\) on S such that for all \(x\in S\), \(xx'x=x\) and \(x'xx'=x'\). Such an operation is called an inverse unary. By a regular unary semigroup one means an algebra \((S,\circ,')\), such that (S,\(\circ)\) is a regular semigroup and \('\) is an inverse unary operation on (S,\(\circ)\). We denote the variety of all regular unary semigroups by \({\mathcal R}{\mathcal U}{\mathcal S}\), and the e-variety of all regular semigroups by \({\mathcal R}{\mathcal S}\). For each e-variety \({\mathcal V}\) of regular semigroups the class \({\mathcal V}'=\{S,\circ,')\in {\mathcal R}{\mathcal U}{\mathcal S}:\) (S,\(\circ)\in {\mathcal V}\}\) is a variety of regular unary semigroups. An e-variety \({\mathcal V}\) is strongly determined by the set of identities B if B is a basis of Id(\({\mathcal V}')\). For each set C of \({\mathcal R}{\mathcal U}{\mathcal S}\)-identities the class \({\mathcal E}(C)=\{(S,\circ)\in {\mathcal R}{\mathcal S}:\) (S,\(\circ)\) satisfies \(C\}\) is called an equational class. Each e-variety is an equational class but the converse is not true. We say that the e-variety \({\mathcal V}\) is weakly determined by a set of identities C if \({\mathcal E}(C)={\mathcal V}\). The author gives sets of identities strongly [weakly] determining many of the known e-varieties of regular semigroups.

Reviewer: A.Tishchenko

MSC:

20M07	Varieties and pseudovarieties of semigroups
08B05	Equational logic, Mal’tsev conditions
20M05	Free semigroups, generators and relations, word problems

Keywords:

existence variety; Birkhoff-type theorem; regular unary semigroups; variety of regular unary semigroups; sets of identities; varieties of regular semigroups

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