Solid varieties of semigroups. (English) Zbl 0797.20045

Denecke, K.; Wismath, S. L.

Semigroup Forum 48, No. 2, 219-234 (1994).

Let \(\tau = (n_ i)_{i \in I}\) be a type and \((f_ i)_{i \in I}\) operation symbols such that the arity of \(f_ i\) is \(n_ i\). Let \(W_ \tau(X)\) be the set of all terms of type \(\tau\) in an alphabet \(X\). A map \(\sigma: (f_ i)_{i \in I} \to W_{\tau}(X)\) is a hypersubstitution. If \(t \approx t'\) is an equation, then \(\Xi[t \approx t']\) denotes the set of all equations obtained from \(t \approx t'\) by hypersubstitution. The equation \(t \approx t'\) is a type \(\tau\) hyperidentity of an algebra \(A\) if \(A\) satisfies all the equations in \(\Xi[t \approx t']\). Similarly, an equation is a hyperidentity of a variety \(V\) if it is a hyperidentity of every algebra in \(V\). A variety \(V\) of type \(\tau\) is solid if every identity of \(V\) is also a type \(\tau\) hyperidentity in \(V\).
It was shown by K. Denecke, D. Lau, R. Pöschel and D. Schwiegert [Contrib. Gen. Algebra 7, 97-118 (1991; Zbl 0759.08005)] that a variety is solid if and only if it is a hypervariety. This paper gives other results of this nature for algebras in general, and then turns to a study of solid varieties of semigroups. Many new examples of such varieties are given, and a picture of the bottom of the lattice of such varieties is produced.

Reviewer: S.Oates-Williams (St.Lucia)

MSC:

20M07	Varieties and pseudovarieties of semigroups
08B15	Lattices of varieties
08B05	Equational logic, Mal’tsev conditions

Keywords:

lattice of varieties; type; operation symbols; hypersubstitution; hyperidentity; hypervariety; solid varieties of semigroups

Citations:

Zbl 0759.08005

References:

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