Solid varieties of semigroups. (English) Zbl 0797.20045

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.


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


Zbl 0759.08005
Full Text: DOI EuDML


[1] [DK93] Denecke, K. and J. Koppitz,Hyperassociative semigroups, preprint, 1993.
[2] [DL91] Denecke, K., D. Lau, R. Pöschel and D. Schweigert,Hyperidentities, hyperequational classes and clone congruences, Contributions to General Algebra7(1991), 97–118. · Zbl 0759.08005
[3] [GS90] Graczynka, E. and D. Schweigert,Hyperidentities of a given type, Algebra Universalis27 (1990), 305–318. · Zbl 0715.08002
[4] [Kop93] Koppitz, J.,On equational descriptions of solid semigroup varieties, preprint, 1993.
[5] [Per69] Perkins, P.,Bases for equational theories of semigroups, J. Algebra11 (1969), 298–314. · Zbl 0186.03401
[6] [Sch92] Schweigert, D.,Hyperidentities, Universitat Kaiserslautern Preprint No. 220, 1992. · Zbl 0795.08003
[7] [Tay81] Taylor, W.,Hyperidentities and hypervarieties, Aequationes Mathematicae23(1981), 30–49. · Zbl 0491.08009
[8] [Wis90] Wismath, S.,Hyperidentities for some varieties of semigroups, Algebra Universalis27(1990), 111–127. · Zbl 0692.08008
[9] [Wis91] Wismath, S.,Hyperidentities for some varieties of commutative semigroups, Algebra Universalis28(1991), 245–273. · Zbl 0741.20043
[10] [Wis93] Wismath, S.,Hyperidentily Bases for rectangular bands and other semigroup varieties, to appear in Journal of the Australian Math. Society (A).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.