Pre-solid varieties of semigroups. (English) Zbl 0842.20049

An identity \(t \approx t'\) of terms of type \(\tau\) is called a hyperidentity [pre-hyperidentity] of a variety \(V\) of this type if substituting terms of the appropriate arity [which are different from variables for operation symbols] the resulting identities are valid in \(V\). A variety \(V\) of algebras of type \(\tau\) is called solid [pre-solid] if every of its identities is a hyperidentity [pre-hyperidentity]. All pre-solid varieties of a given type \(\tau\) form a complete sublattice of the lattice of all varieties of type \(\tau\) which contains the lattice of all solid varieties of this type. Let \(V_{HS}\) be the greatest solid semigroup variety. It is proved that \(V_{HS}\) is also the greatest pre-solid semigroup variety. Let \(S(V_{HS})\) \([Sp(V_{HS})]\) be the set of all solid [pre-solid] varieties of semigroups and \(V_{PS} = \text{var}\{(xy)z \approx x(yz),\;xyxzxyx \approx xyzyx\), \(x^2 \approx y^2\), \(x^3 \approx y^3\}\), \(Z = \text{var}\{(xy)z \approx x(yz)\), \(xy \approx zt\}\). The set \(Sp(V_{HS}) \setminus S(V_{HS})\) forms a sublattice of \(Sp(V_{HS})\) and \(V_{PS}\) is the greatest pre-solid semigroup variety which is not solid. A semigroup variety \(V\) is solid iff \(V \vee Z\) is solid and either \(V \supseteq Z\) or \(V\) coincides with one of the three varieties: \(RB\) (all rectangular bands), \(\text{Reg }B\) (all regular bands), \(NB\) (all normal bands). There is given a description of all pre-solid medial semigroup varieties which are not solid.


20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
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