## 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.

### MSC:

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