Pre-solid varieties. (English) Zbl 0841.08006

An identity \(t\approx t'\) of terms of any type \(\tau\) is called a hyperidentity for an algebra of type \(\tau\) if \(t\approx t'\) holds identically for every choice of \(n\)-ary term operations to represent \(n\)-ary operation symbols occurring in \(t\) and \(t'\). A variety of algebras of type \(\tau\) for which every identity is a hyperidentity is called a solid variety. On the one hand, although the concept of a hyperidentity is very strong, it is known that there are countably infinitely many solid varieties of semigroups. On the other hand, the commutative law fails to be a hyperidentity in any nontrivial variety with a binary operation symbol because a hyperidentity must be satisfied at least for the projections. The simplest way to weaken this notion could be to substitute only term operations different from projections. By this way, the author comes to the concepts of a pre-hyperidentity and a pre-solid variety. After developing the theory of pre-hyperidentities and pre-solid varieties, he applies the results to semigroups and proves that the variety \(\text{Mod} \{(xy)z \approx x(yz)\), \(xy\approx yx\), \(xy^2\approx x^2 y\), \(x^2\approx y^2\}\) is the greatest pre-solid variety of commutative semigroups.


08B99 Varieties
20M07 Varieties and pseudovarieties of semigroups
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