Dualities in lattices of semigroup varieties
The subvariety lattice of a variety \({\mathfrak V}\) of universal algebras is denoted by L(\({\mathfrak V})\). Varieties \({\mathfrak V}\) and \({\mathfrak V}'\) are said to be dual to one another if L(\({\mathfrak V})\) and L(\({\mathfrak V}')\) are dual lattices; \({\mathfrak V}\) is selfdual if L(\({\mathfrak V})\) is selfdual. Let \({\mathfrak X}\) be a class of semigroup varieties, then a semigroup variety \({\mathfrak V}\) is said to admit duality in \({\mathfrak X}\) if there is \({\mathfrak V}'\in {\mathfrak X}\) which is dual to \({\mathfrak V}\); \({\mathfrak C}\) admits duality if \({\mathfrak X}\) is the class of all varieties of semigroups. If all subvarieties of \({\mathfrak V}\) admit duality in \({\mathfrak X}\) (admit duality) then \({\mathfrak V}\) is said to hereditarily admit duality in \({\mathfrak X}\) (hereditarily admit duality). The abbreviations h.a.d. in \({\mathfrak X}\) and h.a.d. will be used for these properties. Similarly \({\mathfrak V}\) is hereditarily selfdual if all its subvarieties are selfdual. This paper investigates h.a.d. and hereditarily selfdual varieties. Two results on hereditarily selfdual varieties of universal algebras are given, together with three equivalents to semiduality in a variety of semigroups. In the case of h.a.d. varieties, attention is restricted to semigroup varieties. Results are given for h.a.d. varieties in general and also for varities h.a.d. in \({\mathcal K}\), the class of all varieties of semigroups in which every periodic group is locally finite. In the latter case we have the interesting result that, if one of two additional conditions is satisfied, a variety which is h.a.d. in \({\mathcal K}\) is also hereditarily selfdual.
scientific article; zbMATH DE number 4125706
scientific article