Antivarieties of unars
A class of algebras is said to be an antivariety if it is definable by anti-identities (an anti-identity is a disjunction of a finite number of negated formulas). In the paper, antivarieties of unars (\(=\) monounary algebras) are dealt with. They form a lattice \({\mathbb L}({\mathcal U})\) with respect to inclusion. It is shown that the unique atom of \({\mathbb L}({\mathcal U})\) is the antivariety defined by \((\forall x)\neg(f(x)=x)\) and that \({\mathbb L}({\mathcal U})\) satisfies neither the ascending nor the descending chain condition. The main result is as follows: There is a continuum of members of \({\mathbb L}({\mathcal U})\) having an independent basis of anti-identities, and, similarly, not having such a basis. Further, a finite unar has an independent basis of anti-identities iff it is definable by the empty set of anti-identities, and iff the unar possesses a one-element cycle.
scientific article; zbMATH DE number 6116200