Interior algebras and varieties
The authors consider rings and semigroups with an additional unary operation which mimics the operation of taking the interior of a subset in topological spaces. More precisely, they call a ring or semigroup \(R\) an interior algebra if \(R\) is equipped with a unary operation \(a\mapsto I(a)\) such that \(I(a)I(b)=I(b)I(a)\), \(aI(a)=I(a)\), \(I(I(a))=I(a)\), and \(I(a)I(b)=I(ab)I(b)\) for all \(a,b\in R\). The axioms easily imply that the set \(\text{Im}(I)=\{I(a)\mid a\in R\}\) is a multiplicative subsemilattice of \(R\). A fully interior algebra \(R\) is a ring or semigroup in which \(\text{Im}(I)=E(R)\), the set of all idempotents of \(R\). A ring or semigroup \(R\) is a fully interior algebra if and only if \(E(R)\) is a subsemilattice of \(R\) and, for any \(r\in R\), there exists a largest (with respect to the natural order on \(E(R)\)) element \(e\in E(R)\) with \(re=e\). The authors describe ring and semigroup varieties \(V\) such that each \(R\in V\) verifies the latter property. For rings, this is equivalent for \(V\) to satisfy the identities \(x^n=x^{2n}\), \(x^ny^n=y^nx^n\), \(x^ny=x^ny^{n+1}\) for some positive integer \(n\) (Theorem 2), and for semigroups, the corresponding identities are \(x^n=x^{n+1}\), \(x^ny^n=y^nx^n\) (Theorem 3).
scientific article; zbMATH DE number 1384437
scientific article