Expanding varieties by monoids of endomorphisms
Let \({\mathcal V}(M)\) denote the variety obtained from a variety \({\mathcal V}\) and a monoid M by adding a unary operation to the language of \({\mathcal V}\) for each element of M and by adding axioms to guarantee that the new functions form a submonoid (isomorphic to M) of the endomorphism monoid of each algebra of \({\mathcal V}\). This paper examines various universal algebraic properties for the expanded variety \({\mathcal V}(M)\). The main results include: (1) \({\mathcal V}\) is equivalent to a subvariety of \({\mathcal V}(M)\); (2) for \({\mathcal V}\) nontrivial, \({\mathcal V}(M)\) is locally finite [resp. a discriminator variety] if and only if \({\mathcal V}\) is locally finite [resp. a discriminator variety] and M is finite [resp. a finite group]; (3) for \({\mathcal V}\) nontrivial, if \({\mathcal V}(M)\) is finitely generated, then \({\mathcal V}\) is finitely generated and M is finite; (4) \({\mathcal V}(M)\) is abelian if and only if \({\mathcal V}\) is abelian. A structure theorem is also obtained when \({\mathcal V}\) is a congruence modular abelian variety. Associated with such a variety are a ring R(\({\mathcal V})\) and a variety of left R(\({\mathcal V})\)-modules. After describing a generating set for R(\({\mathcal V}(M))\), the authors show that R(\({\mathcal V}(M))\) is isomorphic to the ring of all R(\({\mathcal V})\)-valued functions on M which are 0 on all except finitely many arguments.
scientific article; zbMATH DE number 3966165