Almost equational classes of algebras (Q1090355)

A class \({\mathbb{C}}\) of algebras of type \(\tau '<\tau\) is said to be almost equational with respect to an equational class \({\mathbb{K}}\) of type \(\tau\) iff \(Hom_{{\mathbb{C}}}({\mathfrak A},{\mathfrak B})=Hom_{{\mathbb{K}}}({\mathfrak A},{\mathfrak B})\) for all \({\mathfrak A},{\mathfrak B}\in {\mathbb{K}}\). A good example is the class \({\mathbb{C}}\) of all groups (G;.) and the variety \({\mathbb{K}}\) of all groups \((G;.,^{-1},1)\). This notion is intensively investigated in this paper. In particular, the following characterization is given: A class \({\mathbb{C}}\) is almost equational iff it is closed under taking of isomorphisms, direct products, quotient algebras over ''improper'' congruences and all \({\mathbb{C}}\)-free algebras on arbitrary sets do exist. Here an ''improper'' congruence of \({\mathfrak A}\in {\mathbb{C}}\) is an equivalence relation on the carrier A of \({\mathfrak A}\), such that it admits any function \(A^ n\to A\), which commutes with all homomorphisms \(h\in Hom_{{\mathbb{C}}}({\mathfrak A},{\mathfrak B})\) for all \({\mathfrak B}\in {\mathbb{C}}\).