Affine complete varieties
For a congruence relation \(\theta\) on an algebra \({\mathcal A}=(A;F)\), a k- ary function f on A is said to be compatible with \(\theta\) if \[ (\forall (x\quad 1,...,x\quad k),(y\quad 1,...,y\quad k)\in A\quad k)(((\forall i)(x\quad i\theta y\quad i))\to f(x\quad 1,...\quad,x\quad k)\theta f(y\quad 1,...,y\quad k)). \] f is compatible if it is compatible with all \(\theta\in con(A)\), \({\mathcal A}\) is affine complete if each compatible function is a polynomial function of \({\mathcal A}\) and a variety is affine complete if all its members are. This paper investigates the properties of affine complete varieties; some of the results obtained are: (1) the subdirectly irreducible algebras in an affine complete variety are finite; (2) the subdirectly irreducible algebras in a congruence distributive affine complete variety have no proper subalgebras; (3) an arithmetical variety of finite type is affine complete if and only it is generated by a single finite algebra having no proper subalgebras. The paper mentions a number of open problems, and ends with some results on variants of the notion of affine completeness.
scientific article; zbMATH DE number 4043915