On the existence of congruence-uniform structures on universal algebras (Q1363445)

Let \(\mathcal A=(A,F)\) be an algebra and \(\mathcal N\) a uniform structure on \(A\). \(\mathcal N\) is called a proper congruence-uniform structure (pcus) if there exists a fundamental system of entourages consisting of congruence relations and this uniform structure is neither discrete nor indiscrete. A question which arises is whether or not for a given \(\mathcal A\) there exists a pcus. Sample result: There exists a pcus on \(\mathcal A\) iff Con\(\mathcal A\) is not atomic or Con\(\mathcal A\) is atomic but the greatest element of \(\mathcal L\) is not compact. In particular, if Con\(\mathcal A\) is distributive, then there exists a pcus iff Con\(\mathcal A\) is not atomic or it has infinitely many of atoms. The author gives a complete solution for multioperator groups and for lattices.