Efficient realizations of closure systems

As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity. In particular, it is shown that an algebraic closure system with arity n in which the empty set is closed and every finitely generated closed set is countable can be realized by a single \((n+1)\)-ary operation. The algebraic closure system of cosets on any group is realized by the single ternary Mal’cev term \(xy^{-1}z\). It is shown that the closure system of cosets on an Abelian group A can be realized by a single binary operation if and only if A has at most one element of order 2. Similar results are obtained for modules over an arbitrary ring.