Pseudocompact Malʹtsev spaces

A theorem due to Comfort and Ross asserts that the product of any family of pseudocompact topological groups is pseudocompact. We generalize this theorem to the case of Malʹtsev spaces. A Malʹtsev operation on a space X is a continuous function ƒ:X3 → X satisfying the identity ƒ(x, y, y) = ƒ(y, y, x) = x for all x, y ϵ X. A topological space is Malʹtsev if it admits a Malʹtsev operation. We prove that every Malʹtsev operation on a pseudocompact space X can be extended to a Malʹtsev operation on βX. It follows that: 1. X is a pseudocompact Malʹtsev space, then βX is Dugundji; e product of any family of pseudocompact Malʹtsev spaces is pseudocompact.