DENUMERABLE PRODUCT SPACES OF PSEUDOQUOTIENTS I

A space of pseudoquotients ß(X,G) is defined as the set of equivalence classes of pairs (x, g), where x ∈ X, an arbitrary non-empty set, and g ∈ G, a commutative semigroup acting on X such that (x, g)~(y, h) if hx = gy. In this paper, we shall construct the pseudoquotient space ß(ΠXi,ΠGi) where X is replaced by a cartesian product of countably infinite non-empty sets Xi and G by a direct product denumerable commutative semigroups Gi, i ∈ I an indexing set, such that ΠGi acts injectively on ΠXi.