Products of Many Sorted Algebras

For simplicity we follow the rules: I, J denote sets, A, B denote many sorted sets of I, i, j, x are arbitrary, and S denotes a non empty many sorted signature. A set has common domain if: (Def.1) For all functions f , g such that f ∈ it and g ∈ it holds dom f = dom g. Let us mention that there exists a set which is functional and non empty and has common domain. The following proposition is true (1) {∅} is a functional set with common domain. Let X be a functional set with common domain. The functor DOM(X) yielding a set is defined as follows: (Def.2) (i) For every function x such that x ∈ X holds DOM(X) = domx if X 6= ∅, (ii) DOM(X) = ∅, otherwise. We now state the proposition (2) For every functional set X with common domain such that X = {∅} holds DOM(X) = ∅.