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Algebraic Operation on Subsets of Many Sorted Sets

Let S be a non empty 1-sorted structure. One can verify that the 1-sorted structure of S is non empty. We now state three propositions: (1) For every non empty set I and for all many sorted sets M , N indexed by I holds M+·N = N. (2) Let I be a set, M , N be many sorted sets indexed by I, and F be a family of many sorted subsets indexed by M . If N ∈ F, then ⋂ |:F :| ⊆ N. (3) Let S be a non void non empty many sorted signature, M1 be a strict non-empty algebra over S, and F be a family of many sorted subsets indexed by the sorts of M1. Suppose F ⊆ SubSorts(M1). Let B be a subset of M1. If B = ⋂ |:F :|, then B is operations closed.