More on the Lattice of Many Sorted Equivalence Relations
Robert Milewski · 1996
For simplicity we adopt the following convention: I will be a non empty set, M will be a many sorted set indexed by I, x will be arbitrary, and r1, r2 will be real numbers. We now state several propositions: (1) For every set X holds x ∈ the carrier of EqRelLatt(X) iff x is an equivalence relation of X. (2) idM is an equivalence relation of M . (3) [[M,M ]] is an equivalence relation of M . (4) ⊥EqRelLatt(M) = idM . (5) ⊤EqRelLatt(M) = [[M,M ]]. Let us consider I, M . Note that EqRelLatt(M) is bounded. One can prove the following propositions: (6) Every subset of the carrier of EqRelLatt(M) is a family of many sorted subsets of [[M,M ]]. (7) Let a, b be elements of the carrier of EqRelLatt(M) and let A, B be equivalence relations of M . If a = A and b = B, then a ⊑ b iff A ⊆ B.