Lattice of Congruences in Many Sorted Algebra

For simplicity we adopt the following convention: I, X denote sets, M denotes a many sorted set indexed by I, R1 denotes a binary relation on X, and E1, E2, E3 denote equivalence relations of X. We now state the proposition (1) (E1 ⊔ E2) ⊔ E3 = E1 ⊔ (E2 ⊔ E3). Let X be a set and let R be a binary relation on X. The functor EqCl(R) yielding an equivalence relation of X is defined as follows: (Def. 1) R ⊆ EqCl(R) and for every equivalence relation E2 of X such that R ⊆ E2 holds EqCl(R) ⊆ E2. One can prove the following propositions: (2) E1 ⊔E2 = EqCl(E1 ∪ E2). (3) EqCl(E1) = E1. (4) ∇X ∪R1 = ∇X .