On the Trivial Many Sorted Algebras and Many Sorted Congruences

In this paper a, I denote sets and S denotes a non empty non void many sorted signature. The scheme MSSExD deals with a non empty set A and a binary predicate P , and states that: There exists a many sorted set f indexed by A such that for every element i of A holds P [i, f (i)] provided the following condition is met: • For every element i of A there exists a set j such that P [i, j]. Let I be a set and let M be a many sorted set indexed by I. One can check that there exists an element of Bool(M) which is locally-finite. Let I be a set and let M be a non-empty many sorted set indexed by I. Note that there exists a many sorted subset indexed by M which is non-empty and locally-finite. Let S be a non empty non void many sorted signature, let A be a non-empty algebra over S, and let o be an operation symbol of S. Note that every element of Args(o,A) is finite sequence-like. Let S be a non void non empty many sorted signature, let I be a set, let s be a sort symbol of S, and let F be an algebra family of I over S. Observe that every element of (SORTS(F))(s) is function-like and relation-like. Let S be a non void non empty many sorted signature and let X be a non-empty many sorted set indexed by the carrier of S. One can verify that FreeGenerator(X) is free and non-empty. Let S be a non void non empty many sorted signature and let X be a non-empty many sorted set indexed by the carrier of S. Observe that Free(X) is free. Let S be a non empty non void many sorted signature and let A, B be non-empty algebras over S. Note that [:A, B :] is non-empty. The following propositions are true: