The Correspondence Between Homomorphisms of Universal Algebra & Many Sorted Algebra

The articles [22], [25], [26], [28], [8], [9], [11], [21], [23], [3], [12], [10], [1], [19], [6], [27], [18], [15], [2], [5], [4], [16], [7], [24], [13], [14], [17], and [20] provide the notation and terminology for this paper. For simplicity we follow the rules: U1, U2, U3 denote universal algebras, n denotes a natural number, A denotes a non empty set, and h denotes a function from U1 into U2. The following propositions are true: (1) For all functions f , g and for every set C such that rng f ⊆ C holds (g C) · f = g · f. (2) For every set I and for every subset C of I holds C ∗ ⊆ I. (3) For every function f and for every set C such that f is function yielding holds f C is function yielding. (4) For every set I and for every subset C of I and for every many sorted set M indexed by I holds (M C)# = M# C. Let us consider A, n and let a be an element of A. Then n 7→ a is a finite sequence of elements of A. Let S, S be non empty many sorted signatures. The predicate S ≤ S ′ is defined by the conditions (Def.1).