Part I: Signature Reduct of an Algebra

One can prove the following propositions: (1) Let I be a set, f be a function, and F , G be many sorted functions indexed by I. If rng f ⊆ I, then (G ◦ F ) · f = (G · f) ◦ (F · f). (2) Let S be a non empty non void many sorted signature, o be an operation symbol of S, V be a non-empty many sorted set indexed by the carrier of S, and x be a set. Then x is an argument sequence of Sym(o, V ) if and only if x is an element of Args(o,Free(V )). Let S be a non empty non void many sorted signature, let V be a non-empty many sorted set indexed by the carrier of S, and let o be an operation symbol of S. Note that every element of Args(o,Free(V )) is decorated tree yielding. Next we state two propositions: (3) Let S be a non empty non void many sorted signature and A1, A2 be algebras over S. Suppose the sorts of A1 are transformable to the sorts of A2. Let o be an operation symbol of S. If Args(o,A1) 6= ∅, then Args(o,A2) 6= ∅.