Terms over Many Sorted Universal Algebra 1

Pure terms (without constants) over a signature of many sorted universal algebra and terms with constants from algebra are introduced. Facts on evaluation of a term in some valuation are proved.] provide the notation and terminology for this paper. 1. Terms over a Signature and over an Algebra Let I be a non empty set, let X be a non-empty many sorted set indexed by I, and let i be an element of I. Note that X(i) is non empty. In the sequel S is a non void non empty many sorted signature and V is a non-empty many sorted set indexed by the carrier of S. Let us consider S, V. The functor S-Terms(V) yields a subset of FinTrees(the carrier of DTConMSA(V)) and is deened by: (Def. 1) S-Terms(V) = TS(DTConMSA(V)): Let us consider S, V. Observe that S-Terms(V) is non empty. Let us consider S, V. A term of S over V is an element of S-Terms(V). In the sequel A denotes an algebra over S and t denotes a term of S over V. Let us consider S, V and let o be an operation symbol of S. Then Sym(o; V) is a nonterminal of DTConMSA(V). Let us consider S, V and let s 1 be a nonterminal of DTConMSA(V). A nite sequence of elements of S-Terms(V) is said to be an argument sequence of s 1 if: (Def. 2) It is a subtree sequence joinable by s 1. The following proposition is true (1) Let o be an operation symbol of S and a be a nite sequence. Then h ho; the carrier of Si i-tree(a) 2 S-Terms(V) and a is decorated tree yielding if and only if a is an argument sequence of Sym(o; V).