Free-algebraic characterizations of primal and independent algebras.

Abstract

In the study of many-valued logics, one is led to consider a (finitary) algebra (A, 0j, * , ?On), or simply A, with a finite number of primitive operations that generate by composition all functions in AAm for each m < X . Such algebras are called primal. For example, the algebra ({0, 1 }, A, --) of truth-values in 2-valued logic and, more generally, ({0, * * *, n-1}, min{x, y}, x+1 (mod n)) in nvalued Post logics are primal algebras. The truth-values of the Lukasiewicz-Tarski logics ({ 0, * * *, n-1 }, C, N), where Cxy =max{0, y-x}, Nx=n-1-x, do not form a primal algebra, but if the 0-ary (constant) operation 1 is admitted, then ({ 0, * * *, n-i }, C, N, 1) becomes a primal algebra. Note that any primal algebra is finite, for if it were infinite, then the set of functions would be uncountable, while the set of generated operations would be countable at most. If 01, , On are the operations symbols in the language for the operations 01, * * *, O? of a fixed species (or similarity type), then by the absolutely-free algebra (of the given species), (4k, 01, , * n) with k generators x1, * * *, Xk, is meant the set of all formal expressions defined inductively as follows: 1. X1, * * * Dk; 20. for each i = 1, , n, if 41, ** , kkE C4k, then also Oi(?01, * * * 4 0k;)Ek; with the operations defined by setting