In this paper S is an order sorted signature. Let S be an order sorted signature and let U0 be an order sorted algebra of S. A subset of U0 is called an order sorted generator set of U0 if: (Def. 1) For every OSSubset O of U0 such that O = OSCl it holds the sorts of OSGenO = the sorts of U0. The following proposition is true (1) Let S be an order sorted signature, U0 be a strict non-empty order...

In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers [1] [2] and [3]. We first define the free clone T pL, Cq of terms of a first order language L over a set C of parameters in a standard way. The free right algebra FpL, Cq of formulas over the clone T pL, Cq of terms is then generated by atomic formulas...

We extend Lawvere-Pitts prop-categories (aka. hyperdoctrines) to develop a general framework for providing “algebraic” semantics nonclassical first-order logics. This includes natural notion of substitution, which allows logics be considered as structural closure operators just propositional are in abstract algebraic logic. then establish an extension the homomorphism theorem from universal alg...