Full Formulas Induced by Full Terms

Abstract

The algebraic system is a well-established structure of classical universal algebra. An algebraic system is a triple consisting a nonempty set together with the sequence of operation symbols and the sequence of relation symbols. To express the primary properties of algebraic systems one needs the notion of formulas. The paper is devoted to studying of the structures related to full formulas which are extensional concepts constructed from full terms. Defining a superposition operation on the set of full formulas one obtains a many-sorted algebra which satisfies the superassociative law. In particular, we introduce a natural concept of a full hypersubstitution for algebraic systems which extends the concept of full hypersubstitutions of algebras, i.e., the mappings which send operation symbols to full terms of the same arities and relation symbols to full formulas of the corresponding arities. Together with one associative operation on the collection of full hypersubstitutions for algebraic systems, we obtain a semigroup of full hypersubstitutions for algebraic systems.