Definable Functions of Universal Algebras and Definable Equivalence Between Algebras

Abstract

Research into universal algebras (for different classifications included) is generally confined to working with termal (or polynomial) functions of these algebras. Attempts to go beyond this scope while staying within the frames of functions naturally definable on the algebras under consideration led the author to the idea of studying conditional termal functions (and their different generalizations such as positively, elementarily conditional termal, implicit, and abstract functions). As a continuation of studies in naturally definable functions on universal algebras, we propose to consider L-definable functions, where L is some logical language. This most general approach turns out to be connected with a scheme for defining conditional termal functions and their generalizations, as well as with various derivative structures of universal algebras. Here we present L-definable functions on universal algebras and outline their basic properties. On this basis, also, we introduce the notion of L-definably equivalent algebras, which is a generalization of the concept of being rationally equivalent for algebras.