THE CANONICAL STRUCTURE AS A MINIMUM STRUCTURE

Abstract

In this paper a structure of a system is defined as a mathematical structure (ℒ; Σ ), where ℒ is a first-order logic language and Σ is a set of sentences of the given first-order logic. It is shown that a canonical structure determined by Σ which is similar to those used in proving the Gödei's completeness theorem, satisfies a universality in the sense of category theory when homomorphisms are used as morphisms, and a freeness in the sense of universal algebra when Σ-morphisms, which preserve Σ are used. The universality and the freeness give the minimality of the canonical structure. As an example, a structure of a stationary system is defined as a pair (ℒe-sta, Σe-sta)-Its canonical structure is actually constructed. In a sense this canonical structure accords with models constructed by Nerode realization.