Bases for Structures and Theories II

Abstract

AbstractIn Part I of this paper (Ketland in Logica Universalis 14:357–381, 2020), I assumed we begin with a (relational) signature $$P = \{P_i\}$$ P = { P i } and the corresponding language $$L_P$$ L P , and introduced the following notions: a definition system$$d_{\Phi }$$ d Φ for a set of new predicate symbols $$Q_i$$ Q i , given by a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of defining $$L_P$$ L P -formulas (these definitions have the form: $$\forall \overline{x}(Q_i(x) \leftrightarrow \phi _i)$$ ∀ x ¯ ( Q i ( x ) ↔ ϕ i ) ); a corresponding translation function$$\tau _{\Phi }: L_Q \rightarrow L_P$$ τ Φ : L Q → L P ; the corresponding definitional image operator$$D_{\Phi }$$ D Φ , applicable to $$L_P$$ L P -structures and $$L_P$$ L P -theories; and the notion of definitional equivalence itself: for structures $$A + d_{\Phi } \equiv B + d_{\Theta }$$ A + d Φ ≡ B + d Θ ; for theories, $$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$ T 1 + d Φ ≡ T 2 + d Θ . Some results relating these notions were given, ending with two characterizations for definitional equivalence. In this second part, we explain the notion of a representation basis. Suppose a set $$\Phi = \{\phi _i\}$$ Φ = { ϕ i } of $$L_P$$ L P -formulas is given, and $$\Theta = \{\theta _i\}$$ Θ = { θ i } is a set of $$L_Q$$ L Q -formulas. Then the original set $$\Phi $$ Φ is called a representation basis for an $$L_P$$ L P -structure A with inverse $$\Theta $$ Θ iff an inverse explicit definition $$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$ ∀ x ¯ ( P i ( x ¯ ) ↔ θ i ) is true in $$A + d_{\Phi }$$ A + d Φ , for each $$P_i$$ P i . Similarly, the set $$\Phi $$ Φ is called a representation basis for a $$L_P$$ L P -theory T with inverse $$\Theta $$ Θ iff each explicit definition $$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$ ∀ x ¯ ( P i ( x ¯ ) ↔ θ i ) is provable in $$T + d_{\Phi }$$ T + d Φ . Some results about representation bases, the mappings they induce and their relationship with the notion of definitional equivalence are given. In particular, we show that $$T_1$$ T 1 (in $$L_P$$ L P ) is definitionally equivalent to $$T_2$$ T 2 (in $$L_Q$$ L Q ), with respect to $$\Phi $$ Φ and $$\Theta $$ Θ , if and only if $$\Phi $$ Φ is a representation basis for $$T_1$$ T 1 with inverse $$\Theta $$ Θ and $$T_2 \equiv D_{\Phi }T_1$$ T 2 ≡ D Φ T 1 .