Bases for Structures and Theories I

In Part I of this paper (Ketland in Logica Universalis 14:357–381, 2020), I assumed we begin with a (relational) signature $$P = \{P_i\}$$ and the corresponding language $$L_P$$ , and introduced the following notions: a definition system $$d_{\Phi }$$ for a set of new predicate symbols $$Q_i$$ , given by a set $$\Phi = \{\phi _i\}$$ of defining $$L_P$$ -formulas (these definitions have the form: $$\forall \overline{x}(Q_i(x) \leftrightarrow \phi _i)$$ ); a corresponding translation function $$\tau _{\Phi }: L_Q \rightarrow L_P$$ ; the corresponding definitional image operator $$D_{\Phi }$$ , applicable to $$L_P$$ -structures and $$L_P$$ -theories; and the notion of definitional equivalence itself: for structures $$A + d_{\Phi } \equiv B + d_{\Theta }$$ ; for theories, $$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$ . 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 $$\Phi = \{\phi _i\}$$ of $$L_P$$ -formulas is given, and $$\Theta = \{\theta _i\}$$ is a set of $$L_Q$$ -formulas. Then the original set $$\Phi $$ is called a representation basis for an $$L_P$$ -structure A with inverse $$\Theta $$ iff an inverse explicit definition $$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$ is true in $$A + d_{\Phi }$$ , for each $$P_i$$ . Similarly, the set $$\Phi $$ is called a representation basis for a $$L_P$$ -theory T with inverse $$\Theta $$ iff each explicit definition $$\forall \overline{x}(P_i(\overline{x}) \leftrightarrow \theta _i)$$ is provable in $$T + d_{\Phi }$$ . 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 $$T_1$$ (in $$L_P$$ ) is definitionally equivalent to $$T_2$$ (in $$L_Q$$ ), with respect to $$\Phi $$ and $$\Theta $$ , if and only if $$\Phi $$ is a representation basis for $$T_1$$ with inverse $$\Theta $$ and $$T_2 \equiv D_{\Phi }T_1$$ .