# Bases for Structures and Theories II

Logica Universalis 14 (4):461-479 (2020)

# Abstract

In Part I of this paper, I assumed we begin with a 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 \leftrightarrow \phi _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} \leftrightarrow \theta _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} \leftrightarrow \theta _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$$T 1 is definitionally equivalent to$$T_2$$T 2, with respect to$$\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.

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