Bases for Structures and Theories I

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature P={Pi}i∈IP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P = \{P_i\}_{i \in I_P}$$\end{document} be given. For a set Φ={ϕi}i∈IΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi = \{\phi _i\}_{i \in I_{\Phi }}$$\end{document} of LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-formulas, we introduce a corresponding set Q={Qi}i∈IΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q = \{Q_i\}_{i \in I_{\Phi }}$$\end{document} of new relation symbols and a set of explicit definitions of the Qi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_i$$\end{document} in terms of the ϕi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _i$$\end{document}. This is called a definition system, denoted dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\Phi }$$\end{document}. A definition system dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\Phi }$$\end{document} determines a translation functionτΦ:LQ→LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau _{\Phi } : L_Q \rightarrow L_P$$\end{document}. Any LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-structure A can be uniquely definitionally expanded to a model A+⊧dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{+} \models d_{\Phi }$$\end{document}, called A+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + d_{\Phi }$$\end{document}. The reduct A+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A + d_{\Phi }$$\end{document} to the Q-symbols is called the definitional imageDΦA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\Phi }A$$\end{document} of A. Likewise, a theory T in LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document} may be extended a definitional extension T+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T + d_{\Phi }$$\end{document}; the restriction of this extension T+dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T + d_{\Phi }$$\end{document} to LQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_Q$$\end{document} is called the definitional imageDΦT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{\Phi }T$$\end{document} of T. If T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document} and T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2$$\end{document} are in disjoint signatures and T1+dΦ≡T2+dΘ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1 + d_{\Phi } \equiv T_2 + d_{\Theta }$$\end{document}, we say that T1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document} and T2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2$$\end{document} are definitionally equivalent (wrt the definition systems dΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\Phi }$$\end{document} and dΘ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{\Theta }$$\end{document}). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories.


Introduction
Sometimes theories are formulated with different sets of primitives and yet are definitionally equivalent. The non-logical primitives of a formalized language L are called its signature. There are many examples of theories (often involving formalized systems of arithmetic and set theory) formulated in very different signatures, which are nonetheless "equivalent". To take a simple example, consider the theory T 1 of a reflexive relation: A similar relationship holds between the theory of formalized syntax and PA. Suppose S A is the theory of concatenation for strings from alphabet A, with |A| ≥ 2, and with the appropriate induction principle. Then S A is definitionally equivalent to PA. 2 There are other examples-from mathematics, logic and philosophy of science. 3 To return to the broader point, the transformations between such equivalent formulations are rather like "basis transformations" in linear algebra and other parts of mathematics. In this paper and the follow-up, an analogous idea is investigated. 4

Syntax, Structures and Theories
Throughout, everything we consider is 1-sorted, relational and first-order. Definition 1. Let P = {P i } i∈I be a set and let a : P → N. The pair S = (P, a) is called a one-sorted signature, and a is called the arity function for S. The multiset t = (a(P i ) | P i ∈ P ) is called the similarity type of the signature S. If a(P i ) = 0, then P i is called a sentence letter (or a propositional atom). The alphabet of S is P .

Definition 2.
(P c , a c ) is a copy of (P, a) iff the similarity types of (P, a) and (P c , a c ) are the same. (P c , a c ) is a disjoint copy of (P, a) when, in addition, P ∩ P c = ∅. Definition 3. L P is the first-order language over the signature S = (P, a), where each symbol P i is a primitive relation symbol of arity a(P i ). We will sometimes call L P "the P -language". 2 A weak theory of concatenation without the induction principle is now usually called TC (Grzegorczyk [8]), and the precise interpretability relationship of TC and Robinson arithmetic Q has recently been clarified. Although similar systems had been studied before (Quine [15]; Tarski et al. [17]), the undecidability of TC is demonstrated in Grzegorczyk [8]; in Grzegorczyk and Zdanowski [9], the essential undecidability of TC is demonstrated through an interpretation of TC into Robinson arithmetic Q. Subsequently, Visser and Sterken [18], Ganea [7] andŠvejdar [16] demonstrated the interpretability of Q into TC. 3 As another example, there is the theory CEM, of "classical extensional mereology", in a language with basic binary relation symbol (x y means "x is part of y"). There is a definitionally equivalent theory I shall call F (for "fusions") with a basic binary operation symbol ⊕ (where x ⊕ y can be read "the fusion of x and y"). The detailed formulation and proof of definitional equivalence are given in Ketland and Schindler [12]. 4 Often, we have translations/interpretations T 1 → T 2 and T 2 → T 1 , for theories which appear, on the surface, to be quite different. But it is not automatically true that T 1 and T 2 are definitionally equivalent: the translations involved must be mutual inverses of each other. (See our Theorem 2 below.) A valuable discussion of this point, and a criterion for it to hold, is Friedman and Visser [6], who show that when two theories are "bi-interpretable via identity-preserving interpretations", then they are definitionally equivalent (Sect. 5 of their paper). Moreover, they give an example of two finitely axiomatized sequential theories that are bi-interpretable but not definitionally equivalent (Sect. 7 of their paper). An example of a pair of theories which "define each others' models" but which are not definitionally equivalent is given in Andréka et al. [1].

Definition 4. Let
be a set of L P -formulas, which will be called defining formulas. Given the φ i , a corresponding set of new relation symbols Q i is introduced, such that the arity of each Q i matches the arity of its corresponding φ i . The new language L Q will sometimes be called "the Q-language". The combined language is then called L P,Q .
A theory T in L is a set of L-sentences. When we require deductive closure, we say so. We use a deductive system such that, if Δ α, then Δ ∀xα, so long as x doesn't appear free in any formulas in Δ. 5 T α means: there exists a derivation of α from the axioms/rules of T . An L-structure A will always interpret all variables. So, we can always write A |= α, even where α has free variables, since A will assign a value x A to each variable x ∈ FV(α). 6 The Completeness Theorem holds in the usual form: So far as I can tell, nothing in this paper uses either proof theoretic methods or model theoretic methods beyond what is taught at intermediate logic. 7 We do introduce specific new terminology for the following notions:

Definition 5.
A structure A for the language L P specifies a non-empty domain, dom(A); and interprets each variable x of L P as an element x A ∈ dom(A); and interprets each n-ary relation symbol P i as a n-ary relation (P i ) A ⊆ (dom(A)) n . Definition 6. Given signature P , let P c be a disjoint copy of P . Let A be an L P -structure. Then the disjoint copy A c of A in L P c is defined by setting dom(A c ) = dom(A) and, for each P i , setting (P c i ) A c = (P i ) A . Let T be an L P -theory. Then the disjoint copy T c of T in L P c is defined by replacing every occurrence of P i , in any sentence in T by the new symbol P c i . 5 The deductive system I usually have in mind is the Hilbert system set out in Machover [14] or in Enderton [5]. 6 For any variable x, the denotation of x in A is x A . We define A x a to be the structure just like A, except that, for the variable x, we have x A x a = a. 7 The terminology and definitions we use largely follow those of Machover [14], Hodges [10], or Enderton [5].

Bases for Structures and Theories I 361
We may next give inductive definitions of the denotation function t → t A (specifying what any term t in L P refers to in A) and the satisfaction relation |= between A and L P -formulas.

Definition 7.
If A 1 and A 2 are L P -structures, then an isomorphism is a bijection from dom(A 1 ) to dom(A 2 ) satisfying the preservation condition that f [(P i ) A1 ] = (P i ) A2 , for each relation symbol P i in the signature P . This is written

Definition 8. If
A is an L P -structure and T, T 1 , T 2 are sets of L P -sentences: (1) Th LP (A) := the set of L P -sentences true in A.
Th If T 1 and T 2 are deductively closed theories, then T 1 T 2 iff T 2 ⊆ T 1 . The Completeness Theorem tells us that T 1 ≡ T 2 iff T 1 T 2 and T 2 T 1 .

Definition 9.
Let A be an L P -structure and A + be an L P,Q -structure. Then A + is an expansion of A iff for all P i , (P i ) A + = (P i ) A . This is equivalent to saying that A is a reduct of A + . If A + is an L P,Q -structure, its reduct to P is denoted A + P (an L P -structure) and its reduct to Q is denoted A + Q (an L Q -structure) and we have: The central property of expansions is that the truth value of a formula in the smaller language L P remains invariant as we pass from an L P -structure to an expanded structure for L P,Q : if α ∈ L P and an L P,Q -structure A + is an expansion of an L P -structure A, then A + |= α iff A |= α.

Definition 10.
A theory T + is an extension of T iff T is a subset of T + . Let signatures P, Q, and corresponding languages L P , L Q and L P,Q be given. An extension T + in L P,Q of T in L P is called a conservative extension of T with respect to L P -formulas iff, for any L P -formula α,

Definition
Definition 11. Given the set Φ = {φ i } i∈I of L P -formulas, we introduce a disjoint set Q = {Q i } i∈I of new relation symbols, with card Q = card Φ, and with the arity of Q i matching the arity of φ i , and let n i be a(φ i ). The definition system over Φ, which we write as, d Φ is the set of explicit definitions, Given any L P -structure A, it is clear that there is a unique definitional expansion A + |= d Φ . We introduce the following notation for this expansion: We are going to treat the definitional expansion map as a unary operator +d Φ (indexed by Φ), taking us from L P -structures to L P,Q -structures. It is clear that it is well-defined (i.e., unique, given Φ). It also satisfies the following useful "right cancellation" law (this amounts, in essence, to taking a reduct): for some formula θ in the language of the subsignature P \P i . We say that θ is a defining formula for P i in T .
Vol. 14 (2020) Bases for Structures and Theories I 363 Definition 16. A relation symbol P i in the signature P is implicitly definable in T in L P just in case, given any pair of models A, B |= T , with dom(A) = dom(B) and which assign the same extension to all P j except P i , we have Beth's Theorem states that a relation symbol Analogous to what we did with structures, we are going to treat the definitional extension map for theories as a unary operator +d Φ (indexed by Φ), taking us from L Ptheories to L P,Q -theories. Again, it is well-defined (i.e., unique, given Φ) and satisfies an analogous "right cancellation" law for L P -theories: Lemma 2. The following are straightforward consequences of the definitions: (2) Before moving on to translations, we give three standard lemmas about definitional and conservative extensions (the converse of Lemma 3 is far from being true): Lemma 5. Let T in L P and T + in L P,Q be such that T ⊆ T + . Suppose that, for any model A |= T , there is an expansion A + |= T + . Then T + is a conservative extension of T for L P -formulas.

Translation
Definition 18. Let a definition system d Φ be given. Define the translation, induced by Φ is any binary connective, q is a quantifier and (φ i ) is the result of ensuring that the free variables appearing φ i are relabelled, to match those of Q i (x). We call τ Φ the translation induced by Φ. It maps from the new language L Q back to the original language L P . 10 Proof. We will prove this using a lemma below.
Lemma 6 is a general property of translations, but its converse is not true in general.
Corresponding to a translation τ Φ : L Q → L P is its "lift" τ + Φ : L P,Q → L P from the combined language L P,Q down to L P : Definition 19. Let Φ be given, along with definition system d Φ . Define the lifted translation τ + Φ induced by Φ τ + Φ : L P,Q → L P as follows. For symbols Q i , P i , variables x, y, x: Along with the requirement that τ + Φ commutes with the logical operators on the full language L P,Q . Thus, the translation τ Φ is the restriction to L Q of its lift, τ + Φ . Note that because the translations we are interested in always act as the identity on equations, it is always the case that α ↔ τ Φ (α) if α is an equation. Thus, in inductive proofs establishing biconditionals of the form α ↔ τ Φ (α), we only need to check the condition holds for atomic formulas which are not identity formulas.
Vol. 14 (2020) Bases for Structures and Theories I 365 Lemma 7. We have: Proof. For (2), we reason by induction. Let α be an atomic L P,Q -formula. As noted above, for equations (x = y), the translation The other cases are shown by induction on the construction of α.
For (1), the result follows from (2), by restricting to L Q -formulas (since For (4), reasoning by induction, let α be an atomic L P , Q-formula. If α is atomic, then the condition is trivial. Suppose α has the form P i (x). Again, he condition is trivial, For (3), the result follows from (4), by restricting to L Q -formulas.
Definition 20. Let τ Φ : L Q → L P be the translation induced by d Φ . If T 2 is a theory in L Q , then the image of T 2 under τ Φ is the set of L P -sentences: If T 1 is a theory in L P , then the pre-image of T 1 under τ Φ is the set of L Qsentences: Similarly, if Θ = {θ i } i∈IP is a set of L Q -formulas and d Θ is the corresponding definition system over Θ (for the primitives P i of L P ), we can define a translation τ Θ : L P → L Q by requiring that τ Θ commute with the logical operators and, for atomic L Pformulas: Likewise, we can also define the lifted translation τ + Θ : L P,Q → L Q .
Definition 21. Let τ : L Q → L P be a translation. Let T 1 be a theory in L P and T 2 be a theory in L Q . Then we say: (1) (2) τ faithfully interprets One may compare the condition T 1 ≡ τ [T 2 ] with Visser's definition of faithful interpretability: We write K : U faith V for: K is a faithful interpretation of U in V . This means that: for all U -sentences A, we have: (Visser [19], p. 6). Thus τ Φ : T 2 faith T 1 holds iff, for all α ∈ L T2 , we have: ]. This establishes: Definition 22. Let τ Φ : L Q → L P and τ Θ : L P → L Q be translations induced by d Φ and d Θ . Let T 1 be an L P theory. Let T 2 be an L Q theory. Then τ Θ is an right inverse of τ Φ in T 1 iff, for any α ∈ L P , We write this more suggestively as: Likewise, we write this more suggestively as: The following two lemmas are easy to prove, and yet hold to the key to much that follows. Both lemmas use "invertibility conditions", of the form: As we see later, these conditions express a very strong constraint on the set Φ of defining L P -formulas involved-the property of being a "representation basis" for A (or T ) with inverse Θ.

Lemma 9. Suppose that
Vol. 14 (2020) Bases for Structures and Theories I 367 Proof. Claims (1) and (2) are already established in Lemma 7(1,2) and do not need the side condition. (They are included for convenience of reference.) For (3), the proof is analogous to the proof of Lemma 7(1), but using the fact that A+d Φ |= d Θ . Reasoning by induction, let α be an atomic L P -sentence, say P i (x). Then its translation τ Θ (α) is ). The other cases are shown by induction on the construction of α.
For (9), the proof is analogous to the proof of (2), but is applied to the "lift" τ + Θ : L P,Q → L Q of τ Θ . Reasoning by induction, let α be an atomic L Psentence, say P i (x). Then its translation τ The other cases are shown by induction on the construction of α.
The following lemma, and the corresponding proofs, is a near repetition of the previous one, except that it deals with theories: Proof. Essentially, a repetition of the proofs for Lemma 9.
Lemma 11. Suppose T 1 is an L P -theory and T 2 is an L Q -theory. Then: Proof. For (1), we assume τ Θ is a right inverse of τ Φ in T 1 . I.e., for any α ∈ L P , We obtain (2) by relabelling everything (T 2 is now a theory in L Q ).
Lemma 11 says that if τ Θ is a right-inverse of τ Φ , relative to T , then every relation symbol P i from the original language can be explicitly defined from the θ i . In a sense, the original definition system, d Φ is a kind of inverse of d Θ .

Definitional Images
Definition 23. Let A be an L P -structure. Then the L Q -structure D Φ A is defined by: Immediately, we see that the following three conditions provide an equivalent characterization of D Φ A: Vol. 14 (2020) Bases for Structures and Theories I 369 Although A + d Φ is by construction a definitional expansion of A, it is by no means automatically true that A + d Φ is a definitional expansion of D Φ A. The requirement for this to hold is that each primitive relation (P i ) A be definable in D Φ A, by some formula, say θ i .
Turning to theories, we introduce analogous concepts: Definition 24. The definitional image of T , with respect to Φ, is the restriction of the deductive closure of T + d Φ to the new language L Q . The definitional image of T with respect to Φ is denoted D Φ T . That is, The definitional image D Φ T of a theory T in L P is, essentially, the pre- Definition 25. Let A be an L P -structure, B an L Q -structure, T 1 an L P -theory and T 2 an L Q -theory. Then we say: The second of these, (2), amounts to saying that T 2 D Φ T 1 and D Φ T 1 T 2 . The third is equivalent to saying that Note that the restriction LQ is taken after the models are extracted from the definitional extension T 1 + d Φ . If the restriction is taken first, we get the rather different set Mod((DedCl(T 1 + d Φ )) LQ ) of models: i.e., Mod(D Φ T 1 ). Indeed, this is generally a superset of

Some Book-Keeping Lemmas
We next provide several groups of "book-keeping" lemmas about translations and definitional images. The first, Lemma 14, primarily concerns theories. The second group, Lemma 15, concerns semantics and models. The third group (in Log. Univers. particular, Lemma 16) establishes five calculationally useful equivalences for the "definition invertibility condition" on structures: The fourth group establishes some analogous results for theories, including the main equivalence (Lemma 19) for the "definition invertibility condition" on theories: Lemma 14. Each of the following is true: Proof. To establish (1), note that D Φ T is simply the restriction of the deductive closure of T + d Φ to L Q -sentences. So if T + d Φ α, where α ∈ L Q , then D Φ T α, as required. Statement (2) simply expresses the relationship between images and pre-images. Statement (3) is an immediate corollary of (2). Statement (4) is an immediate corollary of (3).
For (7), suppose first that Instead suppose that for all β ∈ L Q , we have: . Then reason as follows: So, T 2 ≡ D Φ T 1 , as required.

Lemma 15. Each of the following holds:
Vol. 14 (2020) Bases for Structures and Theories I 371 Proof. (1): By induction. Suppose α = Q i (x). Thus, The case of equality, and the induction steps are routine.
(2): Let A |= T and D Φ T α, for some α ∈ L Q . We want to show D Φ A |= α. We do this using the translation τ Φ . Since D Φ T α, it follows, by Lemma 14 (4) (4): Suppose that for any β such that T 2 β, we have A + d Φ |= β. We want to show that for any β such that T 2 β, we have D Φ A |= β. So, let Neither the converse of Lemma 15(2) nor the converse of the inclusion in Lemma 15(3) is true. Lemma 16. The following are equivalent: Proof. For (1) ⇔ (2). This is simply unwinding the definitions.
Then, from (2), we have, for all P i , for all Q j , So, since φ j (A + d Φ ) = (Q j ) DΦA+dΘ , we have, for all P i , for all Q j , But we also have Thus, for all P i , for all Q j , So, from (3) above, we have: A proof analogous to that of (3) above gives us And so, we may conclude,

And right cancellation gives,
We have, applying Lemma 15(1) twice, for any α ∈ L P , β ∈ L Q : And thus, And θ i ∈ L Q , and so, from Lemma 7(1), Turning next to theories: Proof. We reason by induction on the construction of α. Let α = P i (x). Then we have: ). The equality case and compound cases proceed routinely.
(Notice this is analogous to Lemma 10(1).) Then, for any α ∈ L Q : (1) Proof. For (1), we have, for any β ∈ L Q , if T + d Φ β, then D Φ T β, since D Φ T is simply the restriction of the set of theorems of T + d Φ to L Q . By Lemma 10(4), we have For (2), from Lemma 17, we have, for any α ∈ L Q : The next lemma is the most important result needed for Theorem 2 given in Sect. 8: The following are equivalent First, for (a), let And therefore, since A was arbitrary, That is, for for any L P,Q -formula α, First, note that we may relabel Lemma 9(2) in terms of some L Q -structure B and definition system d Θ , rather than A and d Φ , to obtain: for any α ∈ L P,Q , For a contradiction, suppose we have some α ∈ L P,Q such that Since T + d Φ d Θ , we have, from Lemma 10(9), that T + d Φ α ↔ τ + Θ (α) and since T + d Φ α, we have: Vol. 14 (2020) Bases for Structures and Theories I 375 . And since τ + Θ (α) ∈ L Q , we have: . So, from (iv), we infer: Contradiction.
Proof. If we examine the definitions of D Φ T and D Θ T 2 (where T 2 is in L Q ), we get Together, these imply that D Θ D Φ T α iff T τ Φ (τ Θ (α)).
Proof. For (1), suppose T + d Φ d Θ . By Lemma 18 (2), for any α ∈ L Q , Thus, Thus, We want to show that, for any α ∈ L P , we have: D Θ D Φ T α iff T α. By Lemma 20, we have Vol. 14 (2020) Bases for Structures and Theories I 377

Definitional Equivalence
We next explain what it means for structures and theories to be definitionally equivalent. 12 Intuitively, a pair of structures are definitionally equivalent when they have a common definitional expansion. And a pair of theories are definitionally equivalent when they have a common definitional extension. Throughout the next two definitions, A is an L P -structure, Φ = {φ i } i∈I1 is a set of L P -formulas; Q is the new disjoint signature corresponding to Φ; and B is an L Q -structure. Similarly, Θ = {θ i } i∈I2 is a set of L Q -formulas. d Φ is the definition system of the Q i primitives in terms of the φ i , and d Θ is the definition system the P i primitives in terms of the θ i . Similarly, T 1 is an L P -theory and T 2 is an L Q -theory. Please note that the definitions of these notions given below assume disjoint signatures.
Definition 26. Structures A and B are definitionally equivalent wrt d Φ and d Θ iff If this is so, we write: Definition 27. Theories T 1 and T 2 are definitionally equivalent wrt d Φ and d Θ iff To express this, we write: These definitions require that the signatures of A and B (or T 1 and T 2 ) be disjoint. But is not A obviously definitionally equivalent to itself? Is not a theory definitionally equivalent to itself? Well, one can always arrange for a pair of structures A and B in overlapping signatures to be reformulated as copies A c and B c in entirely disjoint signatures (see Definitions 2, 6 above). If A c is a disjoint copy of A in L P c , then clearly A c is definitionally equivalent to A with respect to the trivial definition systems: That is, A similar copying procedure can be adopted for theories too. If we have a theory T in L P , and T c is a disjoint copy of T in L P c , then clearly T c is definitionally equivalent to T with respect to the definitions: So, one can give a more general definition of definitional equivalence by first applying this copying procedure to both structures, and then applying the definition above. 13

Main Results
We finally give two main results, which characterize definitional equivalence. Theorem 1. The following are equivalent: Thus, by Lemma 16(3) (switching labels in the second case), we have: And by right cancellation,