## Duality theories

A duality theory is a theory with 3 types E, E' and K (or just both types E, E' when taking for K the pair of booleans); and at least the following structures and axioms:

• The duality operation 〈 , 〉 from E × E' to K ;
• Both separation axioms allowing to interpret E as a second-order structure over E', and vice versa:
• x,x'∈E, (∀yE', 〈x,y〉=〈x',y〉) ⇒ x=x'
• y,y'∈E', (∀xE, 〈x,y〉=〈x,y'〉) ⇒ y=y'

Morphisms between models of a duality theory, will be conceived as follows: let us first introduce a notion of "morphism", then of "semi-morphism".

#### Morphisms

First, take a fixed set K (with fixed internal structures in K, that will not be used here).
Models of a duality theory with the same set K, will be called K-duality systems (E,E',〈 , 〉), omitting the mention of K that is kept implicit.
Then, we shall call morphism from a K-duality system (E,E',〈, 〉) to another (F, F',〈 , 〉'), a pair of functions (f,g) where f : EF and g: F'E', such that ∀xE, ∀yF', 〈f(x),y〉' = 〈x,g(y)〉. We shall write (f,g)∈Mor((E,E'),(F,F')).

The separation axioms ensure that each of f and g determines the other: the fact E separates E' (i.e. E' is a second-order structure over E) ensures the uniqueness of g, as
fFE,∀g, g'E'F',(∀xE, ∀yF',〈x,g(y)〉 = 〈f(x),y〉' = 〈x,g'(y)〉) ⇒ (∀yF', g(y) = g'(y)) ⇒ g=g'
Based on this uniqueness, we shall say that g is the transpose of f, and written g = tf.
It can be more explicitly understood as follows:
E separates E', means that E' can be seen as a subset of KE (as the map from E' to KE defined by currying 〈 , 〉 is injective). In this view, g is simply defined by ∀yF', g(y) = yf .
Then we have ∀fFE, (∃gE'F', (f,g)∈Mor((E,E'), (F, F'))) ⇔ (∀yF'yfE').

This condition on f, will be abbreviated by saying that f is a morphism, with the simple notation f∈Mor(E,F), keeping implicit the data of E' and F', provided that they are clearly fixed by the context — especially if they are seen by the reverse curried view, that is EKE' and FKF', for which it means that it is the transpose of some function from F' to E'.

This notion of morphism, shares with the previous one the following properties:
• IdE ∈ Mor(E,E)
• (f∈Mor(E,F)∧g∈Mor(F,G)) ⇒ gf∈Mor(E,G)
Indeed, we have then t(gf)= tftg.

Defining endomorphisms of a duality system (E,E'), as the particular case of morphisms from such a model to itself, we can see that the monoid of these endomorphisms acts differently on E and E' : choosing the convention that it is a left action on E, it is then a right action on E'.

#### Duality theories with structures

Let R be a relational language, that is interpreted in K. We may even see it as R⊂RelK.

For any set E, we have a Galois connection between interpretations of R in E and subsets E'KE, defined as follows:
• Any E'KE defines an interpretation of each n-ary rR in E as rE={(x1,..., xn)∈En|∀yE' , r(y(x1),...,y(xn))}. This is the only one making (∏yE' y) an embedding from E to KE'.
• Any interpretation of R in E gives a set E*= MorR(E,K) ⊂ KE.
Now let us consider duality theories, or kinds of K-duality systems (E,E',〈 , 〉), that interpret R in E as defined from E' in this way. (This forms a first-order theory, while the claim of using of E* would form a second order theory).

By the properties of closures of the above Galois connection we have:
• E' ⊂ E*
• As ∏yE' y is an embedding of E into KE' (injective as E is assumed to be separated by E'),
yE* y is also an embedding of E into KE* (which means that the used R-structure on E is closed), i.e. (E,E*) is another K-duality system giving the same R-structure on E.

Thus, as the system E is a subset of an exponentiation of K (namely KE'), every axiom of the form (∀ variables) (F1 ∧ ... ∧ Fn) ⇒ G (where (F1 ,..., Fn, G) are relation symbols applied to variables) that is true in K, is also true in E.

Theorem. For any two K-duality systems (E,E',〈 , 〉) and (F,F',〈 , 〉') interpreting R in E and F in this way,
• Mor((E,E'),(F,F')) ⊂ MorR(E,F)
• Mor((E,E*),(F,F')) = MorR(E,F)
Proof.
As (∏yF' y) is an R-embedding from F to KF', we have
 ∀f∈FE, f ∈ MorR(E,F) ⇔ (∏y∈F' y)०f ∈ MorR(E,KF') ⇔ ∀y∈F', y०f ∈ MorR(E,K)=E* ⇔ f ∈ Mor((E,E*),(F,F'))
As E'E*, the condition f∈Mor((E,E'),(F,F')), that is ∀yF', yfE', is more restrictive, thus the inclusion.
Now, let S ⊂ Pol R ⊂ OpK. As seen above, in any K-duality system (E,E') we have E* ∈ SubS(KE).
As the R-structure in E stays unchanged whether it is defined from E' or from E*, it remains also unchanged using any X such that E'XE*. This is the case for X = E'S = the S-algebra generated by E', that is the smallest sub-S-algebra of E* (or equivalently of KE) such that E'X.

#### Semi-morphisms

The notion of semi-morphism, generalizes the above by allowing the type K to be changed but connected by a bijection φ (that will have to be an isomorphism for internal structures given in K by the considered theory). Even if K is the same as a set (and even as a system), this still generalizes the above, as φ may be an automorphism other than IdK.
So, a semi-morphism from a model (E,E',K1) to a model (F, F', K2) will consist in a triple (f,g,φ) where f : EF , g: F'E' and φ:K1K2, such that ∀xE, ∀yF', 〈f(x),y〉' = φ〈x,g(y)〉.

(More developments will be added later)

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