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, (∀y∈E',
- ∀y,y'∈E', (∀x∈E,
Morphisms between models of a duality theory, will be conceived as
follows: let us first introduce a notion of "morphism", then of
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 : E→F and g: F'→E',
such that ∀x∈E, ∀y∈F',
= 〈x,g(y)〉. We shall write
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
= 〈x,g'(y)〉) ⇒ (∀y∈F', 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 ∀y∈F', g(y) = y०f
Then we have ∀f∈FE,
(f,g)∈Mor((E,E'), (F, F')))
⇔ (∀y∈F', y०f ∈E').
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 E ⊂ KE'
and F ⊂ KF', 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
Indeed, we have then t(g०f)= tf
- IdE ∈ Mor(E,E)
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:
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
- Any E' ⊂ KE defines an
interpretation of each n-ary r∈R in E
This is the only one making (∏y∈E'
y) an embedding
from E to KE'.
- Any interpretation of R in E gives a set E*=
MorR(E,K) ⊂ KE.
By the properties of closures of the above Galois connection we
- E' ⊂ E*
- As ∏y∈E' y
is an embedding of E into KE'
(injective as E is assumed to be separated by E'),
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')) ⊂
- Mor((E,E*),(F,F')) = MorR(E,F)
y) is an R-embedding from F to KF',
Now, let S ⊂ Pol R ⊂ OpK. As seen
above, in any K-duality system (E,E') we have
E* ∈ SubS(KE).
As E'⊂E*, the condition f∈Mor((E,E'),(F,F')),
that is ∀y∈F', y०f ∈ E', is
more restrictive, thus the inclusion.
f ∈ MorR(E,F)
y)०f ∈ MorR(E,KF')
|⇔ ∀y∈F', y०f ∈
|⇔ f ∈ Mor((E,E*),(F,F'))
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' ⊂ X ⊂ E*.
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.
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 : E→F
, g: F'→E' and φ:K1↔ K2,
such that ∀x∈E, ∀y∈F',
(More developments will be added later)
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