Relational clones
This section is only a remark, not needed for the next ones.
Truth of formulas in products
Given a product P=∏_{i}_{∈}_{I}
E_{i} of any family of Lsystems E_{i},
and any formula Φ with any number n of free variables, let
us compare the truth of Φ(x) in P for some tuple x∈P^{n}
of values of free variables, with the value (∀i∈I, Φ(x_{i}))
obtained as the product of the structures defined by the same
formula in each E_{i}, where x=∏_{i∈I
}x_{i}, i.e.
∀i∈I, x_{i} = π_{i} ० x
∈E_{i}^{n}.
We will do it with elementary formulas, from which the behavior of
some more complex formulas can eventually be deduced.
There is equivalence with the equality symbol: ∀x,y∈P,
(x=y ⇔∀i∈I, x_{i} =
y_{i}), and also for a formula Φ made of a relation
symbol s with a choice of substitution h of its
arguments by free variables:
Φ(x) ⇔ s_{P}(x०h)
⇔ (∀i∈I,s_{i}(x०h)) ⇔
(∀i∈I,Φ(x_{i}))
which simplifies the remaining needed study, to cases without
substitutions. Now let us see the effects of connectives and
quantifiers (between relation symbols s interpreted in P
as s_{P}(x)⇔
∀i∈I, s_{i}(x_{i})):
(r_{P}(x)
∧s_{P}(x)) ⇔ 
∀i∈I,(r_{i}(x_{i})∧s_{i}(x_{i}))

(r_{P}(x)∨s_{P}(x))
⇒ 
∀i∈I,(r_{i}(x_{i})∨s_{i}(x_{i})) 
(r_{P}(x)
⇒ s_{P}(x)) ⇐ 
∀i∈I,(r_{i}(x_{i})⇒s_{i}(x_{i})) 
(∀y∈P,
r_{P}(x,y)) ⇐ 
∀i∈I, ∀y_{i}∈E_{i},
r_{i}(x_{i},y_{i}) 
(P≠ⵁ ∧
∀y∈P, r_{P}(x,y))
⇒ 
∀i∈I, ∀y_{i}∈E_{i},
r_{i}(x_{i}, y_{i})

(∃y∈P,
r_{P}(x,y,...)) ⇒ 
∀i∈I, ∃y_{i}∈E_{i},
r_{i}(x_{i},y_{i})

(∃y∈P,
r_{P}(x,y,...)) ⇐ 
(AC_{I}∧∀i∈I,
∃y_{i}∈E_{i}, r_{i}(x_{i},y_{i}))

The ⇒ cases (white and in green) can be understood as deduced from π_{i}
∈ Mor_{L}(P,E_{i}) as
seen in 3.1, plus the preservation of ∀ by surjective
morphisms.
The converse implications (⇐, in white and yellow, where AC_{I}
is the axiom of choice over I, see 2.10), describe which
properties are "preserved by taking the product of systems".
For example if each member of a product satisfies an axiom of the
form (∀ variables) (F_{1} ∧ ... ∧ F_{n})
⇒ G where F_{1},..., F_{n}
and G are relation symbols applied to variables, then the
product also satisfies this axiom.
For example, any product of ordered sets is an ordered set; but a
product of totally ordered sets, is a set with an order that is
usually not a total order.
Relational clones
For any set R ⊂ Rel_{E} of relations in E,
the relational clone generated by R, is the set Cl'(R)⊂
Rel_{E} of relations defined by formulas with
symbols in R∪{∃,=,∧, variables}. This is also a closure,
whose closed elements are the relational clones (or clones
of relations).
For any system E and formula Φ, the truth of Φ in E
is written E⊨Φ .
Theorem. The set of invariants Inv S of any S
⊂ Op_{E}, is a relational clone. In other words,
for any R ⊂ Rel_{E} we have Cl'(R) ⊂
Inv Pol R.
Proof 1:
Let R ⊂ Rel_{E} and s ∈
Mor_{R}(E^{n},E)⊂Pol R.
Let m∈ℕ, and a formula Φ with symbols in R∪{∃,=,∧},
variables bound by ∃, and m free variables.
Let r ∈Cl'(R) the mary relation defined
by Φ in E.
We saw that (∀i<n, E⊨Φ(π_{i}०x))
⇒ (E^{n}⊨Φ(x)) using the truth of
the axiom of choice over finite sets.
Then, (s∈Mor_{R}(E^{n},E)
∧ E^{n}⊨Φ(x)) ⇒ (E⊨Φ(s०x)).
We conclude ∀x∈(E^{n})^{m},
(∀i<n, r(π_{i}०x))
⇒ r(s०x), i.e. s ▷r.
Proof 2:
For any S ⊂ Op_{E}, let us
verify that Inv S is a clone of relations, that is, ∀m∈ℕ,
∀r⊂E^{m} defined by a term with
language in {∃,=,∧}∪Inv S, we have r∈Sub_{S}(E^{m}).
We can verify that we stay in Inv S in the following steps
of building formulas:
 Substitutions : for the relation r' defined by a
relation symbol r with a substitution map u
from r's arguments list I to the variables
list J, let f=∏_{i}_{∈}_{I}
π_{u(i)} ∈Mor(E^{J},E^{I}).
Then r∈Sub_{S}(E^{I})
⇒ r'=f^{ 1}(r)∈Sub_{S}(E^{J}).
 ∃: the relation defined by (∃x∈E, r(x,y,...))
is the image of r by the morphism ∏_{i}_{≠0}
π_{i}
 =: this relation is Gr Id_{E} = Im (Id_{E}×Id_{E})
∈ Sub_{S}(E×E)
 ∧: intersections of subalgebras are subalgebras.
Other ways to operate on subalgebras reflect formulas with these
symbols: for example
A∈Sub_{S}(E) ⇒ (Id_{E}×Id_{E})[A]
= {(x,y)∈E^{2}x∈A ∧ x=y}
For each relational clone R, the set Gr*R is a clone
of operations : for example, the graph of the term "x" with
variables x,y, is {(x,y,z)z=x};
if f,g∈E^{E} we have (x,y)∈
Gr(g०f) ⇔ (∃z∈E, (x,z)∈Gr(f)
∧ (z,y)∈Gr(g)).
Duality theories with structures
A duality theory is a theory describing duality systems.
Let R be a relational language, that is interpreted in K.
We may even see it as R⊂Rel_{K}.
For any set E, we have a Galois connection between
interpretations of R in E and subsets E'
⊂ K^{E}, defined as follows:
 Any E' ⊂ K^{E} defines an
interpretation of each nary r∈R in E
as r_{E}={(x_{1},...,
x_{n})∈E^{n}∀y∈E'
, r(y(x_{1}),...,y(x_{n}))}.
This is the only one making (∏_{y}_{∈}_{E'}
y) an embedding
from E to K^{E'}.
 Any interpretation of R in E gives a set E*=
Mor_{R}(E,K) ⊂ K^{E}.
Now let us consider duality theories, or kinds of Kduality
systems (E,E',〈 , 〉), that interpret R in E
as defined from E' in this way. (This forms a
firstorder 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 ∏_{y}_{∈}_{E'} y
is an embedding of E into K^{E'}
(injective as E is assumed to be separated by E'),
∏_{y}_{∈}_{E*} y
is also an embedding of E into K^{E}^{*}
(which means that the used Rstructure on E is
closed), i.e. (E,E*) is another Kduality
system giving the same Rstructure on E.
Thus, as the system E is a subset of an exponentiation of K
(namely K^{E'}), every axiom of the form (∀
variables) (F_{1} ∧ ... ∧ F_{n}) ⇒ G
(where (F_{1} ,..., F_{n}, G)
are relation symbols applied to variables) that is true in K,
is also true in E.
Theorem. For any two Kduality systems (E,E',〈
, 〉) and (F,F',〈 , 〉') interpreting R in E
and F in this way,
 Mor((E,E'),(F,F')) ⊂
Mor_{R}(E,F)
 Mor((E,E*),(F,F')) = Mor_{R}(E,F)
Proof.
As (∏_{y}_{∈}_{F'}
y) is an Rembedding from F to K^{F'},
we have
∀f∈F^{E},
f ∈ Mor_{R}(E,F) 
⇔ (∏_{y}_{∈}_{F'}
y)०f ∈ Mor_{R}(E,K^{F'}) 

⇔ ∀y∈F', y०f ∈
Mor_{R}(E,K)=E* 

⇔ f ∈ Mor((E,E*),(F,F')) 
As E'⊂E*, the condition f∈Mor((E,E'),(F,F')),
that is ∀y∈F', y०f ∈ E', is
more restrictive, thus the inclusion.
Now, let S ⊂ Pol R ⊂ Op_{K}. As seen
above, in any Kduality system (E,E') we have
E* ∈ Sub_{S}(K^{E}).
As the Rstructure 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 Salgebra
generated by E', that is the smallest subSalgebra
of E* (or equivalently of K^{E})
such that E' ⊂ X.