Abstract clones
Abstract clones are the abstract version of clones of operations,
generalizing to all arities the concept of monoid which was the
abstract version of transformations monoids.
An abstract clone is an algebraic system (a multitype
algebra) C=⋃_{n∈ℕ} C_{n},
with types C_{n} (which are pairwise disjoint) and
 C will play the role of an algebraic language where C_{n}
is seen as a set of nary operation symbols
 ∀n∈ℕ, C_{n} has 2 kinds of
structures:
 A family 1_{n} = (π_{i,n})_{i}_{<}_{n}
of n constant symbols (which correspond to the n
variables when elements of C_{n} play the role
of nary operation symbols)
 a structure of Calgebra, which can also be seen as
a sequence of operations : one operation C_{p}
× C_{n}^{p} → C_{n}
for each p∈ℕ
 Each C_{n} satisfies the following axioms:
 ∀x∈C_{n}, x=x⋅1_{n}
(this ensures that 1_{n} is a generating family
of C_{n})
The next two express that C_{p} satisfies C_{n}
: ∀n,p∈ℕ,∀u∈C_{p}^{n},
 1_{n}⋅ u=u , i.e.∀i<n,
π_{i,n} ⋅ u = u_{i}
 (x↦x⋅u)∈Mor_{C}(C_{n},C_{p}),
i.e. ∀m∈ℕ,∀s∈C_{m} ,∀x∈ C_{n}^{m},
(s⋅x) ⋅ u = s ⋅ ((x_{i}
⋅ u)_{i<m})
These axioms generalize the 3 axioms of monoids.
A module or representation of an abstract clone C,
is any Calgebra M satisfying all C_{n},
and where the interpretation of all nary symbols (from C_{n})
coincides with the expression of the satisfaction of C_{n}
by M. Formally, this is just a copy of the last 2 axioms:
∀n∈ℕ,∀u∈M^{n},
 1_{n}⋅ u=u , i.e.∀i<n,
π_{i,n} ⋅ u = u_{i}
 (x↦x⋅u)∈Mor_{C}(C_{n},M),
i.e. ∀m∈ℕ,∀s∈C_{m} ,∀x∈ C_{n}^{m},
(s⋅x) ⋅ u = s ⋅ ((x_{i}
⋅ u)_{i<m})
Correspondences between interpretations of formulas
Formulas can be expressed to relate elements in an abstract clone as
if they operated on an unknown module M, using universal
quantifiers on the variables in M . Namely, a formula R
of equality «t = t'» between 2 nary terms t,t',
can be written (for example with n=2):
∀x,y, R(x,y)
to actually mean the relation in C_{2} that is R(π_{0,2},π_{1,2})
because this indeed implies the above universal formula in any
representation. In particular, it implies R(π_{0,3},π_{1,3}),
i.e. ∀x,y,z, R(x,y).
In fact these are equivalent, but the reason for the converse
implication (R(π_{0,3},π_{1,3})⇒R(π_{0,2},π_{1,2}))
has a subtlety, that is the same already faced to
represent an operation symbol in an algebra with higher arity. In
other words, to deduce (∀x,y, R(x,y))
from (∀x,y,z, R(x,y)) we need to choose
a value for z. We may set it as equal to x or y,
except if n=0 when they are not available, but then we can
use any available symbol of constant instead (there will be, since
terms not using variables must use constants).
Centralizers in abstract clones
We can generalize the concept of centralizer (or commutant),
which was defined on monoids, to the case of abstract clones. It
is directly obtained by abstracting with the above trick, the
definition of the Galois
connection PolPol between sets of operations, where the
condition of commutation between x∈C_{i}
and y∈C_{j} is an equation expressed in C_{ij}
.
So, the centralizer Pol X of any subset X of an
abstract clone C, is a subclone of C.
The center of C, is the centralizer of C
itself, Pol C = ∩_{x}_{∈}_{C}
Pol {x}. We have Pol_{ }Pol C =
C.
For any p<n, Pol C_{n} ⊂ Pol C_{p}
because all pary operations are represented among nary
operations, with equivalence of formulas between pary and
nary views for the above reason.
∀k∈C_{0}, Pol{k} ∩ C_{0}
= {k}.
The multiplication monoid in an abstract clone
Elements of C_{1} will be called scalars.
The unit element of the clone, is the scalar π_{0,1}
that will be simply denoted 1. The operation that applies any scalar
a to any element x of any representation, (a,x)↦a⋅x
will be called multiplication. The multiplication between
scalars forms a monoid, with 1 as identity element:
∀a∈C_{1},

a⋅1=a 
∀x,

1⋅x = x 
∀a,b∈C_{1},∀x,

(a⋅b)⋅x = a⋅(b⋅x) 
The multiplication is right distributive over any operation : for
example if #∈C_{2},
∀a,b∈C_{1},∀x,
(a#b)⋅x =(a⋅x)#(b⋅x)
However, the left distributivity formula (∀x,y, a⋅(x#y)
= (a⋅x)#(a⋅y)) means that # commutes
with a.
Similarly for constants (nullary operations) : as any constant
symbol k∈C_{0} can be interpreted both in C_{1}
and in any other representation, both are related by
∀x, k⋅x = k
The same works for any constant, i.e. defined by a term that does
not depend on any "variable". For example, if #∈C_{2}
and k,k'∈C_{0}, we have k#k'∈C_{0},
thus
∀x, (k#k')⋅x = k#k'.
Conversely, if C_{0}≠Ø and an element k ∈C_{1}
satisfies ∀x∈C_{1}, k⋅x = k
then for any k'∈C_{0} , the element k⋅k'
∈ C_{0} satisfies k⋅k' = k in
C_{1}, so that k is the image in C_{1}
of k⋅k'∈C_{0}.
As already argued above on the correspondence between
interpretations of formulas, if two constants k,k'∈C_{0}
commute by multiplication in C_{1} then they are
equal in C_{1} (k = k⋅k' = k'⋅k
= k'∈C_{1}) and thus also in C_{0}
(k = k⋅k = k'⋅k = k'∈C_{0}).
Every term in the language C (i.e. without ⋅ nor 1) with
only one variable, is reducible to a scalar multiplying the variable
: for example, ∀#∈C_{2}, ∀x,
∀k∈C_{0},

x#k = (1⋅x)#(k⋅x)
= (1#k)⋅ x 

x#x = (1⋅x)#(1⋅x)
= (1#1)⋅ x 
Proposition. C_{0} ∩ Pol C_{1}
⊂ Pol C.
Proof. Let k∈C_{0}∩ Pol C_{1 },
i.e. ∀a∈C_{1}, a⋅k=k⋅a=k.
Let n∈ℕ and x∈C_{n}. The commutation
relation between k and x is verified as
x(k,...,k) = x(1⋅k,...,1⋅k)
= x(1,...,1)⋅k = k
∎
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