Abstract clones
Clones
Putting all arities together gives an algebraic language called the clone of C:
L = ∐_{n∈ℕ} L_{n}
where each set L_{n}
of nary symbols is an object of C with a chosen basis of n elements
1_{n} = (e_{i,n})_{i<n}
∈ L_{n}^{n}. Thus, ∀n∈ℕ,
L_{n} = ^{C}∐_{i<n}
L_{1}.Depending on C, a clone may exist or not,
but it is essentially unique, serving as the maximal algebraic language for objects of C.
Each object E of C, gets the Lalgebra structure
φ_{E} = ∐_{n∈ℕ}
φ_{E}^{(n)}, where
φ_{E}^{(n)} : L_{n} × E^{n}
→ E is defined taking as ⃖φ_{E}^{(n)} the inverse of
Mor(L_{n},E) ∋ f ↦
f⚬1_{n} which is bijective to E^{n} because
1_{n} is a basis in C. This definition is expressible by both axioms
which extend to clones those defining the action of an egg in a category:
 Each e_{i,n} acts as the ith
projection from E^{n} to E, i.e. ∀i<n,
∀u∈E^{n},
e_{i,n} ⋅ u = u_{i}.
 ∀n∈ℕ, ⃖φ_{E}^{(n)} :
E^{n} → Mor(L_{n},E).
This implies (for all objects E, F),
Mor(E,F) ⊂ Mor_{L}(E,F)
∀n∈ℕ, Mor(L_{n},E) =
Mor_{L}(L_{n},E).
Let us recall the proof of Mor_{L}(L_{n},E)
⊂ Mor(L_{n},E) previously seen for n=1. From
1_{n} being a basis of L_{n} and
Id_{Ln} ∈ Mor(L_{n},L_{n}), comes
 ∀x∈L_{n}, x = x⋅1_{n}
Thus ∀f∈Mor_{L}(L_{n},E),
(∀x∈L_{n}, f(x) = f(x⋅1_{n}) = x⋅
(f⚬1_{n}) ∴ f =
⃖φ_{E}^{(n)}(f⚬1_{n})
∈ Mor(L_{n},E).∎
3. implies that 1_{n} generates L_{n} in the sense that
〈1_{n}〉_{Ln} = L_{n} ∴
〈1_{n}〉_{L} = L_{n}.
Abstract clones and varieties
A clone L may be conceived independently of C
(like monoids are independent of their actions), as the small concrete category made of the sequence of objects
(L_{n})_{n∈ℕ}, called an abstract clone. It
defines on L a structure of typed algebra with types L_{n} and operations
 Constant symbols e_{i,n}
 An Lalgebra structure on each L_{n}, that is
a sequence of operations we shall abusively denote the same
• : L_{p}
× L_{n}^{p} → C_{n}
for each p∈ℕ.
Generalizing the concept of action to such
a typed algebra, an Lmodule (or representation of L),
is an Lalgebra M satisfying the above 1. and 2. where 2. uses
Mor_{L} :
∀n,p∈ℕ, ∀s∈L_{p},
∀x∈L_{n}^{p}, ∀u∈M^{n},
(s•x) ⋅ u = s ⋅ ((x_{i}
⋅ u)_{i<p})
The variety of L is the category of all Lmodules with Mor = Mor_{L}.
A typed algebra L = ∐_{n∈ℕ} L_{n} with such symbols, is
an abstract clone if each L_{n} satisfies the above axioms 1., 2. and 3. extending to L
the axioms of monoid
for L_{1}:
 1. and 2. make L_{n} an Lmodule.
 As 3. was seen to imply ∀M, ∀f∈Mor_{L}(L_{n},M),
f =
⃖φ_{M}^{(n)}(f⚬1_{n}), it
completes making 1_{n} a basis of L_{n}, equivalently in
the small category L with its Lmorphisms or in its whole variety.
As will be seen later, the general concept of variety
will use algebraic languages that do not need to be abstract clones, and not even
typed algebras. Roughly speaking, a variety is a category of algebras which has a clone
and is equivalent to the variety of this clone. But other equivalent definitions are possible.
With a simple algebraic language L, conditions may be
expressed by a list of axioms made of ∀ over equalities of Lterms.
As will be shown, such categories are still qualified as varieties in the sense of
having a clone ∐_{n∈ℕ} L_{n}, and being
identifiable with the variety of this clone; L may differ from it, but generates it in
the sense that each Lalgebra L_{n} is Lgenerated by
(the image of) its basis, i.e. is the set of all symbols definable by nary Lterms
(where the role of the symbols of variable is played by the symbols
e_{i,n}):
∀n∈ℕ, 〈1_{n}〉_{L} = L_{n}.
The axioms of module of abstract clones,
say that this algebra satisfies all C_{n}, with the interpretation of symbols from
C_{n} coincides with the expression of this satisfaction.
An equivalent condition for an algebraic language to be an abstract clone is that
there exists a class of interpretations (algebras), such that this language is stable by
definitions by terms, while any two
symbols keeping the same interpretation throughout this class are confused.
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
∎
Examples of varieties
The above gives two examples of varieties
 That of all Lalgebras for any given algebraic language L; its clone is the sequence of term algebras
with each finite arity.
 That of all monoids; its clone is made of free monoids.
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