Abstract clones
The definition of
abstract clones was moved to another page.
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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