There are infinitely many maximal primitive positive clones in a diagonalizable algebra M* that serves as an algebraic model for provability logic GL. The paper constructs M* as the subalgebra of infinite binary sequences generated by the zero element (0,0,...). It defines primitive positive clones as sets of operations closed under existential definitions, and shows there are infinitely many maximal clones K1, K2, etc. that preserve the relations x = ¬Δi, x = ¬Δ2, etc. in M*. This provides a simple example of infinitely many maximal primitive positive clones.

1. Andrei RUSUInfinitely many maximal primitive positive clones
Primitive positive clones, Provability logic, Diagonalizable algebra
03F45, 03G25, 06E25
We present a relative simple example of infinitely many maximal primitive
positive clones in a diagonalizable algebra, which serve as an algebraic model
for the provability propozitinal logic GL.
Andrei Rusu, 2013
Andrei Rusu
bd. Mamaia, 124, Constanta, Romˆnia
¸
a
Ovidius University of Constanta
¸
E-mail: agrusu@univ-ovidius.ro
str. Academiei 5a, Chi¸in˘u, Rep. of Moldova
s a
Information Society Development Institute
E-mail: andrei.rusu@idsi.md
February ....., 2013
1

2. Infinitely many maximal primitive positive clones
in a diagonalizable algebra
Andrei RUSU
27 martie 2013
1
Introduction
The present paper deals with clones of operations of a diagonalizable algebra
which are closed under definitions by existentially quantified systems of equations. Such clones are called primitive positive clones [?] (in [?] they are referred
to as clones acting bicentrally, and are also called parametrically closed classes
in [?, ?]). Diagonalizable algebras [?] are known to be algebraic models for the
propositional provability logic GL [?].
In [?] was given the proof that there are finitely many primitive positive
clones in any k-valued logic. In the case of 2-valued boolean functions, i.e.
card(A) = 2, A. V. Kuznetsov state there are 25 primitive positive clones [?],
and A. F. Danilcenko proved there are 2986 primitive positive clones among 3valued functions [?]. In the present paper we construct a diagonalizable algebra,
generated by its least element, which has infinitely many primitive positive
clones, moreover, these primitive positive clones are maximal.
2
Definitions and notations
Diagonalizable algebras. A diagonalizable algebra [?] D is a boolean
algebra A = (A; , ∨, ⊃, ¬, , ) with an additional operator ∆ satisfying the
following relations:
∆(x ⊃ y) ≤ ∆x ⊃ ∆y,
∆x ≤ ∆∆x,
∆(∆x ⊃ x) = ∆x,
∆
= ,
where is the unit of A.
We consider the diagonalizable algebra M = (M ; , ∨, ⊃, ¬, ∆) of all infinite binary sequences of the type α = (µ1 , µ2 , . . . ), µi ∈ {0, 1}, i = 1, 2, . . . . The
boolean operations , ∨, ⊃, ¬ over elements of M are defined component-wise, and
the operation ∆ over element α we define by the equality ∆α = (1, ν1 , ν2 , . . . ),
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3. where νi = µ1 . . . µi . Let M∗ the subalgebra of M generated by its zero element (0, 0, . . . ). Remark the unite of the algebra M∗ is the element (1, 1, . . . ).
As usual, we denote by x ∼ y and ∆2 x, . . . , ∆n+1 x, . . . the corresponding
functions (¬x∨y)(¬y ∨x) and ∆∆x, . . . , ∆∆n x, . . . . Denote by x the function
x∆x and denoted by x the function ¬ ¬ x.
Primitive positive clones. The term algebra T (D) of D is defined as
usual, stating from constants , and variables and using operations , ∨, ⊃
, ¬, ∆. We consider the set T erm of all term operations of M∗ , which obviously
form a clone [?].
Let us recall that a primitive positive formula Φ over a set of operations Σ
of D is of the form
Φ(x1 , . . . , xm ) = (∃xm+1 ) . . . (∃xn )((f1 = g1 ) . . . (fs = gs )),
where f1 , g1 , . . . , fs , gs ∈ T (D) ∪ IdA and the formula (f1 = g1 ) . . . (fs = gs )
contains variables only from x1 , . . . , xn . An n-ary term operation f of T (D)
is (primitive positive) definable over Σ if there is a primitive positive formula
Φ(x1 , . . . , xn , y) over Σ of T (D) such that for any a1 , . . . , an , b ∈ D we have
f (a1 , . . . , an ) = b if and only if Φ(a1 , . . . , an , b) on D [?]. Denote by [Σ] all term
operations of D which are primitive positive definable over Σ of D. They say
also [Σ] is a primitive positive clone on D generated by Σ. If [Σ] contains T (D)
then it is referred to as a complete primitive positive clone on D. A primitive
positive clone C of D is maximal in D if T (D) ⊆ C and for any f ∈ T (D) C
we have T (D) ⊆ [C ∪ {f }].
Let α ∈ D. They say f (x1 , . . . , xn ) ∈ T (D) conserves the relation x = α on
D if f (α, . . . , α) = α. According to [?] the set of all functions that conserves
the relation x = α on an arbitrary k-element set is a primitive positive clone.
3
Preliminary results
We start by presenting some useful properties of the term operations ∆,
and of M∗ .
Let x, y arbitrary elements of M∗ . Then:
x ≥ ∆ , if and only if
x=
(1)
x = , if and only if
x=
(2)
For any x, y, either
x≤
y, or
y≤
∆x = ∆ x
(3)
(4)
= ,
=
x ≥ ∆ if and only if
x=
x
if and only if
(5)
¬x =
(6)
¬x ≥ ∆
(7)
Demonstratie. The proof is almost obvious by construction of the algebra M∗ .
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4. Let us mention the following observation:
Any function f of T (D) is primitive positive definable on D via system of
functions xy, x ∨ y, x ⊃ y, ¬x, ∆y.
Let us consider on D the following functions (??) and (??) of T (D), denoted
by f¬ (x, y) and f∆ (x, y) correspondingly, where αi , ξ ∈ D, αi = ¬∆i , where
ξ = αi and η = αi :
( ¬(x ∼ y)((¬x ∼ y) ∼ ξ)) ∨ ( (x ∼ y)αi )
(8)
( y((∆x ∼ y) ∼ η)) ∨ (¬ yαi )
(9)
Let arbitrary α, β ∈ M∗ . If ¬α = β on M∗ , then
f¬ (α, β) = ξ
on M∗ .
Demonstratie. Since ¬α = β we get α ∼ β = , ¬(α ∼ β) =
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have
(α ∼ β) = , ¬(α ∼ β) = ,
and by (??) we
which implies
f¬ (α, β) = ( ( ∼ ξ)) ∨ ( αi ) = ξ
Let arbitrary α, β ∈ M∗ . If ¬α = β on M∗ , then
f¬ (α, β) = ξ
on M∗ .
Demonstratie. Since ¬α = β we get ¬α ∼ β = , α ∼ β = We distinguish
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two cases: 1) (α ∼ β) = , and 2) (α ∼ β) ≥ ∆ .
In case 1) by (??), (??) and (??) we get ¬(α ∼ β) ≥ ∆ , ¬(α ∼ β) =
, and (α ∼ β) = , which implies
f¬ (α, β) = ( ¬(α ∼ β)((¬α ∼ β) ∼ ξ)) ∨ ( (α ∼ β)αi
= ( ((¬α ∼ β) ∼ ξ) ∨ ( αi ) = (¬α ∼ β) ∼ ξ = ξ,
Thus the first case is already examined.
Now consider the second case, when x ≥ ∆ . Again, since ¬α = β by
(??), (??) and (??) we obtain ¬(α ∼ β) = , ¬(α ∼ β) = , (α ∼ β) = .
Then,
f¬ (α, β) = ( ¬(α ∼ β)((¬α ∼ β)ξ)) ∨ ( (α ∼ β)αi )
= ( ((¬α ∼ β)ξ)) ∨ ( αi ) = αi = ξ.
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5. Let arbitrary α, β ∈ M∗ such that ∆α = β. Then
f∆ (α, β) = η
Demonstratie. Since ∆α ≥ and ∆α = β we have β ≥ ∆ , ∆α ∼ β = and
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by (??) we get β = , ¬ β = . These ones imply the following relations:
f∆ (α, β) = ( β((∆α ∼ β) ∼ η)) ∨ (¬ βαi )
= ( ( ∼ η)) ∨ ( αi ) =
∼η=η
Let arbitrary α, β ∈ M∗ such that ∆α = β. Then
f∆ (α, β) = η
Demonstratie. We consider 2 cases: 1) β = , and 2) β ≥ ∆ .
¸
Suppose β = . In view of (??) we have β = and ¬ β =
quently,
. Subse-
f∆ (α, β) = ( β((∆α ∼ β) ∼ η)) ∨ (¬ βαi )
= ( ((∆α ∼ β) ∼ η)) ∨ ( αi ) = 0 ∨ αi = αi = η
Suppose now
we get
β ≥ ∆ . Let us note ∆α ∼ β =
. Then considering (??)
f∆ (α, β) = ( β((∆α ∼ β) ∼ η)) ∨ (¬ βαi )
= ( ((∆α ∼ β) ∼ η)) ∨ ( αi ) = (∆α ∼ β) ∼ η = η
Let arbitrary α ∈ M∗ . Then
f¬ (α, α) = αi .
Demonstratie. Let us calculate f¬ (α, α). By (??) we obtain imidiately:
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f¬ (α, α) = ( ¬(α ∼ α)((¬α ∼ α)ξ)) ∨ ( (α ∼ α)αi )
= ( ( ξ)) ∨ ( αi ) = αi .
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6. Let arbitrary α ∈ M∗ and
α = . Then
f∆ (α, α) = αi .
Demonstratie. Taking into account (??) we have
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f∆ (α, α) = ( α((∆α ∼ α) ∼ η)) ∨ (¬ ααi )
= ( ((∆α ∼ α) ∼ η)) ∨ ( αi ) =
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∨ αi = αi .
Important properties of some primitive positive clones
Consider an arbitrary value i, i = 1, 2, . . . . Let Ki the primitive positive
clone of M∗ consisting of all functions of M∗ which conserves the relation x =
¬∆i on M∗ . For example, K1 is defined by the relation x = (0, 1, 1, 1, . . . ).
The functions x, xy, x ∨ y, ¬∆i ∈ Ki , and ¬x, ∆x ∈ Ki .
Since Ki is a primitive positive clone it follows from the above statement the
functions ¬x and ∆x are not primitive positive definable via functions of Ki on
M∗ , so T (M∗ ) ⊆ Ki and thus clone Ki is not complete in M∗ .
By propositions ?? and ?? we have the earlier defined functions f¬ (x, y) and
f∆ (x, y) are in Ki .
Suppose an arbitrary f (x1 , . . . , xk ) ∈ T (M∗ ) and f ∈ Ki . Then functions
∆x and ¬x are primitive positive definable via functions of Ki ∪{f (x1 , . . . , xk )}
Demonstratie. Let us note since f ∈ Ki we have f (¬∆i , . . . , ¬∆i ) = ¬∆i .
¸
Now consider next term operations f¬ and f∆ defined by terms (??) and (??):
( ¬(x ∼ y)((¬x ∼ y) ∼ f (¬∆i , . . . , ¬∆i ))) ∨ ( (x ∼ y)¬∆i )
i
i
i
( y((∆x ∼ y) ∼ f (¬∆ , . . . , ¬∆ ))) ∨ (¬ y¬∆ )
(10)
(11)
and examine the primitive positive formulas containing only functions from
Ki ∪ {f }:
(f¬ (x, y) = f (¬∆i , . . . , ¬∆i )) and (f∆ (x, y) = f (¬∆i , . . . , ¬∆i ))
Let us note by propositions ?? and ?? we have (¬x = y) if and aonly if
(f¬ (x, y) = f (¬∆i , . . . , ¬∆i )) and according to propositions ?? and ?? we
get (∆x = y) if and only if f∆ (x, y) = f (¬∆i , . . . , ¬∆i )).
Lemma is proved.
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7. 5
Main result
There are infinitely many maximal primitive positive clones in the diagonalizable algebra M∗ .
Demonstratie. The theorem is based on the example of an infinite family of
¸
maximal primitive positive clones presented below.
The classes K1 , K2 , . . . of term operations of T (M∗ ), which preserve on
algebra M∗ the corresponding relations x = ¬∆ , x = ¬∆2 , . . . , constitute a
numerable collection of maximal primitive positive clones in M∗ .
Really, it is known [?] that these classes of functions represent primitive
positive clones. According to observation ?? each clone Ki is not complete in
M∗ . In virtue of lemma ?? these primitive positive clones are maximal. It
remains to show these clones are different. The last thing is obvious since
¬∆j ∈ Kj and ¬∆j ∈ Ki , when i = j.
The theorem is proved.
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Conclusions
We can consider the logic LM∗ of M∗ , which happens to be an extension of
the propositional provability logic GL, and consider primitive positive classes of
formulas M1 , M2 , . . . of the propositional provability calculus of GL preserving
on M∗ the corresponding relations x = ¬∆ , x = ¬∆2 , . . . .
The classes of formulas M1 , M2 , . . . constitutes an infinite collection of primitive positive classes of formulas in the extension LM∗ of the propositional provability logic GL.
Demonstratie. The statement of the theorem is just another formulation of the
¸
theorem ?? above in terms of formulas of the claculus of GL, which follows the
usual terminology of [?].
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