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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
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a
Ovidius University of Constanta
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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
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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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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.
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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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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 ≥ ∆ .
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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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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 .
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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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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
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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
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theorem ?? above in terms of formulas of the claculus of GL, which follows the
usual terminology of [?].
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