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    Andrei rusu-2013-amaa-workshop Andrei rusu-2013-amaa-workshop Document Transcript

    • 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
    • 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 , . . . ), 2
    • 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∗ . ¸ 3
    • 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) = ¸ 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 ¸ 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. 4
    • 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 ¸ 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: ¸ f¬ (α, α) = ( ¬(α ∼ α)((¬α ∼ α)ξ)) ∨ ( (α ∼ α)αi ) = ( ( ξ)) ∨ ( αi ) = αi . 5
    • Let arbitrary α ∈ M∗ and α = . Then f∆ (α, α) = αi . Demonstratie. Taking into account (??) we have ¸ f∆ (α, α) = ( α((∆α ∼ α) ∼ η)) ∨ (¬ ααi ) = ( ((∆α ∼ α) ∼ η)) ∨ ( αi ) = 4 ∨ 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. 6
    • 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. 6 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 [?]. Bibliografie [1] Burris S., Willard R. Finitely many primitive positive clones // Proceedings of the American Mathematical Society, 101, no. 3, 1987, pp. 427–430. [2] Szabo L., On the lattice of clones acting bicentrally // Acta Cybernet., 6 (1984), 381–388. [3] Kuznetsov A. V., On detecting non-deducibility and non-expressibility in: Locical deduction, Nauka, Moscow (1979), 5–33 (in russian) ˘ [4] Danil’cenco A. F. Parametric expressibility of functions of three-valued logic // Algebra i Logika, 16 (1977), pp.397–416 (in russian) 7
    • [5] Magari R., The diagonalizable algebras (the algebraization of the theories which express Theor.: II) // Boll. Unione Mat. Ital. , 12 (1975) (suppl. fasc 3) pp. 117–125. [6] Solovay R.M., Provability interpretations of modal logic // Israel J. Math., 1975, 25, p. 287 - 304. ´ [7] Szendrei A , Clones in universal algebra, S´minaire de Math´matiques e e Sup´rieures, 99, Les Presses de l’Universit´ de Montr´al, 1986. e e e ´ [8] Szabo L., On algebras with primitive positive clones, // Acta Sci. Math. (Szeged), 73 (2007), 463–470 ˘ [9] Danil’cenco A. F. On parametrical expressibility of the functions of k-valued logic // Colloq. Math. Soc. Janos Bolyai, 28, North-Holland, 1981, pp. 147–159. 8