AN ALGORITHM FOR FINDING FINITE AXIOMATIZATIONS OF FINITE INTERMEDIATE LOGICS BY MEANS OF JANKOV FORMULAS. Abstract

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1 Bulletin of the Section of Logic Volume 31/1 (2002), pp. 1 6 Eugeniusz Tomaszewski AN ALGORITHM FOR FINDING FINITE AXIOMATIZATIONS OF FINITE INTERMEDIATE LOGICS BY MEANS OF JANKOV FORMULAS Abstract In the paper a new algorithm for finding finite axiomatizations of varieties of Heyting algebras generated by a single finite algebra is given. The novelty of this algorithm is that it produces axiomatizations which utilize only Jankov characteristic formulas. 1. Introduction It was proved constructively by D. H. J. de Jongh that every finite intermediate logic (i.e. logic determined by a variety of Heyting algebras generated from a finite Heyting algebra) is finitely axiomatizable. Another constructive proof of this fact was given by A. S. Troelstra, and an elegant algorithm for finding such axiomatizations was constructed by A. Wroński in [4]. In this paper I would like to present yet another algorithm, which essentially utilizes the approach adopted by the last author. The new algorithm shows, however, that we can confine ourselves to Jankov formulas in order to obtain the desired axiomatization. This means that the class of Jankov formulas is rich enough to axiomatize every finite intermediate logic, which is not true in general, i.e. when we consider all intermediate logics. The result discussed in the present paper was presented at the XLVII History of Logic Conference, Kraków, October 23-24, 2001.

2 2 Eugeniusz Tomaszewski To start with, I would like to introduce some notions and remind some facts I will make use of later on. Our general aim is to determine a variety of Heyting algebras by means of a set of formulas. Let V, W be varieties of Heyting algebras, and V SI, W SI subdirectly irreducible algebras of these varieties respectively. H will denote the variety of all Heyting algebras. As every variety is determined by its subdirectly irreducible members, we will confine our considerations to subdirectly irreducible algebras (sub-irr algebras for short). Thus, to determine any variety V of Heyting algebras, it means to find a suitable set Σ V of formulas in intuitionistic language, such as V SI = Σ V and for any sub-irr algebra A V SI there exists a formula σ Σ V so as A = σ. It happens that Jankov formulas posses characteristics which can prove useful when constructing a set Σ V. As Jankov formulas are strictly connected with finite sub-irr Heyting algebras, we should have in mind the following useful characterization (due to A. Day): a Heyting algebra A is sub-irr iff there is a greatest element in A {1}, where 1 is the greatest element in A. Such element is usually denoted by. Definition 1. Let A be a finite subdirectly irreducible Heyting algebra. Choose a 1-1 map from its universe onto a subset of variables (thus a x is a variable assigned by this map to an element x). Characteristic (Jankov) formula over A χ(a) is defined as follows: (ax a y ) a x y x, y A ) (ax a y ) a x y x, y A ) (ax a y ) a x y x, y A ) ax a x x A ) a Fact 1. By defining a suitable valuation we get easily the following fact: For any finite sub-irr algebra A, A = χ(a). The next theorem, proved by V. Jankov in [2] characterises the entire class of algebras which refute the formula χ(a) for a given algebra A.

3 An Algorithm for Finding Finite Axiomatizations... 3 Theorem 1. (V. Jankov) For any algebra B and any finite sub-irr algebra A the following holds B = χ(a) iff A SH(B). From Jankov s theorem it follows that if a finite sub-irr algebra A does not belong to a variety V we want to axiomatise, then we can add χ(a) to the set Σ V. To see this, let us suppose that there is an algebra B V such that B = χ(a). This would mean, however, that A SH(B) and thus A is in V as well. Furthermore, let W be the variety axiomatized by characteristic formulas on all finite sub-irr algebras not in V. It is obvious that V W and for any finite sub-irr algebra M, M V iff M W (i.e. V F SI = W F SI ). Thus, it is a straightforward task to determine the variety V with respect to its finite members. The real challenge is to deal with infinite sub-irr algebras or to get a finite axiomatization. We will need one more specific fact concerning Jankov formulas. If V (α) denotes the variety determined by a formula α (i.e. V (α) = {A H : A = α}) then we get the following. Fact 2. Let A be a finite sub-irr algebra. Then W = V (χ(a)) is the largest variety not containing the algebra A. Proof. If A V then V = χ(a) (by Jankov s theorem), so for every such variety, V W. A W (since A = χ(a)), so W is the largest variety not containing A. In our algorithm, an essential role will be played by linear Heyting algebras. Following A. Day [1], let (H : C n ) (where C n is an n-element linear Heyting algebra for n 2) denote the class {A H : C n S(A)}. As the finite chains are projective Heyting algebras, (H : C n ) is a variety for every n 2. In fact, it is the greatest variety not containing C n. In his manuscript, A. Day has shown that all these varieties are locally finite. In the proof, the following characterisation of sub-irr algebras in (H : C n ) was used. (A denots the algebra obtained from an algebra A by adding a new greatest element and suitably expanding Heyting operations.)

4 4 Eugeniusz Tomaszewski Lemma 1 For n 2, (H : C n+1 ) SI = {A : A (H : C n )}. Proof. It is easy to prove that C n S(A) iff C n+1 S(A ). As every sub-irr algebra B has the form A for a suitable algebra A, we get: A (H : C n+1 ) SI iff C n+1 S(A ) iff C n S(A) iff A (H : C n ), which concludes the proof. Theorem 2. For every n 2, the variety (H : C n ) is locally finite. Proof. We will show that in every variety (H : C n ), the free generated algebra F (k) on k generators is finite for each k, which is an equivalent condition for (H : C n ) being locally finite. (n = 2) (H : C 2 ) is locally finite as it is the variety of all trivial algebras. (n > 2) Let F (k) be the k-generated free algebra in (H : C n+1 ). F (k) P S ({A i } i I ), where A i are sub-irr, k-generated algebras belonging to (H : C n+1 ). Thus, every A i = B i for some B i (H : C n ) by Lemma 1. Now, as B i is k-generated, by the Induction Hypothesis we get that it is a finite algebra and there are only finitely many such algebras in (H : C n ). Therefore, F (k) V ({A i } i I ) where I is finite and every A i is a finite algebra. Hence, F (k) is finite as well. 2. Algorithm Now, we are ready to construct an algorithm which for every finite intermediate logic produces suitable finite axiomatization utilizing only Jankov formulas. As this algorithm is a generalization of the algorithm used in [4], I introduce the idea of the latter first. As every finite intermediate logic is determined by a variety of Heyting algebras generated from a single finite algebra, say A, this variety contains only finitely many sub-irr algebras and all of them are finite. This is an immediate consequence of the fact that the variety of Heyting algebras is congruence-distributive and the well known Jonsson s theorem concerning such varieties. Thus, there is an integer number n such that every sub-irr algebra in V (A) has at most n elements. Now, there is a single axiom γ n = (a i a j i, j = 1,..., n + 1, i j) which is true exactly in these sub-irr algebras that have less than n + 1 elements, as the following lemma due to C. G. McKay [3] states.

5 An Algorithm for Finding Finite Axiomatizations... 5 Lemma 2. Let A be a sub-irr Heyting algebra. Then A = γ n iff A n. In order to find a finite axiomatization of the variety V (A), it suffices to adopt the following procedure. Firstly, list all sub-irr algebras not in V (A) which have n elements at most. We get a finite set of finite sub-irr algebras, say {A 1,..., A k } and a finite set of Jankov formulas defined on them: {χ(a 1 ),..., χ(a k )}. Secondly, add to these Jankov formulas the formula γ n. Then {χ(a 1 ),..., χ(a k ), γ n } constitutes a suitable axiomatization of V (A). As a matter of fact, we do not need to take all algebras {A 1,..., A k }. If A i V (A j ) then we can use the axiom χ(a i ) instead of {χ(a i ), χ(a j )}. The interesting issue, however, is the question how to replace the formula γ n with a finite set of Jankov formulas. To do this, we will need the following lemma. Lemma 3. Let W be a locally finite variety of Heyting algebras. Then each its subvariety V generated from a single finite algebra can be determined, relatively to W, by a finite set of characteristic formulas. Proof. If V is a subvariety of W generated by a single finite algebra, then there is some integer m such that all subdirectly irreducible algebras in V have at most m elements. We list all algebras A 1,..., A k, such that A i W V and A i m. Obviously, for every such algebra A j, A j = k i=1 χ(a i) and V = k i=1 χ(a i). Now, let B be a sub-irr algebra from W V and B m + 1. Every such algebra B has an m + 1-generated subalgebra which is finite (as W is locally finite) and subdirectly irreducible (as any finite subalgebra of a subdirectly irreducible algebra is again subdirectly irreducible). We know also that there are only finitely many m + 1-generated subdirectly irreducible algebras in W, because of the local finiteness of W. Now, let B 1,..., B l be all these algebras. Then, for every sub-irr B W V such that B m+1 there is i {1,..., l} such that B i S(B). Thus B = χ(b i ). Obviously, V = l i=1 χ(b i). Thus, the variety V is determined relatively to W by means of a finite set of characteristic formulas on algebras {A 1,..., A k, B 1,..., B l }. Combining the above lemma with the results of A. Day, it is easy to prove that every variety generated by a finite Heyting algebra can actually be axiomatized by a finite set of Jankov formulas.

6 6 Eugeniusz Tomaszewski Theorem 3. Let V be a variety of Heyting algebras generated by a finite algebra. Then V is finitely axiomatizable by means of Jankov formulas. In fact, there is an algorithm, which for every such variety produces desired axiomatization. Proof. To prove the theorem we only need to find a locally finite variety W such as V W and W can be axiomatized by a finite set of Jankov formulas. Let m be the maximal number of elements in any sub-irr algebra in V. Then the linear algebra C m+1 does not belong to V, which means that V is a subvariety of the locally finite variety (H : C m+1 ). As (H : C m+1 ) is the greatest variety not containing C m+1, it is axiomatized by means of χ(c m+1 ) Jankov formula according to Fact 2. Given this axiomatization as well as local finiteness of (H : C m+1 ) we get that it is a decidable variety and the number of elements of any k-generated free algebra in it can be effectively determined. Thus V can be axiomatized by Jankov formulas on algebras {C m+1, A 1,..., A k, B 1,..., B l }, where A 1,..., A k, B 1,..., B l are to be found as in Lemma 3 which can be done effectively. References [1] A. Day, Varieties of Heyting algebras I, Manuscript. [2] V. A. Jankov, Conjunctively indecomposable formulas in propositional calculi, Izvestiya. Mathematics. Presidium Russ. Acad. Sci., Moscow, 3 (1969), [3] C. G. McKay, On finite logics, Indagationes Mathematicae, 29 (1967), [4] A. Wroński, An algorithm for finding finite axiomatizations of finite intermediate logics, Zeszyty Naukowe Uniwersytetu Jagiellońskiego, Prace z logiki, 7 (1972), Jagiellonian University Department of Logic Grodzka 52, Kraków, Poland tomaszew@if.uj.edu.pl