Involutive symmetric Gödel spaces, their algebraic duals and logic

It is introduced a new algebra (A,⊗,⊕,∗,⇀,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$\end{document} called LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document}-algebra if (A,⊗,⊕,∗,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, \otimes , \oplus , *, 0, 1)$$\end{document} is LP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_P$$\end{document}-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and (A,⇀,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A,\rightharpoonup , 0, 1)$$\end{document} is a Gödel algebra (i.e. Heyting algebra satisfying the identity (x⇀y)∨(y⇀x)=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)$$\end{document}. The lattice of congruences of an LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document} -algebra (A,⊗,⊕,∗,⇀,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$\end{document} is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra (A,⊗,⊕,∗,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(A, \otimes , \oplus , *, 0, 1)$$\end{document}. The variety LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {L_PG}$$\end{document} of LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document} -algebras is generated by the algebras (C,⊗,⊕,∗,⇀,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C, \otimes , \oplus , *, \rightharpoonup , 0, 1)$$\end{document} where (C,⊗,⊕,∗,0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(C, \otimes , \oplus , *, 0, 1)$$\end{document} is Chang MV-algebra. Any LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document} -algebra is bi-Heyting algebra. The set of theorems of the logic LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document} is recursively enumerable. Moreover, we describe finitely generated free LPG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_PG$$\end{document}-algebras.


Symmetry in sentential calculus
Some non-classical logics also capture the dual principle holding in the classical propositional logic. Indeed if in a propositional calculus any connective has its dual, we say that such calculus is symmetric. The symmetric property of a logic can induce a property for its algebraic models. We say that a poset (X , R) is symmetric if (X , R) is isomorphic to its dual (X ,Ȓ), whereȒ is the converse relation of R. If we have an B G. Vitale vitale.gaetano.27@gmail.com 1 University of Salerno: Universita degli Studi di Salerno, Fisciano, Italy 2 Tbilisi State University, Georgian Technical University, Tbilisi, Georgia algebra A such that A has a lattice order R, then we say that A is symmetric if (A, R) is isomorphic to (A,Ȓ). Consequently, we have that: the classical propositional calculus is symmetric, and its algebraic models are (algebraically) symmetric.
We recall further examples of symmetric logic below. Heyting-Brouwer logic (alias symmetric Intuitionistic logic I nt 2 ) was introduced by C. Rauszer as a Hilbert calculus with an algebraic semantics [22]. Notice that the variety of Skolem (Heyting-Brouwerian, bi-Heyting) algebras are algebraic models for symmetric Intuitionistic logic I nt 2 [15,22].
Recall that Gödel logic is the extension of Intuitionistic logic Int by the axiom (α β) ∨ (β α). The main idea (principle) of Int asserts that the truth of a mathematical statement can be established only by producing a constructive proof of the statement. So, the intending meaning of the intuitionistic connectives is defined in terms of proofs and constructions and these notions are regarded as primary.
• A proof of the proposition α ∧ β consists of a proof of α and a proof of β; • a proof of a proposition α ∨ β is given by presenting either a proof of α or a proof of β; • a proof of α → β is a construction which, given a proof of α, returns a proof of β; • ⊥ has no proof and a proof of ¬α is a construction which, given a proof of α, would return a proof of ⊥.

Priestley spaces
A Priestley space is a triple (X , R, ), where (X , ) is a Stone space and R is an order relation on X such that, for all x, y ∈ X with xR y, there exists a clopen up-set V with x ∈ V and y / ∈ V . A morphism between Priestley spaces is a continuous order-preserving map. We denote the category of Priestley spaces plus continuous order-preserving maps by PS. For details on Priestley duality see Priestley [21] and Davey and Priestley [6]. Note that for simplicity we will often refer to Priestley space by (X , R) or by its underlying set X .
Priestley duality relates the category of bounded distributive lattices to the category of Priestley spaces by mapping each bounded distributive lattice L to its ordered space F(L) of prime filters, and mapping each Priestley space X to the bounded distributive lattice L(X ) of clopen up-sets of X . Notice that there exist distributive lattices and Priestley spaces that are not symmetric.

Heyting spaces and Gödel spaces
The sets R(U ) and R(x) are defined dually. More precisely, (X , R) is Heyting space [13,15]  If Heyting algebra satisfies the identity (x y)∨(y x) = 1, then it is called Gödel algebra. A Heyting space (X , R) is Gödel space if (X , R) is a root system [17].
Notice that there exist Heyting algebras (Gödel algebras, as well) and Heyting spaces (Gödel spaces) that are not symmetric.

Symmetric heyting and Gödel spaces
A Heyting space (X , R) is symmetric [14] if (X ,Ȓ) is also Heyting space, where x Ry ⇔ yȒx. Heyting space is symmetric iff R −1 (clU ) = cl R −1 (U ) [14]. Let (X , R) and (X , R ) be symmetric Heyting spaces. a map f : X → X is said to be interval [14] if Denote the category of Skolem algebras and Skolem homomorphism by HA 2 and the category of symmetric Heyting spaces and interval maps by HS 2 .
For any symmetric Heyting space (X , R) and U , V ∈ H(X )(= the set of all clopen up-sets of X ) define: Then the algebra H((X , R)) = (H(X ), ∨, ∧, , , ∅, X ) is a Skolem algebra [14]. Furthermore, for any morphism f : On the other hand, for each Skolem algebra A, the set F(A) of all prime filters of A with the binary relation R on it,which is the inclusion between prime filters, and topologised by taking the family of supp(a) = {F ∈ F(A) : a ∈ F}, for a ∈ A, and their complements as a subbase, is an object of HS 2 ; and for each Skolem algebra homomorphism h : A → B, F(h) = h −1 is a morphism of HS 2 . Therefore we have two contravariant functors F : HA 2 → HS 2 and H : HS 2 → HA 2 . These functors establish a dual equivalence between the categories HA 2 and HS 2 [14]. Proposition 1.1 [14]. Let T be a Skolem algebra and (X , R) be the symmetric Heyting space corresponding to it. Then the lattice ϑ(T ) of congruences of the algebra T is anti-isomorphic to the lattice of all closed bicones (i.e. the sets that are simultaneously up-set and down-set) of the symmetric Heyting space (X , R).
Recall that a Skolem algebra A is said to be G 2 − algebra (or symmetric Gödel algebra) if it satisfies the linearity conditions: (a b) ∨ (b a) = 1, (a b) ∧ (b a) = 0 for all a, b ∈ A. G 2 -algebras represent the algebraic models for logic G 2 . It is well known that the Heyting spaces for Gödel algebras form root systems. A.
Horn [17] showed that Gödel algebras can be characterized among Heyting algebras in terms of the order on prime filters (co-ideals). Specifically, a Heyting algebra is a Gödel algebra iff its set of prime filters lattice is a root system (ordered by inclusion). We say that X is a Gödel space iff it is a Heyting space such that R(x) is a chain for any x ∈ X . For a G 2 -algebra A we can assert that its G 2 -space (or symmetric Gödel space) H(A) is a root system in both directions, i.e. R(x) and R −1 (x) are chains for every x ∈ H(A). So, it holds Moreover, we have Theorem 1.2 There exist two contravariant functors F : G 2 → GS 2 and H : GS 2 → G 2 . These functors establish a dual equivalence between the categories G 2 of G 2algebras and GS 2 of G 2 -spaces (symmetric Gödel spaces).

Proof
The proof immediately follows from the fact that G 2 is a full subcategory of HA 2 .
If f : A → B is an injective homomorphism between G 2 -algebras A and B, then is a surjective interval mapping. If A and B are Heyting algebras we have a partition of F(B) by ker(F( f )) −1 on closed classes, which gives the corresponding equivalence relation, say E, on F(B), i.e. E is an equivalence relation on F(B) [16].
We refer to [16] for the definition of correct partition. A correct partition of a G 2 -space (Y , R) adapted to G 2 -algebras is an equivalence relation E on Y , such that (1) E is a closed equivalence relation, i.e. the E-saturation 1 of any closed subset is closed; (2) the E-saturation of any bicone is a bicone; and an interval map f : Y → Z such that ker f = E.
As follows from the duality, there is a one-to-one correspondence between subalgebras of a G 2 -algebra T and correct partitions corresponding to its G 2 -space (X , R) Figure 1 shows partitions of the corresponding equivalence relation E, where ovals represent blocks of equivalent elements. The partitions in (a), (b) and (c) are correct partitions since the E-saturation of any bicone is a bicone. On the other hand (d) is not a correct partition since the E-saturation of the bicone {g, h} is not a bicone: Observe that symmetric Gödel algebras and symmetric Gödel spaces are symmetric.

Looking for symmetry in ukasiewicz sentential calculus
M V -algebras are the algebraic counterpart of the infinite-valued Łukasiewciz sentential calculus, as Boolean algebras are for the classical propositional logic. In contrast  As it is well known, M V -algebras form a category which is equivalent to the category of abelian lattice ordered groups ( -groups, for short) with a strong unit [20]. Let us denote by the functor implementing this equivalence. In particular, each perfect M V -algebra is associated with an abelian -group with a strong unit. Moreover, the category of perfect M V -algebras is equivalent to the category of abelian -groups, see [12].
The class of perfect M V -algebras does not form a variety and contains non-simple subdirectly irreducible M V -algebras. It is worth stressing that the variety generated by all perfect M V -algebras, denoted by MV(C), is also generated by a single M Vchain, the M V -algebra C defined by Chang in [4]. We name M V (C)-algebras the algebras from the variety generated by C. Notice that the Lindenbaum algebra of L P is an M V (C)-algebra (a detailed description of free M V (C)-algebras is presented in [10]).
Let us define C m as follows: Notice that Rad * (C 2 ) as a lattice is not a Heyting lattice because there is no Heyting implication (c, 0) (0, 0). We can also observe that this space is a Gödel space, and it is not a dual Gödel space, i.e. the Gödel space is not symmetric, and nevertheless, this perfect algebra is symmetric.
M V -algebras and Gödel algebras are both distributive lattices. In Gödel algebras for the lattice operation ∧ there exists its adjoint operation, Heyting implication → G . We cannot say the same for the distributive lattice of an MV-algebra, i.e. there exists an MV-algebra where we have no adjoint operation for ∧. Analogically, there exists an MV-algebra where we have no adjoint operation for ∨. Distributive lattices where exists adjoint operation → Br for ∨ form Brouwerian algebras. Distributive lattices where exist both Heyting implication → G and Brouwerian implication → Br are named Heyting-Brouwerian algebras (C. Rauszer, Esakia). We are interested in the M V -algebras' distributive lattice which contains both Heyting implication → G and Brouwerian implication → Br besides M V -algebra implication (x * ⊕ y) and its dual (x ⊗ y * ). In other words, we have a full symmetric case, where M V -negation transforms each implication into its dual like in the Boolean case. So, if we add to the M V -algebras signature the Heyting implication → G , we automatically obtain Brouwerian implication → Br . and (A, ∨, ∧, , 0, 1) is a Gödel algebra, i.e. Heyting algebra satisfying the identity (x y)∨(y x) = 1. In other words we have a fusion of two algebras, an M V -algebra and a Gödel algebra. Moreover, notice that the variety L P G of L P G-algebras is the subvariety of the variety GMV [11] of G M V -algebras axiomatized by the perfect M V -algebra identity 2x 2 = (2x) 2 . Taking into account that we can also express the Łukasiewicz implication x → y = x * ⊕ y, hence we have two distinct residuations.

Proof
The variety generated by perfect M V -algebras is generated by chain perfect M V -algebras. In any chain perfect M V -algebras (a * b * ) * is a co-implication (relative pseudo-difference) and (¬ a * ) * is co-negation.
Theorem 2.2 [11]. Let F be an M V -Skolem filter of an L P G-algebra A. The equiv- Proof x ≡ y is a congruence because F is an M V -filter, since it is a congruence with respect to M V -algebra operations, and, at the same time, it is a Skolem filter.
In section 1.6 we define the M V (C)-algebras C i , i ∈ Z + , where C 1 (= C) is the Chang algebra. Now we enrich the signature of the algebras C i (i ∈ Z + ) by the Heyting operations ∨, ∧, and denote them by C G i = (C G i , ⊗, ⊕, * , ∨, ∧, , 0, 1). Now we define the algebra C G 1 similar to Chang.
by the following operations (consider 0 = 0c): Notice that in any bounded chain the Heyting implication is defined in the same manner: The formulas of the logic G L are built up by means of the propositional variables p 1 , p 2 , ..., logical connectives →, , ∼ in the usual way. We have some abbreviations Notice that the L P G-logic is the extension of the logic G M V [11] by the axiom (α∨α)&(α∨α) ↔ (α&α)∨(α&α). The axioms are the axioms of the Łukasiewicz logicŁ the axioms of the Gödel logic axioms connecting Łukasiewicz and Gödel logics and the axiom corresponding to perfectness , where ∨ is the strong disjunction and & is the strong conjunction defined as follows: . Inference rule modus ponens: α, α → β ⇒ β. From the axiom(GL2) it follows Gödel logic modus ponens: α, α β ⇒ β. It is clear that this axiom system is not the most economical one; axioms are given in the above form for their intuitive contents.
We introduce the equivalence relation on the set of formulas of the logic L P G: α ≡ β iff the formulas α ↔ β and ∼ ¬ α ↔∼ ¬ β. Taking into account that M V -algebras and Gödel algebras axioms correspond to Łukasiewicz logic and Gödel logic axioms, respectively, and L P G-logic axioms GL1 -GL6 are the translation of L P G-algebras axioms to the logical ones and the same for the axiom (L P G), it holds Theorem 2. 3 The Lindenbaum algebra of the logic L P G is an L P G-algebra.

Theorem 2.4 (Completeness theorem). For any formula α, α is a theorem of the logic L P G iff α is a tautology.
Proof It is obvious that if α is a theorem, then α is a tautology. Let us suppose that α is not a theorem. Then α/ ≡ = 1 in the Lindenbaum algebra A(L), but the Lindenbaum algebra A(L) is isomorphic to a subdirect product of chain L P G-algebras. Hence the element α/ ≡ can be represented as a sequence (x 1 , ..., x i , ...), where the elements of this sequence belong to A(L) and x i = 1 for some i. Let h be a natural homomorphism from the algebra of formulas F L onto A(L), i.e. h(α) = α/ ≡ for every α ∈ F L . Then we can consider the map g = π i • h where π i is a projection of A(L) onto i-th chain component D of the subdirect product of chain L P G-algebras g is a value function of the algebra D such that g(α) = 1. Therefore α is not a tautology.

Generating algebras for L P G
Notice that if A is a finitely generated M V -algebra, then A is subdirectly irreducible iff A is a chain. Similarly, if A is an L P G-algebra, then A is subdirectly irreducible iff A is a chain, and hence it is simple, like in Boolean algebras.
In [12] it is shown that the variety generated by C 1 contains any perfect algebra. So, C 1 and C n generate the same variety. Here we give similar results for L P G. More precisely, we will show that the variety generated by C G 1 coincides with the variety generated by C G n . We prove this assertion for n = 2 which is easy generalized for any positive integer that is greater than 2.
Let C = (C G 1 , =, ⊗, ⊕, * , ∨, ∧, , 0, 1) be the model for L P G-theory and let T h(C) be the set of all true sentences in C. Let At(x) = (∀y)(y ≤ x ⇒ (y = 0 ∨ y = x)) that means that x is an atom. Notice that the following sentences are true in C: We add to the signature the new constants c 1 and c 2 , and we define the first order formula n = (At(c 1 ) ∧ c 2 2 = 0 ∧ c 2 = nc 1 ) (n ∈ Z + ). Let us consider a theory So, in this theory we have terms nc 1 , n = 1, 2, 3, ....

Lemma 3.1 Every finite subtheory T 0 ⊆ T is satisfiable.
Proof T 0 contains finite number of axioms of the kind n 1 , n 2 , ..., n k . Let c 2 be interpreted in the model C as any mc 1 such that m > max{n 1 , ..., n k }.
According to the theorem of compactness, there exists a model M | T , that contains atom, that we denote by c M 1 . The model M has the following properties:

for every natural number n;
Let us take the elements c M 1 , c M 2 ∈ M and consider the subalgebra D generated by these elements, which is isomorphic to C G 2 . Taking into account that we can enrich the signature with n (> 2) constants and the corresponding n (n ∈ Z + ), following the lemma above, as a consequence, we have Proof The proof of the theorem immediately follows from the fact that C G 1 , C G 2 , ..., C G n is a complete list of all n-generated subdirectly irreducible L P G-algebras.

Decidability of the logic L P G
In the sequel Form(L) denotes the set of all formulas of the logic L and Th(L) the set of all theorems of the logic L.
A set X is called recursive (or decidable) if there is an algorithm which, given an object x from the class under consideration, recognizes whether x ∈ X or not. X is said to be recursively enumerable if one of the following equivalent conditions is satisfied: (1) X is the domain of a partial recursive function; (2) X is either the range of a total recursive function or empty.

Proposition 4.1 Suppose Y is a recursive set and X ⊂ Y . Then X is recursive iff both X and Y − X are recursively enumerable.
We enumerate formulas. Every formula in Form(L) may be regarded as a word (a string of symbols) in the alphabet p, , &, ¬, , |, (, ), where | is a symbol for generating subscripts: p 0 is represented as p, p 1 as p|, p 2 as p||, etc. Of course, using two or more special signs instead of | we could write as p||, etc. So, for any finite alphabet, we can effectively determine whether a given string of symbols is a formula. Writing down all possible strings, first of length 1, then of length 2, etc., and discarding those that are not formulas, we can effectively enumerate all formulas in Form(L). Thus we obtain Proposition 4.2 [3]. (1) Form(L) is recursively enumerable (without repetitions).
(2) The set Th(L) of theorems of a logic L with a recursively enumerable set of axioms is also recursively enumerable.
Proof (2). Notice first that every derivation in L may be regarded as a word in the alphabet of L's language with the extra symbol "," used for separating formulas in derivations. So we have a recursive enumeration of L's axioms, say, ϕ 0 , ϕ 1 , ... and a recursive enumeration w 0 , w 1 , w 2 , ... of all words in the alphabet. Now, for every n > 0 we select from w 0 , ..., w n all those derivations in L which use only axioms in the list ϕ 0 , ..., ϕ n (to check whether a formula ψ is a substitution instance of an axiom ϕ, it suffices to write down all the substitution instances of ϕ of length not greater than ones of ψ and compare them with ψ). Since every derivation uses only finitely many axioms (axiom schemes), sooner or later it will be found. Thus we recursively enumerate all the derivations of theorems in L and thereby the theorems L itself.

Proposition 4.3 [5] (Craig's theorem). For every logic L the following conditions are equivalent: (i) L has a recursively enumerable set of axioms; (ii) L has a recursive set of axioms; (iii) Th(L) is recursively enumerable.
Say that an algebra is recursive if its universe is a recursive set and the operations are realized by some algorithms. Thus a recursive algebra may be considered a suitable collection of algorithms. A class of recursive algebras is called recursively enumerable if there is an algorithm enumerating the collections of algorithms corresponding to those algebras. Proposition 4.4 [3]. If theorems of a logic L is characterized by a recursively enumerable class C of recursive algebras then the set of formulas that are not theorems in L is also recursively enumerable.
Using this Proposition we obtain the most general criterion of decidability.

Proposition 4.5 [3]. A logic is decidable iff it is recursively axiomatizable and characterized by a recursively enumerable class of recursive algebras.
Notice that in C G 1 the operation is algorithmically defined.

Theorem 4.1 The logic L P G is decidable.
Proof It is obvious that C G 1 is recursively enumerable and moreover it is recursive. Indeed, we can effectively enumerate the elements of C G 1 in the following way: 0, 1 − 0, c, 1 − c, 2c, 1 − 2c, .... It is also obvious that the operations ⊕, ⊗, * , , 0, 1 are realized by some algorithms. So, the class C G 1 is recursively enumerable. Notice that the variety L P G-algebras, which is the counterpart of the logic L P G, is generated by the L P G-algebra C G 1 so we can conclude that the logic L P G is decidable.

Belluce's functor
On each M V -algebra A, a binary relation ≡ is defined by the following stipulation: Dually we can define a binary relation ≡ * by the following stipulation: x ≡ * y iff supp(x) = supp(y), where supp(x) is defined as the set of all prime filters of A containing the element x. Then, ≡ * is a congruence with respect to ⊗ and ∨. The resulting set β * (A)(= A/ ≡ * ) of equivalence classes is a bounded distributive lattice, which we call also the Belluce lattice of A. Notice, that if some assertion is true for the functor β, then the same is true for the functor β * .
For each x ∈ A let us denote by β * (x) the equivalence class of x. Let f : A → B be an M V -homomorphism. Then β * ( f ) is a lattice homomorphism from β * (A) to β * (B) defined as follows: β * ( f )(β * (x)) = β * ( f (x)). We stress that β * defines a covariant functor from the category of M V -algebras to the category of bounded distributive lattices (see [1]).
We can easily extend the domain of β * on the variety of L P G-algebras. In this case β * becomes covariant functor from L P G to the variety G 2 of G 2 -algebras. Indeed, we can reformulate the Proposition 6 in [10] for G 2 -algebras: Proposition 5.1 [10] Let {A i } i∈I be a family of L P G-algebras. Then

Lemma 5.1 Let A be M V -algebra. If A is a M V -subalgebra of the M V -algebra B and β * (B) is a Heyting lattice, then β * (A) is also a Heyting lattice.
Proof Let ε : A → B be the injective homomorphism corresponding to the subalgebra A of B. Then by [8] (Lemma 13) there exists a strongly isotone surjective morphism P( f ) : P(B) → P(A). Therefore, since β * (B) is a Heyting algebra and P(B) is a Heyting space, P(A) is a Heyting space and hence A is a Heyting algebra.
From this Lemma, we deduce the following Corollary 5.2 Let A be an L P G-algebra. If A is an L P G-subalgebra of the L P Galgebra B, then β * (A) is a G 2 -subalgebra of the G 2 -algebra B.  20)). Adapting this assertion to L P G-algebras we get the proof of the theorem.

A duality
We have defined two contravariant functors: the functor F : G 2 → GS 2 from the category of G 2 -algebras to the category of G 2 -spaces and the functor H : GS 2 → G 2 from the category of G 2 -spaces to the category of G 2 -algebras which establish that these two categories are dually equivalent.
We desire to establish a duality 1 between a class of L P G-algebras and the corresponding category that we name involutive M V -spaces (I MV S-spaces) IMVS. More precisely, we construct the functors P : L P G → IMVS, which is full, and H : IMVS → L P G which is faithful. Notice that the objects of IMVS coincides with the set of objects of the category GS 2 and morphisms are interval mapping.
Let L P G Q = LSP{C G n : n ∈ ω} be the class of the algebras generated by {C G n : n ∈ ω} via the operators of direct product, subalgebras and direct limit. It is clear that L P G Q ⊆ L P G. Notice that the set of the images of the set of objects of the category L P G Q coincides with the set of objects of the category G 2 . Taking into account that G 2 is locally finite and any algebra can be represented as a direct limit of finitely generated subalgebras, we have that G 2 = LSP{β * (C G n ) : n ∈ ω}, where β * (C G n ) is the n-element G 2 -algebra. Adapting the Theorem 16 in [8] it holds Theorem 6.1 [8] (Theorem 16) If R 1 and R 2 are finite root systems (i.e. finite cardinal sum of chains) and f : R 1 → R 2 is an interval map, then there exist L P G-algebras A 1 , A 2 ∈ L P G Q and an L P G-homomorphism h : Theorem 6.2 Let us consider the two categories L P G Q and IMVS. Then there exist contravariant functors P : L P G Q → IMVS and H : IMVS → L P G Q such that H(P(A)) ∼ = A for any object A ∈ L P G Q and P(H(X )) ∼ = X for any object X ∈ IMVS, i.e. the functors P and H are dense.
Moreover, the functor P : L P G Q → IMVS is full, but not faithfull and the functor H : IMVS → L P G Q is faithfull, but not full.
Proof Let A be an algebra in L P G Q . Then A is isomorphic to the direct limit of a direct system of finitely generated subalgebras {A i , ϕ i j }, where A i is a subdirect product of algebras from the family {C G n : n ∈ ω} and ϕ i j : A i → A j is an injective homomorphism, i ≤ j (more precisely A i is a subalgebra of A j ). Identify A with its direct limit. We know that any β * (A i ) is a G 2 -algebra. By [8] (Theorem 11) we also know that β * preserves direct limits, so, β * (A), which is the direct limit of the direct system {β * (A i ), β * (ϕ i j )} of G 2 -algebras, where β * (ϕ i j ) is a G 2 -algebra homomorphism, is also G 2 -algebra. We correspond the L P G-space P(A) = P(β * (A)) to the L P G-algebra A ∈ L P G Q . So, we have contravariant functor P from the category L P G Q to the category of IMVS : P : L P G Q → IMVS.
Let (X , R) be an I MV C-space. So, a G 2 -algebra H(X ), corresponding to the I MV C-space (X , R), is a G 2 -algebra, say G. It is known that the variety of G 2algebras is locally finite. Therefore G is isomorphic to the direct limit of a direct system of finite subalgebras Let us identify G with its direct limit. According to the duality between the category of G 2 -algebras and the category of G 2 -spaces, X = P(G) is the inverse limit of the inverse system {P(G i ), P(ψ i j )}, where P(G i ) is a finite cardinal sum of chains and P(ψ i j ) : P(G j ) → P(G i ) is an interval onto map. Then there exists G 2 -algebras A i ∈ L P G Q such that P(β * (A i )) ∼ = P(G i ) and an injective M V -homomorphism f i j : A i → A j such that β * (A i ) ∼ = G i for every i ∈ I and P(β * ( f i j )) = P(ψ i j ). So, we have a direct system of L P G-algebras {A i , f i j }, where f i j : A i → A j is an injective homomorphism for i ≤ j. Let A be the direct limit of this direct system. Then P(A) ∼ = P(G) ∼ = X .
From the construction of the functors P and H we conclude that H(P(A)) ∼ = A for any object A ∈ L P G Q and P(H(X )) ∼ = X for any object X ∈ IMVS, i.e. the functors P and H are dense.
If we have an interval map f : X 1 → X 2 between the G 2 -spces X 1 and X 2 , then there exist an algebras A 1 , A 2 ∈ L P G Q and an M V -algebra homomorphism h : A 2 → A 1 such that P(A 1 ) = X 1 , P(A 2 ) = X 2 (up to isomorphism) and P(h) : P(A 2 ) → P(A 1 ) is an interval. So, P is full. Now, let us consider two different . So, P is not faithfull. It is obvious that if we have two different morhisms g 1 : X 1 → X 2 and g 1 : X 1 → X 2 , then we have two different L P G-homomorphisms H(g 1 ) : H(X 2 ) → H(X 1 ) and H(g 1 ) : H(X 2 ) → H(X 1 ). So, H is faithfull. For the identity map f : P(C G 1 ) → P(C G 1 ), we have identity L P G-homomorphism from C G 1 to C G 1 . But for non-trivial injective homomorphism h : C G 1 → C G 1 , such that h(c) = 3c, there is no (not identity) interval map g : P(C G 1 ) → P(C G 1 ) such that H(g) = h. So, H is not full.
The category GS 2 of G 2 -spaces is dual equivalent to the category G 2 of G 2algebras, i.e. there exist two functors G 2 : G 2 → GS 2 and H : GS 2 → G 2 . So, we have composition of two contravariant functors H • P : L P G Q → G 2 and H • G 2 : G 2 → L P G Q .
From the above, we have the following

Free L P G-algebras
The set of elements B of an L P G-algebra A which are idempotent with respect to the operations ⊕ or ⊗ are precisely those elements which satisfy the law of the excluded middle with respect to the operations ∨ or ∧.
Theorem 7.1 [11]. Let B be the set of elements x of a G M V -algebra A such that x ⊕ x = x. Then B is closed under the operations ⊕, ⊗ and * where x ⊕ y = x ∨ y, x ⊗ y = x ∧ y and x * = ¬x = x for all x, y ∈ B. Furthermore, the system (B, ⊕, ⊗, * , 0, 1) is the largest subalgebra of A which is at the same time a Boolean algebra with respect to the same operations ⊕, ⊗, * .

Denote by B(A) the largest Boolean subalgebra of the G M V -algebra A.
Theorem 7.2 [11]. Let A be a G M V -algebra and F be a Skolem M V -filter of A.

If h : A → B is a homomorphism from G M V -algebra A to G M V -algebra B, then we have homomorphism B(h) : B(A) → B(B).
It holds Theorem 7.3 [11]. Corollary 7.1 [11].
(i) Any L P G-algebra is a subdirect product of chain L P G-algebras.
(ii) If A is a subdirect product of the family {A/F i } i∈I , then B(A) is a subdirect product of the family {B(A/F i )} i∈I .
(iii) Any L P G-algebra is semi-simple.
Theorem 7.4 [11]. Let F be a proper Skolem M V -filter of an G M V -algebra A, then the following conditions are equivalent: Adapting the Theorem V.1 from [18] for varieties of algebras we have the following Proposition 7.1 If F is a free algebra in K with free generators g 1 , ..., g m ∈ F and satisfy the identity is true in K.
Conversely, let g 1 , ..., g m generate the algebra F such that the identity (1) holds on the elements g 1 , ..., g m ∈ F, the identity (2) is true in K. Then F is a free algebra in K with free generators g 1 , ..., g m ∈ F.

Skolem Duality
Now we describe another dual objects for L P G-algebras. For any L P G-algebra A let SK(A) be the set of maximal M V -Skolem filters of A equipped with spectral topology, i.e. the base of the topology is supp(a) = {F : a ∈ F}, for a ∈ A, and their complements. Notice that for every L P G-algebra A we have a Boolean algebra B(A) (the largest Boolean subalgebra of A) corresponding to it. It is easy to prove that the spaces SK(A) and ST (B(A)) are homeomorphic. Observe that SK(C G n ) and ST (B(C G n )) are one element sets with discrete topology.
The involutive M V -space of the one-generated free L P G-algebra F L P G (1) is depicted in the Fig.3a and the Skolem-space of the one-generated free L P G-algebra F L P G (1) is depicted in the Fig.3b. It is isomorphic to the Stone space of the Boolean algebra B(F L P G (1)). Notice that both spaces are symmetric. Now we will give the description of n-generated free L P G-algebra F L P G (n) for n > 1. Let c ε 0 1 , c ε 1 2 , ..., c ε n n be n-element sequence, where c i ∈ C G i , i = 1, ..., n, ε 1 , ε 2 , ..., ε n is the sequence of 0 and 1 and c ε i i = c i if ε i = 1 and c ε i i = c * i if ε i = 0. The number of such kind of elements is equal to 2 n . We can represent the mentioned sequences as a matrix M depicted below. Notice that c 1 , c 2 , ..., c n are generators of the L P G-algebra C G n . Let us denote the sequence c ε i i of the elements of i-th column of the matrix M by g i , i = 1, ..., n.
Theorem 7.6 The n-generated free L P G-algebra F L P G (n) is isomorphic to (C G 0 ) 2 n × (C G 1 ) 2 n × ... × (C G n ) 2 n with free generators g 1 , ..., g n .
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