Involutive symmetric Gödel spaces, their algebraic duals and logic

1.1 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 \((X,\breve{R})\), where \(\breve{R}\) is the converse relation of R. If we have an algebra A such that A has a lattice order R, then we say that A is symmetric if (A, R) is isomorphic to \((A,\breve{R})\). Consequently, we have that: the classical propositional calculus is symmetric, and its algebraic models are (algebraically) symmetric.

A "symmetric" formulation of intuitionistic propositional calculus, suggested by various authors (e.g. G. Moisil, A. Kuznetsov, C. Rauszer), presupposes that any connective &, \(\vee , \rightharpoonup , \top , \bot \) has its dual \(\vee ,\) &, \(\rightharpoondown ,\) \( \bot , \top \), and the duality principle of the classical logic is restored. J. McKinsey and A. Tarski in [19] introduced the notion of double-Browerian algebras, based on the idea considered by T. Skolem in 1919. In [14] double-Browerian algebras were named Skolem algebras.

We recall further examples of symmetric logic below.

Heyting-Brouwer logic (alias symmetric Intuitionistic logic \(Int^2\)) 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 \(Int^2\) [15, 22].

An algebra \((T,\vee , \wedge , \rightharpoonup , \rightharpoondown , 0,1)\) is a Skolem algebra [13] (or Heyting-Browerian algebra), if \((T,\vee , \wedge , 0,1)\) is a bounded distributive lattice, \(\rightharpoonup \) is an implication (relative pseudo-complement), \(\rightharpoondown \) is a co-implication (relative pseudo-difference) on T.

An algebra \((T, \vee , \wedge , \rightharpoonup , \rightharpoondown , 0,1)\) is said to be a \(G^2\)-algebra (or symmetric Gödel-algebra), if

1. (i)

   \((T,\vee ,\wedge ,\rightharpoonup , 0,1)\) is a G-algebra (or Gödel-algebra), corresponding to Gödel logic;

2. (ii)

   \((T, \vee , \wedge , \rightharpoondown , 0,1)\) is a dual G-algebra (alias Browerian algebra with linearity condition: \((p \rightharpoondown q) \wedge (q \rightharpoondown p) = 0\)).

\(G^2\)-algebras, which are algebraic models of the logical system \(G^2\), represent a proper subclass of Skolem algebras.

Let T be an \(G^2\)-algebra. A subset \(F \subset T\) is said to be a Skolem filter [15, 22], if F is a filter (i.e. \(1\in F\), if \(x\in F\) and \(x\le y\), then \(y\in F\), if \(x, y \in F\), then \(x \wedge y \in F\)) and if \(x\in F\), then \(\lnot \ {_\ulcorner } x\in F\), where \(_\ulcorner a = (\lnot \ a^{*})^{*} = 1 \rightharpoondown a\). The equivalence relation \(x \equiv y \Leftrightarrow (x \rightharpoonup y) \wedge (y \rightharpoonup x) \in F\) is a congruence relation for Skolem algebra of T and there is a lattice isomorphism between the lattices of all congruences of a Skolem algebra and all Skolem filters of the Skolem algebra (see [22]).

Recall that Gödel logic is the extension of Intuitionistic logic Int by the axiom \((\alpha \rightharpoonup \beta ) \vee (\beta \rightharpoonup \alpha )\). 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 \(\alpha \wedge \beta \) consists of a proof of \(\alpha \) and a proof of \(\beta \);

• a proof of a proposition \(\alpha \vee \beta \) is given by presenting either a proof of \(\alpha \) or a proof of \(\beta \);

• a proof of \(\alpha \rightarrow \beta \) is a construction which, given a proof of \(\alpha \), returns a proof of \(\beta \);

• \(\bot \) has no proof and a proof of \(\lnot \alpha \) is a construction which, given a proof of \(\alpha \), would return a proof of \(\bot \).

Brouwer (1932) and Heyting (1956) introduced this interpretation.

1.2 Stone spaces

M. H. Stone in [23] ( [24]) gives a topological representation of Boolean algebras (and, afterward, distributive lattices). More precisely, he has established that any Boolean algebra can be represented as some field of sets that are the base of zero-dimensional compact, Hausdorff topological space. This space afterward was named Stone space (or Boolean space). In other words, any Boolean algebra B has an associated topological space denoted by \({{\mathcal {S}}}{{\mathcal {T}}}(B)\). The points of \({{\mathcal {S}}}{{\mathcal {T}}}(B)\) are ultrafilters on B, or equivalently the homomorphisms from B to the two-element Boolean algebra. The topology of \({{\mathcal {S}}}{{\mathcal {T}}}(B)\) is generated by the clopen (closed and open) basis consisting of all sets of the form \(\{F\in {{\mathcal {S}}}{{\mathcal {T}}}(B): b\in F\}\), where \(b\in B\). Conversely, given any topological space X, the collection of subsets of X that are clopens is a Boolean algebra. Moreover, the category \(\textbf{BOOL}\) of Boolean algebras and Boolean homomorphisms (as morphisms) is dually equivalent to the category of Stone spaces and continuous maps (as morphisms). There exists contravariant functor \({\mathcal {S}}\) from the Boolean algebras category to the Stone spaces category.

Notice that any Stone space is symmetric.

1.3 Priestley spaces

A Priestley space is a triple \((X,R,\Omega )\), where \((X,\Omega )\) is a Stone space and R is an order relation on X such that, for all \(x, y \in X\) with \(x{\bar{R}}y\), there exists a clopen up-set V with \(x\in V\) and \(y\notin V\). A morphism between Priestley spaces is a continuous order-preserving map. We denote the category of Priestley spaces plus continuous order-preserving maps by \({{\mathcal {P}}}{{\mathcal {S}}}\). 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 \({\mathcal {F}}(L)\) of prime filters, and mapping each Priestley space X to the bounded distributive lattice of clopen up-sets of X. Notice that there exist distributive lattices and Priestley spaces that are not symmetric.

1.4 Heyting spaces and Gödel spaces

A Heyting space (or Esakia space) X is a Priestley space such that \(R^{-1}(U)\) is open for every open subset U of X. Recall that \(R^{-1}(U)= \{ y \in X: (\exists u \in U)\ y R u\}\) and that \(R^{-1}(\{x\})\) is abbreviated to \(R^{-1}(x)\). The sets R(U) and R(x) are defined dually. More precisely, (X, R) is Heyting space [13, 15] if

1. (1)

   X is a Stone space,

2. (2)

   R(x) is closed for every \(x\in X\),

3. (3)

   \(R^{-1}(U)\) is open for every open subset U of X,

4. (4)

   \(R(cl U) =cl R(U)\) for every subset U of X, where cl is a closure operator of the topological space X.

A morphism between Heyting spaces, called a strongly isotone map (or Heyting morphism in other terminology), is a continuous map \(\varphi : X \rightarrow Y\) such that \(\varphi (R(x)) = R(\varphi (x))\) for all \(x\in X\).

The restricted Priestley duality for Heyting algebras states that a bounded distributive lattice A is the underlying lattice of a Heyting algebra if and only if the Priestley dual of A is a Heyting space, and a \(\{0,1\}\)-lattice homomorphism h between Heyting algebras preserves the operation \(\rightharpoonup \) if and only if the Priestley dual of h is a Heyting morphism. We denote the category of Heyting spaces plus Heyting morphisms by \(\mathcal {{HS}}\).

If Heyting algebra satisfies the identity \((x \rightharpoonup y) \vee (y \rightharpoonup x) = 1\), 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.

1.5 Symmetric heyting and Gödel spaces

A Heyting space (X, R) is symmetric [14] if \((X,\breve{R})\) is also Heyting space, where \(xRy \Leftrightarrow y\breve{R}x\). Heyting space is symmetric iff \(R^{-1}(cl U) =cl R^{-1}(U)\) [14]. Let (X, R) and \((X',R')\) be symmetric Heyting spaces. a map \(f: X \rightarrow X'\) is said to be interval [14] if \(xR' f(z) R' y \Leftrightarrow (\exists x', y') (x' R z R y', f(z) = x, f(y') = y)\). Denote the category of Skolem algebras and Skolem homomorphism by \({{\textbf {H}}}{{\textbf {A}}}^2\) and the category of symmetric Heyting spaces and interval maps by \({{\mathcal {H}}}{{\mathcal {S}}}^2\).

For any symmetric Heyting space (X, R) and \(U,V\in {{\mathcal {H}}}(X)\)(= the set of all clopen up-sets of X) define:

Then the algebra \({{\mathcal {H}}}((X,R))= ({{\mathcal {H}}}(X),\vee ,\wedge ,\rightharpoonup , \rightharpoondown , \emptyset ,X)\) is a Skolem algebra [14]. Furthermore, for any morphism \(f:(X_1,R_1)\rightarrow (X_2,R_2)\) in \({{\mathcal {H}}}{{\mathcal {S}}}^2\), \({{\mathcal {H}}}(f)= f^{-1}\) is a Skolem algebra homomorphism from \({{\mathcal {H}}}((X_2,R_2))\) into \({{\mathcal {H}}}(X_1,R_1)\). On the other hand, for each Skolem algebra A, the set \({{\mathcal {F}}}(A)\) 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 \(supp(a)=\{F\in {{\mathcal {F}}}(A):a\in F\}\), for \(a\in A\), and their complements as a subbase, is an object of \({{\mathcal {H}}}{{\mathcal {S}}}^2\); and for each Skolem algebra homomorphism \(h:A\rightarrow B\), \({{\mathcal {F}}}(h)=h^{-1}\) is a morphism of \({{\mathcal {H}}}{{\mathcal {S}}}^2\). Therefore we have two contravariant functors \({{\mathcal {F}}}: \textbf{HA}^2\rightarrow {{\mathcal {H}}}{{\mathcal {S}}}^2\) and \({{\mathcal {H}}}: {{\mathcal {H}}}{{\mathcal {S}}}^2\rightarrow \textbf{HA}^2\). These functors establish a dual equivalence between the categories \(\textbf{HA}^2\) and \({{\mathcal {H}}}{{\mathcal {S}}}^2\) [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 \(\vartheta (T)\) 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: \((a \rightharpoonup b) \vee (b \rightharpoonup a) = 1, \ (a \rightharpoondown b) \wedge (b \rightharpoondown a) = 0\) for all \(a,b \in A\). \(G^2\)-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 \(x\in X\). For a \(G^2\)-algebra A we can assert that its \(G^2\)-space (or symmetric Gödel space) \({\mathcal {H}}(A)\) is a root system in both directions, i.e. R(x) and \(R^{-1}(x)\) are chains for every \(x \in {\mathcal {H}}(A)\). So, it holds

Theorem 1.1

The \(G^2\)-space of any \(G^2\)-algebra A is a cardinal sum of chain \(G^2\)-spaces.

Moreover, we have

Theorem 1.2

There exist two contravariant functors \({{\mathcal {F}}}: \textbf{G}^2\rightarrow {{\mathcal {G}}}{{\mathcal {S}}}^2\) and \({{\mathcal {H}}}: {{\mathcal {G}}}{{\mathcal {S}}}^2\rightarrow \textbf{G}^2\). These functors establish a dual equivalence between the categories \(\textbf{G}^2\) of \(G^2\)-algebras and \({{\mathcal {G}}}{{\mathcal {S}}}^2\) of \(G^2\)-spaces (symmetric Gödel spaces).

Proof

The proof immediately follows from the fact that \(\textbf{G}^2\) is a full subcategory of \(\textbf{HA}^2\). \(\square \)

If \(f: A \rightarrow B\) is an injective homomorphism between \(G^2\)-algebras A and B, then \({{\mathcal {F}}}(f): {{\mathcal {F}}}(B) \rightarrow {{\mathcal {F}}}(A)\) is a surjective interval mapping. If A and B are Heyting algebras we have a partition of \({{\mathcal {F}}}(B)\) by \(ker({{\mathcal {F}}}(f))^{-1}\) on closed classes, which gives the corresponding equivalence relation, say E, on \({{\mathcal {F}}}(B)\), i.e. E is an equivalence relation on \({{\mathcal {F}}}(B)\) [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. (1)

   E is a closed equivalence relation, i.e. the E-saturation^{Footnote 1} of any closed subset is closed;

2. (2)

   the E-saturation of any bicone is a bicone;

3. (3)

   \((\forall x, y \in Y)(E(x) \cap R^{-1}(E(y)) \ne \emptyset \Rightarrow E(x) \subseteq R^{-1}(E(y))\); \((\forall x, y \in Y)(E(x) \cap R(E(y)) \ne \emptyset \Rightarrow E(x) \subseteq R(E(y))\);

4. (4)

   there is a \(G^2\)-space (Z, Q) and an interval map \(f : Y \rightarrow Z\) 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: \(E(\{g,h\}) = \{d,e,g,h\}\) that is not a bicone.

Observe that symmetric Gödel algebras and symmetric Gödel spaces are symmetric.

1.6 Looking for symmetry in Łukasiewicz sentential calculus

MV-algebras are the algebraic counterpart of the infinite-valued Łukasiewciz sentential calculus, as Boolean algebras are for the classical propositional logic. In contrast with what happens for Boolean algebras, there are MV-algebras that are not semisimple, i.e. the intersection of their maximal ideals Rad(A) (the radical of A) is different from \(\{0\}\). Non-zero elements from the radical of A are called infinitesimals. Perfect MV-algebras are those MV-algebras generated by their infinitesimal elements or, equivalently, generated by their radical [2]. Let \(L_P\) be the logic of perfect MV-algebras which coincides with the set of all Łukasiewicz formulas that are valid in all perfect MV-chains.

As it is well known, MV-algebras form a category which is equivalent to the category of abelian lattice ordered groups (\(\ell \)-groups, for short) with a strong unit [20]. Let us denote by \(\Gamma \) the functor implementing this equivalence. In particular, each perfect MV-algebra is associated with an abelian \(\ell \)-group with a strong unit. Moreover, the category of perfect MV-algebras is equivalent to the category of abelian \(\ell \)-groups, see [12].

The class of perfect MV-algebras does not form a variety and contains non-simple subdirectly irreducible MV-algebras. It is worth stressing that the variety generated by all perfect MV-algebras, denoted by \(\mathbf {MV(C)}\), is also generated by a single MV-chain, the MV-algebra C defined by Chang in [4]. We name MV(C)-algebras the algebras from the variety generated by C. Notice that the Lindenbaum algebra of \(L_P\) is an MV(C)-algebra (a detailed description of free MV(C)-algebras is presented in [10]).

Let us define \(C_m\) as follows:

with generator \((0,1) = c_1 = c\)

with generators \(c_1 (= (0,0,...,1)), ... ,c_m (=(0,1,...,0))\), where \(\times _{lex}\) is the lexicographic product and the number of factors Z is equal to \(m+1\) (with \(m>1\)).

Let us denote \(Rad(A) \cup \lnot Rad (A)\) by \(Rad^*(A)\), where Rad(A) is the radical of A (i.e. the intersection of all maximal ideals) and \(\lnot Rad (A)\) is the intersection of all maximal filters, in other words \(\lnot Rad (A) = \{\lnot x: x\in Rad (A)\}\).

In Fig. 2 is depicted (b), the perfect MV-algebra \(Rad^*(C^2)\), with its spectral space (a). Notice that \(Rad^*(C^2)\) as a lattice is not a Heyting lattice because there is no Heyting implication \((c,0) \rightharpoonup (0,0)\). 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.

MV-algebras and Gödel algebras are both distributive lattices. In Gödel algebras for the lattice operation \(\wedge \) there exists its adjoint operation, Heyting implication \(\rightarrow _G\). 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 \(\wedge \). Analogically, there exists an MV-algebra where we have no adjoint operation for \(\vee \). Distributive lattices where exists adjoint operation \(\rightarrow _{Br}\) for \(\vee \) form Brouwerian algebras. Distributive lattices where exist both Heyting implication \(\rightarrow _G\) and Brouwerian implication \(\rightarrow _{Br}\) are named Heyting-Brouwerian algebras (C. Rauszer, Esakia). We are interested in the MV-algebras’ distributive lattice which contains both Heyting implication \(\rightarrow _G\) and Brouwerian implication \(\rightarrow _{Br}\) besides MV-algebra implication \((x^*\oplus y)\) and its dual \((x \otimes y^*)\). In other words, we have a full symmetric case, where MV-negation transforms each implication into its dual like in the Boolean case. So, if we add to the MV-algebras signature the Heyting implication \(\rightarrow _G\), we automatically obtain Brouwerian implication \(\rightarrow _{Br}\).

An algebra \((A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\) is called \(L_PG\)-algebra if \((A, \otimes , \oplus , *, 0, 1)\) is \(L_P\)-algebra (i.e. an algebra from the variety generated by perfect MV-algebras [12]) and \((A, \vee , \wedge , \rightharpoonup , 0, 1)\) is a Gödel algebra, i.e. Heyting algebra satisfying the identity \((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1\). In other words we have a fusion of two algebras, an MV-algebra and a Gödel algebra. Moreover, notice that the variety \(\mathbf {L_PG}\) of \(L_PG\)-algebras is the subvariety of the variety \(\textbf{GMV}\) [11] of GMV-algebras axiomatized by the perfect MV-algebra identity \(2x^2 = (2x)^2\). Taking into account that we can also express the Łukasiewicz implication \(x \rightarrow y = x^{*} \oplus y\), hence we have two distinct residuations.

More precisely, an \(L_PG\)-algebra is an algebra \((A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\) satisfying the following identities:

1. (1)

   \((x \oplus y) \oplus z = x \oplus (y \oplus z)\); 10) \(x \vee y = (x \otimes y^{*}) \oplus y\);

2. (2)

   \(x \oplus y = y \oplus x\); 11) \(x \wedge y = (x \oplus y^{*}) \otimes y\);

3. (3)

   \(x \oplus 0 = x\); 12) \((x \rightharpoonup y ) \wedge y = y\);

4. (4)

   \(x \oplus 1 = 1\);                        13) \(x \wedge (x \rightharpoonup y ) = x \wedge y\);

5. (5)

   \(0^{*} = 1\); 14) \(x \rightharpoonup (y \wedge z) = (x \rightharpoonup y ) \wedge (x \rightharpoonup z)\);

6. (6)

   \(1^{*} = 0\); 15) \((x \vee y) \rightharpoonup z = (x \rightharpoonup z ) \wedge (y \rightharpoonup z)\);

7. (7)

   \(x \otimes y = (x^{*} \oplus y^{*})^{*}\); 16) \((x \rightharpoonup 0)^{*} \le ((x \rightharpoonup 0)\rightharpoonup 0)\);

8. (8)

   \((x^{*} \oplus y)^{*} \oplus y = (y^{*} \oplus x)^{*} \oplus x\); 17) \((x \rightharpoonup y) \le (x^{*} \oplus y)\).

9. (9)

   \(2x^2 = (2x)^2\);

The axioms (1)– 11) are the MV(C)-algebras axioms, including the perfect algebras identity (9), the axioms (12)–(15) are the Heyting algebras axioms, the axioms (16–(17) are properties connecting the Heyting negation \(x \rightharpoonup 0 \ (= \lnot x)\) with the MV-negation \(x^{*}\) and the Heyting implication with the MV-implication, respectively.

Theorem 2.1

[11]. Let \((A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\) be an \(L_PG\)-algebra. Then A is a bi-Heyting (Heyting-Browerian) algebra, where the pseudo-difference \(b \rightharpoondown a = (a^{*} \rightharpoonup b^{*})^{*}\) and \(_\ulcorner a = (\lnot \ a^{*})^{*} = 1 \rightharpoondown a\).

Proof

The variety generated by perfect MV-algebras is generated by chain perfect MV-algebras. In any chain perfect MV-algebras \((a^{*} \rightharpoonup b^{*})^{*}\) is a co-implication (relative pseudo-difference) and \((\lnot \ a^{*})^{*} \) is co-negation. \(\square \)

Let \((A, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\) be an \(L_PG\)-algebra. A subset \(F \subset A\) is said to be a Skolem MV-filter if: (a) F is an MV-filter, i.e. \(1\in F\), if \(x\in F\) and \(x\le y\), then \(y\in F\), if \(x, y \in F\), then \(x \otimes y \in F\) and (b) if \(x\in F\), then \(\lnot \ {_\ulcorner } x\in F\).

Theorem 2.2

[11]. Let F be an MV-Skolem filter of an \(L_PG\)-algebra A. The equivalence relation \(x \equiv y \Leftrightarrow (x \rightharpoonup y) \wedge (y \rightharpoonup x) \in F\) is a congruence relation on the \(L_PG\)-algebra of A.

Proof

\(x \equiv y\) is a congruence because F is an MV-filter, since it is a congruence with respect to MV-algebra operations, and, at the same time, it is a Skolem filter. \(\square \)

In section 1.6 we define the MV(C)-algebras \(C_i, \ \ i\in Z^+\), where \(C_1 (= C)\) is the Chang algebra. Now we enrich the signature of the algebras \(C_i\) (\(i\in Z^+\)) by the Heyting operations \(\vee , \wedge , \rightharpoonup \) and denote them by \(C_i^G = (C_i^G, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\). Now we define the algebra \(C_1^G\) similar to Chang.

by the following operations (consider \(0 = 0{{\textbf {c}}}\)):

Notice that in any bounded chain the Heyting implication is defined in the same manner: \(x \rightharpoonup y = 1\) if \(x \le y\) and \(x \rightharpoonup y = y\) if \(x > y\).

The formulas of the logic GL are built up by means of the propositional variables \(p_1, p_2, ...\), logical connectives \(\rightarrow , \rightharpoonup , \sim \) in the usual way. We have some abbreviations : \(\bot = \sim (\alpha \rightarrow \alpha ), \ \lnot \alpha = \alpha \rightharpoonup \bot , \ _\ulcorner = \sim (\lnot \sim \alpha ), \ \alpha \rightharpoondown \beta = \sim (\sim \beta \rightharpoonup \sim \alpha ), \alpha \wedge \beta = \lnot ((\lnot \alpha \rightarrow \lnot \beta )\rightarrow \lnot \beta ), \ \alpha \vee \beta = \lnot (\lnot \alpha \wedge \lnot \beta ), \alpha \leftrightarrow \beta = (\alpha \rightarrow \beta ) \wedge (\beta \rightarrow \alpha )\). Notice that the \(L_PG\)-logic is the extension of the logic GMV [11] by the axiom \( (\alpha {\underline{\vee }} \alpha ) \& (\alpha {\underline{\vee }} \alpha ) \leftrightarrow (\alpha \& \alpha ) {\underline{\vee }} (\alpha \& \alpha )\). The axioms are

the axioms of the Łukasiewicz logicŁ

1. (L1)

   \(\alpha \rightarrow (\beta \rightarrow \alpha )\),

2. (L2)

   \((\alpha \rightarrow \beta ) \rightarrow ((\beta \rightarrow \gamma ) \rightarrow (\alpha \rightarrow \gamma ))\),

3. (L3)

   \(((\alpha \rightarrow \beta ) \rightarrow \beta )\rightarrow ((\beta \rightarrow \alpha ) \rightarrow \alpha )\),

4. (L4)

   \((\lnot \alpha \rightarrow \lnot \beta ) \rightarrow (\beta \rightarrow \alpha )\),

the axioms of the Gödel logic

1. (G1)

   \(\alpha \rightharpoonup (\beta \rightharpoonup \alpha )\),

2. (G2)

   \((\alpha \rightharpoonup (\beta \rightharpoonup \gamma )) \rightharpoonup ((\alpha \rightharpoonup \beta ) \rightharpoonup (\alpha \rightharpoonup \gamma ))\),

3. (G3)

   \((\alpha \wedge \beta ) \rightharpoonup \alpha \),

4. (G4)

   \((\alpha \wedge \beta ) \rightharpoonup \beta \),

5. (G5)

   \(\alpha \rightharpoonup (\beta \rightharpoonup (\alpha \wedge \beta ))\),

6. (G6)

   \(\alpha \rightharpoonup (\alpha \vee \beta )\),

7. (G7)

   \(\beta \rightharpoonup (\alpha \vee \vee \beta )\),

8. (G8)

   \((\alpha \rightharpoonup \gamma ) \rightharpoonup ((\beta \rightharpoonup \gamma ) \rightharpoonup ((\alpha \vee \beta ) \rightharpoonup \gamma ))\),

9. (G9)

   \(\bot \rightharpoonup \alpha \)

10. (G10)

    \((\alpha \rightharpoonup \beta ) \vee (\beta \rightharpoonup \alpha )\),

axioms connecting Łukasiewicz and Gödel logics

1. (GL1)

   \(\sim \lnot \alpha \rightarrow \lnot \lnot \alpha \),

2. (GL2)

   \((\alpha \rightharpoonup \beta ) \rightarrow ( \alpha \rightarrow \beta \)),

3. (GL3)

   \(\alpha \vee \sim \lnot _\ulcorner \alpha \),

4. (GL4)

   \(\lnot _\ulcorner \alpha \rightarrow \alpha \),

5. (GL5)

   \((\sim \lnot \alpha \rightarrow \lnot \alpha ) \leftrightarrow \lnot \alpha \),

6. (GL6)

   \((\sim _\ulcorner \alpha \rightarrow _\ulcorner \alpha ) \leftrightarrow _\ulcorner \alpha \),

and the axiom corresponding to perfectness

(\(L_PG\)) \((\alpha {\underline{\vee }} \alpha )\) & \((\alpha {\underline{\vee }} \alpha ) \leftrightarrow (\alpha \)& \(\alpha ) {\underline{\vee }} (\alpha \) & \(\alpha )\),

where \({\underline{\vee }}\) is the strong disjunction and & is the strong conjunction defined as follows: \(\alpha \veebar \beta = \sim \alpha \rightarrow \beta , \ \alpha \) & \(\beta = \sim (\sim \alpha \veebar \sim \beta \)). Inference rule modus ponens: \(\alpha , \alpha \rightarrow \beta \Rightarrow \beta \). From the axiom(GL2) it follows Gödel logic modus ponens: \(\alpha , \alpha \rightharpoonup \beta \Rightarrow \beta \). 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 \(L_PG\): \(\alpha \equiv \beta \) iff the formulas \(\alpha \leftrightarrow \beta \) and \(\sim \lnot _\ulcorner \alpha \leftrightarrow \sim \lnot _\ulcorner \beta \). Taking into account that MV-algebras and Gödel algebras axioms correspond to Łukasiewicz logic and Gödel logic axioms, respectively, and \(L_PG\)-logic axioms GL1 - GL6 are the translation of \(L_PG\)-algebras axioms to the logical ones and the same for the axiom (\(L_PG\)), it holds

Theorem 2.3

The Lindenbaum algebra of the logic \(L_PG\) is an \(L_PG\)-algebra.

Theorem 2.4

(Completeness theorem). For any formula \(\alpha \), \(\alpha \) is a theorem of the logic \(L_PG\) iff \(\alpha \) is a tautology.

Proof

It is obvious that if \(\alpha \) is a theorem, then \(\alpha \) is a tautology. Let us suppose that \(\alpha \) is not a theorem. Then \(\alpha /\equiv \ \ne 1\) in the Lindenbaum algebra \({\mathfrak {A}}(L)\), but the Lindenbaum algebra \({\mathfrak {A}}(L)\) is isomorphic to a subdirect product of chain \(L_PG\)-algebras. Hence the element \(\alpha /\equiv \) can be represented as a sequence \((x_1, ... ,x_i, ...)\), where the elements of this sequence belong to \({\mathfrak {A}}(L)\) and \(x_i \ne 1\) for some i. Let h be a natural homomorphism from the algebra of formulas \(F_L\) onto \({\mathfrak {A}}(L)\), i.e. \(h(\alpha ) = \alpha /\equiv \) for every \(\alpha \in F_L\). Then we can consider the map \(g= \pi _i \circ h\) where \(\pi _i\) is a projection of \({\mathfrak {A}}(L)\) onto i-th chain component D of the subdirect product of chain \(L_PG\)-algebras g is a value function of the algebra D such that \(g(\alpha ) \ne 1\). Therefore \(\alpha \) is not a tautology. \(\square \)

Notice that if A is a finitely generated MV-algebra, then A is subdirectly irreducible iff A is a chain. Similarly, if A is an \(L_PG\)-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 \(\mathbf {L_PG}\). More precisely, we will show that the variety generated by \(C_1^G\) coincides with the variety generated by \(C_n^G\). We prove this assertion for \(n =2\) which is easy generalized for any positive integer that is greater than 2.

Let \({\mathbb {C}} =(C_1^G, =, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\) be the model for \(L_PG\)-theory and let \(Th({\mathbb {C}})\) be the set of all true sentences in \({\mathbb {C}}\). Let \(At(x) = (\forall y)(y \le x \Rightarrow (y = 0 \vee y = x))\) that means that x is an atom. Notice that the following sentences are true in \({\mathbb {C}}\):

1. (i)

   \((\forall x \forall y)(At(x) \wedge At(y) \Rightarrow x=y)\);

2. (ii)

   \((\forall x \forall y)((x \le y) \vee (y \le x))\);

3. (iii)

   \((\forall x \exists y)(x \ne 1 \wedge x< y \wedge \exists z((x< z \wedge z < y) \Rightarrow x=z \vee z=y))\).

We add to the signature the new constants \(c_1\) and \(c_2\), and we define the first order formula \(\Lambda _n = (At(c_1) \ \wedge \ c_2^2 =0 \ \wedge \ c_2 \ne n c_1 )\)\((n\in Z^+)\). Let us consider a theory

So, in this theory we have terms \(nc_1, n = 1,2,3, ...\).

Lemma 3.1

Every finite subtheory \(T_0 \subseteq T\) is satisfiable.

Proof

\(T_0\) contains finite number of axioms of the kind \(\Lambda _{n_1}, \Lambda _{n_2}, ... , \Lambda _{n_k}\). Let \(c_2\) be interpreted in the model \({\mathbb {C}}\) as any \(mc_1\) such that \(m > max\{n_1,...,n_k\}\).

According to the theorem of compactness, there exists a model \({\mathbb {M}}\models T\), that contains atom, that we denote by \(c_1^M\). The model \({\mathbb {M}}\) has the following properties:

1. 1)

   \({\mathbb {C}}\) is embedding into \({\mathbb {M}}\): \(\varepsilon (c_1) = c_1^M\);

2. 2)

   \({\mathbb {M}}\models Th({\mathbb {C}})\);

3. 3)

   \({\mathbb {M}} \ncong {\mathbb {C}}\), in particular \(c_2^M \in M\) such that \(c_2^M \ge nc_1^M\) for every natural number n;

\(\square \)

Let us take the elements \(c_1^M, c_2^M \in M\) and consider the subalgebra D generated by these elements, which is isomorphic to \(C_2^G\).

Taking into account that we can enrich the signature with \(n \ (> 2)\) constants and the corresponding \(\Lambda _n (n\in Z^+)\), following the lemma above, as a consequence, we have

Lemma 3.2

The variety generated by \(C_1^G\) coincides with the variety generated by \(C_n^G\) for any \(n \in Z^+\).

Theorem 3.1

The variety \(\mathbf {L_PG}\) is generated by the \(L_PG\)-algebra \(C_1^G\).

Proof

The proof of the theorem immediately follows from the fact that \(C_1^G, C_2^G, ... , C_n^G\) is a complete list of all n-generated subdirectly irreducible \(L_PG\)-algebras. \(\square \)

In the sequel \({\textbf {Form}}(L)\) denotes the set of all formulas of the logic L and \({{\textbf {T}}}{{\textbf {h}}}(L)\) 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 \(x\in X\) or not. X is said to be recursively enumerable if one of the following equivalent conditions is satisfied:

1. (1)

   X is the domain of a partial recursive function;

2. (2)

   X is either the range of a total recursive function or empty.

Proposition 4.1

Suppose Y is a recursive set and \(X \subset Y\). Then X is recursive iff both X and \(Y - X\) are recursively enumerable.

We enumerate formulas. Every formula in \({\textbf {Form}}(L)\) may be regarded as a word (a string of symbols) in the alphabet \( p, \veebar , \& , \lnot , \rightharpoondown , |, (, )\), 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 \({\textbf {Form}}(L)\). Thus we obtain

Proposition 4.2

[3]. (1) \({\textbf {Form}}(L)\) is recursively enumerable (without repetitions).

(2) The set \({{\textbf {T}}}{{\textbf {h}}}(L)\) 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, \(\varphi _0, \varphi _1, ... \) 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 \(\varphi _0, ... , \varphi _n\) (to check whether a formula \(\psi \) is a substitution instance of an axiom \(\varphi \), it suffices to write down all the substitution instances of \(\varphi \) of length not greater than ones of \(\psi \) and compare them with \(\psi \)). 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. \(\square \)

Proposition 4.3

[5] (Craig’s theorem). For every logic L the following conditions are equivalent:

1. (i)

   L has a recursively enumerable set of axioms;

2. (ii)

   L has a recursive set of axioms;

3. (iii)

   \({{\textbf {T}}}{{\textbf {h}}}(L)\) 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 \({\mathcal {C}}\) 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 \(C_1^G\) the operation \(\rightharpoonup \) is algorithmically defined.

Theorem 4.1

The logic \(L_PG\) is decidable.

Proof

It is obvious that \(C_1^G\) is recursively enumerable and moreover it is recursive. Indeed, we can effectively enumerate the elements of \(C_1^G\) in the following way: \(0, 1- 0, c, 1- c, 2c, 1- 2c, ... \). It is also obvious that the operations \(\oplus ,\otimes , *, \rightharpoonup , 0, 1\) are realized by some algorithms. So, the class \(C_1^G\) is recursively enumerable. Notice that the variety \(L_PG\)-algebras, which is the counterpart of the logic \(L_PG\), is generated by the \(L_PG\)-algebra \(C_1^G\) so we can conclude that the logic \(L_PG\) is decidable. \(\square \)

On each MV-algebra A, a binary relation \(\equiv \) is defined by the following stipulation: \(x \equiv y\) iff \(supp(x) = supp(y)\), where supp(x) is defined as the set of all prime ideals of A not containing the element x. As proved in [1], \(\equiv \) is a congruence with respect to \(\oplus \) and \(\wedge \). The resulting set \(\beta (A)(=A/ \equiv )\) of equivalence classes is a bounded distributive lattice, called the Belluce lattice of A. For each \(x \in A\) let us denote by \(\beta (x)\) the equivalence class of x. Let \(f : A \rightarrow B\) be an MV-homomorphism. Then \(\beta (f)\) is a lattice homomorphism from \(\beta (A)\) to \(\beta (B)\) defined as follows: \(\beta (f)(\beta (x)) = \beta (f(x))\). We stress that \(\beta \) defines a covariant functor from the category of MV-algebras to the category of bounded distributive lattices (see [1]). In [1] (Theorem 20) it is proved that \({\mathcal {M}} (A)\) and \({\mathcal {P}} (\beta (A))\) are homeomorphic, where \({\mathcal {M}} (A)\) is the MV-space (= the space of prime MV-filters) of the MV-algebra A.

Dually we can define a binary relation \(\equiv ^{*}\) by the following stipulation: \(x \equiv ^{*} y\) iff \(supp(x) = supp(y)\), where supp(x) is defined as the set of all prime filters of A containing the element x. Then, \(\equiv ^{*}\) is a congruence with respect to \(\otimes \) and \(\vee \). The resulting set \(\beta ^{*}(A)(=A/ \equiv ^{*})\) 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 \(\beta \), then the same is true for the functor \(\beta ^{*}\).

For each \(x \in A\) let us denote by \(\beta ^{*}(x)\) the equivalence class of x. Let \(f : A \rightarrow B\) be an MV-homomorphism. Then \(\beta ^{*}(f)\) is a lattice homomorphism from \(\beta ^{*}(A)\) to \(\beta ^{*}(B)\) defined as follows: \(\beta ^{*}(f)(\beta ^{*}(x)) = \beta ^{*}(f(x))\). We stress that \(\beta ^{*}\) defines a covariant functor from the category of MV-algebras to the category of bounded distributive lattices (see [1]).

We can easily extend the domain of \(\beta ^{*}\) on the variety of \(L_PG\)-algebras. In this case \(\beta ^{*}\) becomes covariant functor from \(\mathbf {L_PG}\) to the variety \(\mathbf {G^2}\) of \(G^2\)-algebras. Indeed, we can reformulate the Proposition 6 in [10] for \(G^2\)-algebras:

Proposition 5.1

[10] Let \(\{A_i\}_{i\in I}\) be a family of \(L_PG\)-algebras. Then

Corollary 5.1

Let \(\{A_i\}_{i\in I}\) be a family of \(L_PG\)-algebras. If \(\beta ^{*}(A_i)\) is a \(G^2\)algebra, then \(\beta ^{*} (\prod _{i\in I} A_i)\) is also \(G^2\)-algebra.

Proof

Since \(\beta ^{*}(A_i)\) (\(i\in I\)) is a \(G^2\)-algebra, we have that \(\beta ^{*} (\prod _{i\in I} A_i) \ ( \cong \prod _{i\in I} \beta ^{*}(A_i))\) is also \(G^2\)-algebra. \(\square \)

Lemma 5.1

Let A be MV-algebra. If A is a MV-subalgebra of the MV-algebra B and \(\beta ^{*}(B)\) is a Heyting lattice, then \(\beta ^{*}(A)\) is also a Heyting lattice.

Proof

Let \(\varepsilon : A \rightarrow B\) be the injective homomorphism corresponding to the subalgebra A of B. Then by [8] (Lemma 13) there exists a strongly isotone surjective morphism \({\mathcal {P}}(f):{\mathcal {P}}(B) \rightarrow {\mathcal {P}}(A)\). Therefore, since \(\beta ^{*}(B)\) is a Heyting algebra and \({\mathcal {P}}(B)\) is a Heyting space, \({\mathcal {P}}(A)\) is a Heyting space and hence A is a Heyting algebra. \(\square \)

From this Lemma, we deduce the following

Corollary 5.2

Let A be an \(L_PG\)-algebra. If A is an \(L_PG\)-subalgebra of the \(L_PG\)-algebra B, then \(\beta ^{*}(A)\) is a \(G^2\)-subalgebra of the \(G^2\)-algebra B.

Theorem 5.1

Let A be an \(L_PG\)-algebra. Then \({\mathcal {M}} (A)\) and \({\mathcal {P}} (\beta ^{*}(A))\) are homeomorphic.

Proof

For any MV-algebra A the MV-space \({\mathcal {M}} (A)\) and the Priestley space \({\mathcal {P}} (\beta ^{*}(A))\) are homeomorphic ( [1] (Theorem 20)). Adapting this assertion to \(L_PG\)-algebras we get the proof of the theorem. \(\square \)

We have defined two contravariant functors: the functor \({\mathcal {F}}: \mathbf {G^2} \rightarrow {{\mathcal {G}}}{{\mathcal {S}}}^2\) from the category of \(G^2\)-algebras to the category of \(G^2\)-spaces and the functor \({\mathcal {H}}: {{\mathcal {G}}}{{\mathcal {S}}}^2 \rightarrow \mathbf {G^2}\) 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^{Footnote 2} between a class of \(L_PG\)-algebras and the corresponding category that we name involutive MV-spaces (IMVS-spaces) \(\textbf{IMVS}\). More precisely, we construct the functors \({\mathcal {P}}: \mathbf {L_PG} \rightarrow \textbf{IMVS}\), which is full, and \({\mathcal {H}}: \textbf{IMVS} \rightarrow \mathbf {L_PG}\) which is faithful. Notice that the objects of \(\textbf{IMVS}\) coincides with the set of objects of the category \(\mathcal {{{\mathcal {G}}}{{\mathcal {S}}}}^2\) and morphisms are interval mapping.

Let \(\mathbf {L_PG^Q} = {\textbf {{LSP}}}\{C_n^G: n\in \omega \}\) be the class of the algebras generated by \(\{C_n^G: n\in \omega \}\) via the operators of direct product, subalgebras and direct limit. It is clear that \(\mathbf {L_PG^Q} \subseteq \mathbf {L_PG}\). Notice that the set of the images of the set of objects of the category \(\mathbf {L_PG^Q}\) coincides with the set of objects of the category \(\mathbf {G^2}\). Taking into account that \(\mathbf {G^2}\) is locally finite and any algebra can be represented as a direct limit of finitely generated subalgebras, we have that \(\mathbf {G^2} = \mathbf {{LSP}}\{\beta ^{*}(C_n^G): n\in \omega \}\), where \(\beta ^{*}(C_n^G)\) 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 \(f : R_1\rightarrow R_2\) is an interval map, then there exist \(L_PG\)-algebras \(A_1, A_2 \in \mathbf {L_PG^Q}\) and an \(L_PG\)-homomorphism \(h : A_1\rightarrow A_2\) such that Spec \({\mathcal {P}}(A_i)\cong R_i\) \(i = 1,2\).

Theorem 6.2

Let us consider the two categories \(\mathbf {L_PG^Q}\) and \(\textbf{IMVS}\). Then there exist contravariant functors \({\mathcal {P}}: \mathbf {L_PG^Q} \rightarrow \textbf{IMVS}\) and \({\mathcal {H}}: \textbf{IMVS} \rightarrow \mathbf {L_PG^Q}\) such that \({\mathcal {H}}({\mathcal {P}}(A)) \cong A\) for any object \(A\in \mathbf {L_PG^Q}\) and \({\mathcal {P}}({\mathcal {H}}(X)) \cong X\) for any object \(X \in \textbf{IMVS}\), i.e. the functors \({\mathcal {P}}\) and \({\mathcal {H}}\) are dense.

Moreover, the functor \({\mathcal {P}}: \mathbf {L_PG^Q} \rightarrow \textbf{IMVS}\) is full, but not faithfull and the functor \({\mathcal {H}}: \textbf{IMVS} \rightarrow \mathbf {L_PG^Q}\) is faithfull, but not full.

Proof

Let A be an algebra in \(\mathbf {L_PG^Q}\). Then A is isomorphic to the direct limit of a direct system of finitely generated subalgebras \(\{A_i, \varphi _{ij}\}\), where \(A_i\) is a subdirect product of algebras from the family \(\{C_n^G: n\in \omega \}\) and \(\varphi _{ij}:A_i \rightarrow A_j\) is an injective homomorphism, \(i \le j\) (more precisely \(A_i\) is a subalgebra of \(A_j\)). Identify A with its direct limit. We know that any \(\beta ^{*}(A_i)\) is a \(G^2\)-algebra. By [8] (Theorem 11) we also know that \(\beta ^{*}\) preserves direct limits, so, \(\beta ^{*}(A)\), which is the direct limit of the direct system \(\{\beta ^{*}(A_i), \beta ^{*}(\varphi _{ij})\}\) of \(G^2\)-algebras, where \(\beta ^{*}(\varphi _{ij})\) is a \(G^2\)-algebra homomorphism, is also \(G^2\)-algebra. We correspond the \(L_PG\)-space \({\mathcal {P}}(A) = {\mathcal {P}}(\beta ^{*}(A))\) to the \(L_PG\)-algebra \(A \in \mathbf {L_PG^Q}\). So, we have contravariant functor \({\mathcal {P}}\) from the category \(\mathbf {L_PG^Q}\) to the category of \(\textbf{IMVS}\) : \({\mathcal {P}}: \mathbf {L_PG^Q} \rightarrow \textbf{IMVS}\).

Let (X, R) be an IMVC-space. So, a \(G^2\)-algebra \({\mathcal {H}}(X)\), corresponding to the IMVC-space (X, R), is a \(G^2\)-algebra, say G. It is known that the variety of \(G^2\)-algebras is locally finite. Therefore G is isomorphic to the direct limit of a direct system of finite subalgebras \(\{G_i, \psi _{ij}\}\), where \(\psi _{ij}:G_i \rightarrow G_j\) is an injective homomorphism, \(i \le j\) (more precisely \(G_i\) is a subalgebra of \(G_j\)), i.e. \(G = \underrightarrow{lim}\{G_i, \psi _{ij}\}\). 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, \(X = {\mathcal {P}}(G)\) is the inverse limit of the inverse system \(\{{\mathcal {P}}(G_i), {\mathcal {P}}(\psi _{ij})\}\), where \({\mathcal {P}}(G_i)\) is a finite cardinal sum of chains and \({\mathcal {P}}(\psi _{ij}): {\mathcal {P}}(G_j) \rightarrow {\mathcal {P}}(G_i)\) is an interval onto map. Then there exists \(G^2\)-algebras \(A_i \in \mathbf {L_PG^Q}\) such that \({\mathcal {P}}(\beta ^{*}(A_i)) \cong {\mathcal {P}}(G_i)\) and an injective MV-homomorphism \(f_{ij}: A_i \rightarrow A_j\) such that \(\beta ^{*}(A_i) \cong G_i\) for every \(i\in I\) and \({\mathcal {P}}(\beta ^{*}(f_{ij})) = {\mathcal {P}}(\psi _{ij})\). So, we have a direct system of \(L_PG\)-algebras \(\{A_i, f_{ij}\}\), where \(f_{ij}:A_i \rightarrow A_j\) is an injective homomorphism for \(i \le j\). Let A be the direct limit of this direct system. Then \({\mathcal {P}}(A) \cong {\mathcal {P}}(G)\cong X\).

From the construction of the functors \({\mathcal {P}}\) and \({\mathcal {H}}\) we conclude that \({\mathcal {H}}({\mathcal {P}}(A)) \cong A\) for any object \(A\in \mathbf {L_PG^Q}\) and \({\mathcal {P}}({\mathcal {H}}(X)) \cong X\) for any object \(X \in \textbf{IMVS}\), i.e. the functors \({\mathcal {P}}\) and \({\mathcal {H}}\) are dense.

If we have an interval map \(f: X_1 \rightarrow X_2\) between the \(G^2\)-spces \(X_1\) and \(X_2\), then there exist an algebras \(A_1, A_2 \in \mathbf {L_PG^Q}\) and an MV-algebra homomorphism \(h: A_2 \rightarrow A_1\) such that \({\mathcal {P}}(A_1) = X_1\), \({\mathcal {P}}(A_2) = X_2\) (up to isomorphism) and \({\mathcal {P}}(h):{\mathcal {P}}(A_2) \rightarrow {\mathcal {P}}(A_1)\) is an interval. So, \({\mathcal {P}}\) is full. Now, let us consider two different \(L_PG\)-homomorphisms \(f_1, f_2 : C_1^G \rightarrow C_1^G\) such that \(f_1(c)= 2c\) and \(f_2(c) = 3c\). Nevertheless, \({\mathcal {P}}(f_1) = {\mathcal {P}}(f_2): {\mathcal {P}}(C_1^G) \rightarrow {\mathcal {P}}(C_1^G)\). So, \({\mathcal {P}}\) is not faithfull.

It is obvious that if we have two different morhisms \(g_1: X_1 \rightarrow X_2\) and \(g'_1 : X'_1 \rightarrow X'_2\), then we have two different \(L_PG\)-homomorphisms \({\mathcal {H}}(g_1): {\mathcal {H}}(X_2) \rightarrow {\mathcal {H}}(X_1)\) and \({\mathcal {H}}(g'_1) : {\mathcal {H}}(X'_2) \rightarrow {\mathcal {H}}(X'_1)\). So, \({\mathcal {H}}\) is faithfull. For the identity map \(f : {\mathcal {P}}(C_1^G) \rightarrow {\mathcal {P}}(C_1^G)\), we have identity \(L_PG\)-homomorphism from \(C_1^G\) to \(C_1^G\). But for non-trivial injective homomorphism \(h:C_1^G \rightarrow C_1^G\), such that \(h(c) =3c\), there is no (not identity) interval map \(g : {\mathcal {P}}(C_1^G) \rightarrow {\mathcal {P}}(C_1^G)\) such that \({\mathcal {H}}(g) = h\). So, \({\mathcal {H}}\) is not full. \(\square \)

The category \({{\mathcal {G}}}{{\mathcal {S}}}^2\) of \(G^2\)-spaces is dual equivalent to the category \(\mathbf {G^2}\) of \(G^2\)-algebras, i.e. there exist two functors \({\mathcal {G}}^2: \mathbf {G^2} \rightarrow {{\mathcal {G}}}{{\mathcal {S}}}^2\) and \({\mathcal {H}}: {{\mathcal {G}}}{{\mathcal {S}}}^2 \rightarrow \mathbf {G^2}\). So, we have composition of two contravariant functors \({\mathcal {H}} \circ {\mathcal {P}}: \mathbf {L_PG^Q} \rightarrow \mathbf {G^2}\) and \({\mathcal {H}} \circ {\mathcal {G}}^2: \mathbf {G^2} \rightarrow \mathbf {L_PG^Q}\).

From the above, we have the following

Theorem 6.3

Covariant functors \({\mathcal {H}} \circ {\mathcal {P}} : \mathbf {L_PG^Q} \rightarrow \mathbf {G^2}\) and \({\mathcal {H}} \circ {\mathcal {G}}^2 : \mathbf {G^2} \rightarrow \mathbf {L_PG^Q}\) are dense. Moreover, \({\mathcal {H}} \circ {\mathcal {P}}\) coincides with the Belluce functor \(\beta ^{*}\) defined on the \(\mathbf {L_PG^Q}\).

The set of elements B of an \(L_PG\)-algebra A which are idempotent with respect to the operations \(\oplus \) or \(\otimes \) are precisely those elements which satisfy the law of the excluded middle with respect to the operations \(\vee \) or \(\wedge \).

Theorem 7.1

[11]. Let B be the set of elements x of a GMV-algebra A such that \(x \oplus x = x\). Then B is closed under the operations \(\oplus , \otimes \) and \(^{*}\) where \(x \oplus y=x\vee y\), \(x \otimes y=x\wedge y\) and \(x^{*} = \lnot x = \ _\ulcorner x\) for all \(x, y\in B\). Furthermore, the system \((B, \oplus , \otimes , ^{*}, 0, 1)\) is the largest subalgebra of A which is at the same time a Boolean algebra with respect to the same operations \(\oplus , \otimes , ^{*}\).

Denote by \({\mathcal {B}}(A)\) the largest Boolean subalgebra of the GMV-algebra A.

Theorem 7.2

[11]. Let A be a GMV-algebra and F be a Skolem MV-filter of A.

1. 1.

   \({\mathcal {B}}(F) = F \cap {\mathcal {B}}(A)\) is a Boolean filter of \({\mathcal {B}}(A)\).

2. 2.

   If F is maximal, then \({\mathcal {B}}(F)\) is maximal;

3. 3.

   If \(\{F_i\}_{i\in I}\) is the family of all maximal filters of GMV-algebra A, then \(\{\mathcal {{\mathcal {B}}}(F_i)\}_{i\in I}\) is the family of all maximal filters of the Boolean algebra \({\mathcal {B}}(A)\).

4. 4.

   Any Skolem MV-filter F of A is generated by the set \({\mathcal {B}}(F)\).

5. 5.

   If \(h: A \rightarrow B\) is a homomorphism from GMV-algebra A to GMV-algebra B, then we have homomorphism \({\mathcal {B}}(h): {\mathcal {B}}(A) \rightarrow {\mathcal {B}}(B)\).

It holds

Theorem 7.3

[11].

1. (i)

   \({\mathcal {B}}\) is a covariant functor from the category \(\textbf{GMV}\) of GMV-algebras to the category \({\textbf{B}}\) of Boolean algebras.

2. (ii)

   Any chain GMV-algebra (and \(L_PG\)-algebra as well) A is simple.

3. (iii)

   Let \(\{F_i\}_{_{i\in I}}\) be the family of all maximal Skolem MV-filters of the GMV-algebra A. Then A is isomorphic to the subdirect product of the algebras of \(A/F_i\) (\(i \in I\)).

Corollary 7.1

[11].

1. (i)

   Any \(L_PG\)-algebra is a subdirect product of chain \(L_PG\)-algebras.

2. (ii)

   If A is a subdirect product of the family \(\{A/F_i\}_{i\in I}\), then \({\mathcal {B}}(A)\) is a subdirect product of the family \(\{{\mathcal {B}}(A/F_i)\}_{i\in I}\).

3. (iii)

   Any \(L_PG\)-algebra is semi-simple.

Theorem 7.4

[11]. Let F be a proper Skolem MV-filter of an GMV-algebra A, then the following conditions are equivalent:

1. (i)

   F is maximal;

2. (ii)

   F is prime;

3. (iii)

   for every \(x\in A\) either \(x\in F\) or \((\lnot _\ulcorner x)^{*} \in F\).

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 \({\mathbb {K}}\) with free generators \(g_1, ... , g_m \in F\) and satisfy the identity

1. (1)

   \(P(g_1, ... , g_m) = Q(g_1, ... , g_m)\) on the generators \(g_1, ... , g_m\), then the identity

2. (2)

   \(P(x_1, ... , x_m) = Q(x_1, ... , x_m)\)

is true in \({\mathbb {K}}\).

Conversely, let \(g_1, ... , g_m\) generate the algebra F such that the identity (1) holds on the elements \(g_1, ... , g_m \in F\), the identity (2) is true in \({\mathbb {K}}\). Then F is a free algebra in \({\mathbb {K}}\) with free generators \(g_1, ... , g_m \in F\).

Skolem Duality

Now we describe another dual objects for \(L_PG\)-algebras. For any \(L_PG\)-algebra A let \({{\mathcal {S}}}{{\mathcal {K}}}(A)\) be the set of maximal MV-Skolem filters of A equipped with spectral topology, i.e. the base of the topology is \(supp (a) = \{F: a\in F\}\), for \(a\in A\), and their complements. Notice that for every \(L_PG\)-algebra A we have a Boolean algebra \({\mathcal {B}}(A)\) (the largest Boolean subalgebra of A) corresponding to it. It is easy to prove that the spaces \(\mathcal {{{\mathcal {S}}}{{\mathcal {K}}}}(A)\) and \({{\mathcal {S}}}{{\mathcal {T}}}({\mathcal {B}}(A))\) are homeomorphic. Observe that \({{\mathcal {S}}}{{\mathcal {K}}}(C_n^G)\) and \({{\mathcal {S}}}{{\mathcal {T}}}({\mathcal {B}}(C_n^G))\) are one element sets with discrete topology.

Theorem 7.5

The one-generated free \(L_{PG}\)-algebra \(F_{\textbf{L}_{PG}}(1)\) is isomorphic to \((C_0^G)^2 \times (C_1^G)^2\) with free generator \(g=((0,1),(c_1, c_1^{*}))\).

Proof

We have that the only one-generated chain \(L_PG\)-algebras are \(C_0^G\) with generators 1 or 0 and \(C_1^G\) with generators \(c_1\) or \(c_1^{*}\)

It is clear that \((C_0^G)^2\) is the one-generated free Boolean algebra with free generator (0, 1). The element \((c_1, c_1^{*}))\in (C_1^G)^2\) generates the algebra \((C_1^G)^2\). Indeed, from the elements \(2((c_1, c_1^{*})^2) (= (0,1))\) and \((2((c_1, c_1^{*})^2) )^{*} (= (1,0))\) we can obtain all elements of the algebra \((C_1^G)^2\).

Now we have to show that \((C_0^G)^2 \times (C_1^G)^2\) is generated by \(((0,1),(c_1, c_1^{*}))\). Notice, that \( _\ulcorner (g \wedge g^{*}) = ((1,1), (0,0))\) and so we can obtain the element ((0, 0), (1, 1)). By the elements ((1, 1), (0, 0)), ((0, 0), (1, 1)) and \(((0,1),(c_1, c_1^{*}))\) we can obtain every element of the algebra \((C_0^G)^2 \times (C_1^G)^2\).

Another way to show that \((C_0^G)^2 \times (C_1^G)^2\) is generated by \(((0,1),(c_1, c_1^{*}))\) is the following reasoning. The element \(((0,1),(c_1, c_1^{*}))\) generates a subalgebra of \((C_0^G)^2 \times (C_1^G)^2\) which is a subdirect product of two copies of subdirectly irreducible algebras \(C_0^G\) and two copies of subdirectly irreducible algebras \(C_1^G\). But, because any chain \(L_PG\)-algebra is simple, the only subdirect product of the given family of \(L_PG\)-algebras is the algebra \((C_0^G)^2 \times (C_1^G)^2\).

Now suppose that some identity \(P(x) = Q(x)\) is not true in the variety of \(L_PG\)-algebras. Then it is not true in some one-generated chain \(L_PG\)-algebras that are \(C_0^G\) or \(C_1^G\) with corresponding generators. Therefore, there exists corresponding projection from \((C_0^G)^2 \times (C_1^G)^2\) on chosen chain \(L_PG\)-algebra sending the generator g to the corresponding generator of the chosen \(L_PG\)-algebra, that are \(C_0^G\) or \(C_1^G\). By this we deduce that \((C_0^G)^2 \times (C_1^G)^2\) is a free \(L_PG\)-algebra with free generator \(((0,1),(c_1, c_1^{*}))\). \(\square \)

The involutive MV-space of the one-generated free \(L_PG\)-algebra \(F_{\mathbf {L_PG}}(1)\) is depicted in the Fig.3a and the Skolem-space of the one-generated free \(L_PG\)-algebra \(F_{\mathbf {L_PG}}(1)\) is depicted in the Fig.3b. It is isomorphic to the Stone space of the Boolean algebra \({\mathcal {B}}(F_{\mathbf {L_PG}}(1))\). Notice that both spaces are symmetric.

Now we will give the description of n-generated free \(L_PG\)-algebra \(F_{\mathbf {L_PG}}(n)\) for \(n > 1\). Let \(c_1^{\varepsilon _0}, c_2^{\varepsilon _1}, ... , c_n^{\varepsilon _n}\) be n-element sequence, where \(c_i \in C_i^G, \ i = 1, ... ,n\), \(\varepsilon _1, \varepsilon _2, ... , \varepsilon _n\) is the sequence of 0 and 1 and \(c_i^{\varepsilon _i} = c_i\) if \(\varepsilon _i =1\) and \(c_i^{\varepsilon _i} = c_i^{*}\) if \(\varepsilon _i =0\). 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 \(L_PG\)-algebra \(C_n^G\). Let us denote the sequence \(c_i^{\varepsilon _i}\) of the elements of i-th column of the matrix M by \(g_i, \ i = 1, ... , n\).

Theorem 7.6

The n-generated free \(L_PG\)-algebra \(F_{\mathbf {L_PG}}(n)\) is isomorphic to \((C_0^G)^{2^n} \times (C_1^G)^{2^n} \times ... \times (C_n^G)^{2^n}\) with free generators \(g_1, ... , g_n\).

Proof

Observe that \((C_0^G)^{2^n}\) is the n-generated free Boolean algebra. Let \(g_1^{(i)}, ... ,g_n^{(i)}\) be the generators of \((C_i^G)^{2^n}\) where \(i = 0,1, ... , n\), i.e. \(g_1^{(i)}, ... ,g_n^{(i)}\) are projections of \(g_1, ... , g_n\) on \((C_i^G)^{2^n}\). Notice that \(g_1^{(0)}, ... ,g_n^{(0)}\) are free generators of the algebra \((C_0^G)^{2^n}\), where \(C_0^G = (\{0,1\}, \otimes , \oplus , *, \vee , \wedge , \rightharpoonup , 0, 1)\). Observe that \(g_1^{(i)}, ... ,g_n^{(i)} \in C_i^G\) generate \(C_i^G\) for \(i=1, ... ,n\). \(2(g_1^{(i)})^2, \ldots , 2(g_n^{(i)})^2\) generate the largest Boolean subalgebra of \(C_i^G\), and, in particular, we can obtain the following n Boolean elements: (1, 0, ... , 0), (0, 1, ... , 0), ... , (0, 0, ... , 0, 1). So, by these Boolean elements and the generators \(g_1^{(i)}, ... ,g_n^{(i)}\) we can obtain every element of \((C_i^G)^{2^n}\), \(i=1,...,n\). The remainder of the proof is similar to the one in Theorem 7.5\(\square \)