A Categorical Equivalence Motivated by Kalman’s Construction

An equivalence between the category of MV-algebras and the category MV∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm MV^{\bullet}}}$$\end{document} is given in Castiglioni et al. (Studia Logica 102(1):67–92, 2014). An integral residuated lattice with bottom is an MV-algebra if and only if it satisfies the equations a=¬¬a,(a→b)∨(b→a)=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a = \neg \neg a, (a \rightarrow b) \vee (b\rightarrow a) = 1}$$\end{document} and a⊙(a→b)=a∧b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${a \odot (a\rightarrow b) = a \wedge b}$$\end{document}. An object of MV∙\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm MV^{\bullet}}}$$\end{document} is a residuated lattice which in particular satisfies some equations which correspond to the previous equations. In this paper we extend the equivalence to the category whose objects are pairs (A, I), where A is an MV-algebra and I is an ideal of A.


Introduction
In 1958 J. Kalman proved in [7] that if A is a bounded distributive lattice, then is a centered Kleene algebra by defining Later, in 1986, R. Cignoli proved in [3] the following facts: (1) K can be extended to a functor from the category of bounded distributive lattices to the category of centered Kleene algebras, (2) there is an equivalence between the category of bounded distributive lattices and the category of centered Kleene algebras whose objects satisfy an additional condition called "interpolation property", (3) the category of Heyting algebras is equivalent to the category of centered Nelson algebras. In [1], the previous results were extended giving categorical equivalences for some categories of residuated lattices. An equivalence for the category MV of MV -algebras was developed in [2].
Let A ∈ MV and let · be the product of A. The set plays an important role in the construction of a categorical equivalence for MV [2]. Our main goal in this paper is to extend this equivalence by means of a new construction where I is an ideal of A. If I is the zero ideal, then K • (A, I) = K • (A).
In [10] the logic L • was defined, whose algebraic models are the objects of a category called MV • [2]. Let A MV be the Lindenbaum algebra of the infinite valued propositional calculus L of Lukasiewicz and A MV • the Lindenbaum algebra of the calculus L • . In [10,Theorem 3.9] it was proved that there exists an ideal I of A MV such that κ(A MV • ) ∼ = A MV /I, where κ is a unary operation defined on objects of MV • . Also, in [1,Corollary 5] it was proved that κ(A MV • ) ∪ {c} generates A MV • , where c is a center, that is: ∼c = c. So, we have some link between L • and L. Let us try to explain what this link means.
It is a known fact that in any MV -algebra there exists a bijection between congruences and ideals, and that ideals are in bijection with the filters by means of the involution of the algebra. Also, the filters of A MV are in bijection with the theories of L, where a theory is a class of formulas that contains the axioms and is closed by the rule of inference Modus Ponens. Taking into account this bijection, we can say, roughly speaking, that the classes of formulas in L • of the form κ(X) are in correspondence with the classes of formulas of L "modulo" the theory corresponding to the filter For example, the classes of formulas of the form x n ⊕ x n+1 , for every propositional variable x n , belong to that theory.
Let us also remark the following fact: if A ∈ MV and I is an ideal of A, where θ I is the congruence associated to the ideal I (see ([5, Section 1.2]). We think that it is interesting to study the set The paper is organized as follows. In Sect. 2 we give some basic results about the categories considered in [2]. We also introduce and study the category IMV: the objects are pairs (A, I), where A ∈ MV and I is an ideal of A, and the morphisms f : (A, I) → (B, J) are morphisms f : A → B in MV which satisfy the condition I ⊆ f −1 (J). In Sect. 3 we build up an adjunction between IMV and a new category whose objects are algebras. In Sect. 4 we obtain an equivalence for the category IMV, which is a generalization of the equivalence given in [2] for the category MV. Finally, in Sect. 5 we make some remarks about properties of the constructions developed throughout this work.

Preliminary Definitions and Results
Since we are working on ideas and results of the paper [2], we recommend the reader to have the mentioned paper at hand while reading this work. All the residuated lattices considered in this paper are distributive and commutative, so we shall omit mentioning these two conditions in the sequel, assuming them as given. Recall that a residuated lattice is said to be integral if it is bounded above by the unit of the product. All the categories considered in this paper have an underlying class of algebras, so we shall use the same notation for the category and the class of algebras.
Let A, ∧, ∨, ·, →, 0, 1 be an object in the category IRL 0 of integral residuated lattices with bottom. We define the set K • (A) as in the case of MValgebras. For A ∈ IRL 0 , we define the operations ∨, ∧ and ∼ as in the case of K(A). We also define the following binary operations: An involutive residuated lattice is an algebra T = T, ∧, ∨, * , →, ∼, 1 such that 1.
2. ∼ is an involution of the lattice that is a dual automorphism, i.e., ∼∼x = x for every x. 3.
In this case we have that ∼(y * (∼z)) = y → z and ∼x = x → 0. An involutive residuated lattice is said to be centered if it has a distinguished element, called a center, that is, a fixed point for the involution. The adjunction C K • : IRL 0 → DRL restricts to an equivalence C K • : IRL 0 → DRL ([1, Corollary 7.8]), where DRL is the full subcategory of DRL whose objects T satisfy the following condition: (CK • ) For every pair of elements z, w ∈ T such that z, w ≥ c and z * w ≤ c, there exists x ∈ T such that x ∨ c = z and ∼ x ∨ c = w.
If T is an algebra of DRL , there exists a map κ : T → T that satisfies the following two conditions: Conversely, if T is an algebra of DRL in which there exists an operator κ that satisfies (k1) and (k2), then (CK • ) holds on T [2, Theorem 1]. In what follows we denote by DRL the category whose objects have a unary operator κ in its signature, and verify the corresponding equations. In every integral residuated lattice A, ∧, ∨, ·, →, 0, 1 , we define ¬a = a → 0 for every a ∈ A. If we consider an algebra A of IRL 0 , then κ : The category MDRL is the full subcategory of DRL whose objects satisfy the equation Let iIRL 0 be the full subcategory of IRL 0 whose objects satisfy the equation (S) is the morphism in iIRL 0 given by the restriction of g to , which is a morphism in MDRL. For every A ∈ iIRL 0 we have the isomorphism α : A → κ(K • (A)) given by α(a) = (a, ¬a), and for every T ∈ MDRL we have the isomorphism β T : [2,Lemma 7]). There is a categorical equivalence κ K • : iIRL 0 → MDRL [2, Theorem 11].
Recall that an MV -algebra is term equivalent to an integral residuated lattice with bottom A, ∨, ∧, ·, →, ¬, 0, 1 that satisfies (Inv) and the following equations: The category MV • is the full subcategory of MDRL whose objects satisfy the equations There is a categorical equivalence κ K • : MV → MV • [2, Corollary 15]. Consider the following equation in MDRL: In the following diagram we have the relationship among the above mentioned categories, where inc denotes the inclusion functor: In what follows we will give an example of K • (A, I) for an algebra A ∈ MV and an ideal I of A.
Example 1. In 1958 Chang introduced the MV -algebra C [4,5], defined by C = Γ(Z ⊗ Z, (1, 0)), where Z is the set of integer numbers, Z ⊗ Z is the lexicographic product and Γ is the categorical equivalence between -groups with strong unit, and the category MV.
Let I be the following ideal of C: The sets − +, + + and − − are defined in a similar way. We shall prove that K • (C, I) is the union of the quadrants + −, − + and + + that we show in the following graphic of C×C. In fact, if (x, y) ∈ C×C is in + −, the product x.y is positive or ⊥ and the same is true for (x, y) ∈ − +. If (x, y) ∈ + +, then x · y is ⊥.

H H H H H H H H H H H H H H H H H H H H
Inspired by the construction of K • (A, I) we give the following Definition 1. Let IMV be the category whose objects are pairs (A, I), where A ∈ MV and I is an ideal of A, and whose morphisms f :

A Connection Between the Categories MV and IMV
Let A ∈ MV, and let I be an ideal of A. The relation given by It is immediate that the following diagram commutes: We have that f is the unique morphism in IMV with the above mentioned property. Therefore, we have that Q E.

2.
∼ is an involution of the lattice that is a dual automorphism, i.e., ∼∼x = x for every x.
Besides the condition x * y ≤ z if and only if x ≤ ∼(y * (∼z)) for any x, y, z is equivalent to the condition x → y = ∼(x * ∼y) for any x, y. Therefore, IMV • is a variety. Also note that c * x ≤ c, c * c = 0 and (x * y) → z = x → (y → z).
The next goal is to define a functor from IMV • to IMV. The proof of the following technical lemma follows from similar ideas to that given in [2].
Proof. Let us prove (i) by using the item 10. of Lemma 1: Now we prove (ii). We have, in first place, (κx ∨ κy) ∧ c = (x ∧ c) ∨ (y ∧ c) = (x ∨ y) ∧ c. We also prove that (κx ∨ κy) ∨ c = c → (x ∨ y). In order to show it, note that c ≤ c → x, c → y. Then by item 13. of Lemma 1, the problem is reduced to show that ((c → x) ∨ (c → y)) * c = (c → (x ∨ y)) * c. From (i): The condition (iii) follows from the condition (i) because Now we prove the condition (iv). First note that it follows from condition (i) that Besides by item 10. of Lemma 1 and the condition (iii) we have that Thus, we obtain c * κ(x * y) = c * ((λx * κy) ∨ (λy * κx)).
In order to prove the condition (v), suppose that c * κx = c * κy. Taking into account (iii) we obtain x ∧ c = y ∧ c. Hence, by item 4. of Lemma 1 we have that κx = κy.
The item (vi) is consequence of the items (iv) and (v), and the item (vii) follows from Item 5 of Lemma 1.

Hence, by (v) of Corollary 2 we have that κ(T ) satisfies (QHey). It is immediate that the condition (P • ) on T implies the condition (Lin) on κ(T ).
If T ∈ IMV • , we define the following sets: 1. I T = {y ∈ T : y ∧ c = 0}.
Thus, the morphism f is such that the diagram (1) commutes. It follows from the definition of κ that f is the unique morphism in IMV with the above mentioned property. Therefore we conclude the following Theorem 7. There is an adjunction κ K • : IMV → IMV • .

The Categorical Equivalence
In this section we restrict the category IMV • in order to obtain a categorical equivalence between the restricted category and IMV. We find a condition (ICK • ) and we prove that it is equivalent to the existence of a partial function S which is defined by equations.
Moreover, we prove that the existence of S is equivalent to the existence of a new operation defined by equations in κ(T ) × I T . We have that Sx = κt y, where x = t ∨ y, t = x ∧ c ∈ (c], y ∈ I T , where (c] is the set of the elements that are least or equal to c and I T is defined in Sect. 3. But the condition (ICK • ) is also equivalent to the existence of in T ×T , defined by equations, as we show in Proposition 4.
The algebra T, , 0 is a commutative monoid (Proposition 6). Let T ∈ IMV • . We consider the following condition: (ICK • ) For any z, w ≥ c such that z * w ≤ y ∨ c for some y ∈ I T , there exists x such that x ∨ c = z and ∼x ∨ c = w.  Let T ∈ IMV • . We define the following set: Conversely, let x = y ∨ t, with y ∈ I T and t ≤ c. Then we have that x ∈ M T because x ∨ c = y ∨ c. Thus, we obtain that Moreover, the previous decomposition is unique. Let x = y ∨ t = y ∨ t , with y, y ∈ I T and t, t ≤ c. Then, In the same way, Fot T ∈ IMV • we define (if it is possible) a function S : M T → T through the following equalities:  S(a, b) = (b → a, b). We are going to see that S also has an important role in K • (A, I).
We remark that when T has the form K 2. There exists a function S : M T → T which satisfies (S1) and (S2).
Proof. 1. ⇒ 2. Let x ∈ M T . Then there is y such that x ∨ c = y ∨ c and y ∧ c = 0. We define w = ∼x ∨ c and z = ∼x → (x ∨ c). We have that z * w ≤ y ∨ c. Besides w ≥ c, and z ≥ c because ∼x * (c ∧ ∼x) ≤ c.
Thus, there is t such that t ∨ c = z and ∼t ∨ c = w. Put Sx = t. Hence, 2. ⇒ 1. Suppose that there exists S. Let z, w ≥ c be such that z * w ≤ y∨c and y ∧ c = 0 for some y. Since (∼w ∨ y) ∨ c = y ∨ c and y ∧ c = 0 then ∼w ∨ y ∈ M T . Then we define the element x = z ∧ S(∼w ∨ y). We will prove that x ∨ c = z and ∼x ∨ c = w.
The condition z * (∼y ∧ w) ≤ z * w ≤ y ∨ c implies the following inequality Then using (2) we have that Finally we have that Let T ∈ IMV • . Recall that for every x, y ∈ T the operation + is defined by x + y = ∼(∼x * ∼y).
Note that Using the previous lemma, the following can be proved Proof. Suppose that there exists S. In first place, let x ∈ T , x ≤ c and y ∈ I T . We define κx y = S(x ∨ y). Then we have that Besides we have that On the other hand, So, (κx y) ∨ c = x + (y ∨ c).
In second place, for any x ∈ T , we have that κx y = κ(x ∧ c) y. Then it suffices to define κx y = S((x ∧ c) ∨ y) and the proof follows.
Conversely, let x ∈ M T . Then there is y ∈ I T such that x ∨ c = y ∨ c. We define Sx = κx y. Then we have that Remark 9. For (A, I) ∈ IMV we define a map : ((a, b), (d, e)) = (a⊕d, b·e). In order to prove the well definition of this map, let i = a · b and j = d · e. Hence k = i ⊕ j ∈ I, and a ⊕ d ≤ Motivated by the Remark 9, we will extend the domain of to T × T . For T ∈ IMV • we define (if it is possible) a function : T × T → T through the following equalities: Remark 10. The second equation above seems to be "asymmetric" in the variables x and y. But As we can see, in the last equation there is not any distinction between x and y.
Then we have the following (2) ⇒ (3). Let z, w ≥ c such that (z * w) ∨ c = y ∨ c and y ∧ c = 0, for some y ∈ T . We define z = λz and w = λw. It is immediate that z * w = λy. Thus, there is x ∈ T such that λx = z and ∼κx = w. Hence, x ∨ c = z and ∼x ∨ c = w. Thus, we have shown the condition (3).
(3) ⇒ (2). Let z, w ∈ κ(T ) such that z * w = λy and y ∧ c = 0, for some y ∈ T . We define t = z ∨ c and u = w ∨ c. We have that t, u ≥ c. As c * (z ∨ w) = z ∨ w, we obtain and t * u = (z * w) ∨ c = λy ∨ c = y ∨ c. Then, there is x ∈ T such that x ∨ c = z ∨ c and ∼x ∨ c = w ∨ c. It follows from properties of κ the fact that λx = z and ∼κx = w. Hence, we have proved the condition (2).
Therefore we conclude the following Corollary 11. There is an equivalence between the category IMV and the the full subcategory of IMV • whose objects satisfy the conditions (2), (3), (4), (5) or (6) of Proposition 5.
Let f : T → U be a morphism in the category IMV • where the objects satisfy the condition (ICK • ). Then, by the proposition above, there exist the binary operations andˆ on T and U (respectively). Straightforward computations show that for every x, y ∈ T we have that f (x y) ∧ c = (f (x)ˆ f (y))∧c and f (x y)∨c = (f (x)ˆ f (y))∨c, so f (x y) = f (x)ˆ f (y). Then we obtain the following Corollary 12. The class of algebras of IMV • which satisfy (ICK • ) is a variety if we consider in the signature of the algebras. Moreover, the category associated to the previous variety is equivalent to IMV.

Final Remarks
In this section we are looking for answers for the following questions: (1) When does an object of IMV • satisfy the equation x * 1 = x?
(2) What properties are satisfied by the binary operation ?
Proposition 6. Let T ∈ IMV • such that there exists . Then T, , 0 is a commutative monoid.
Thus, using the associativity and commutativity of * we obtain (x (y z)) ∧ c = ((x y) z) ∧ c.
Hence, using again the associativity and commutativity of * we obtain (x (y z)) ∨ c = ((x y) z) ∨ c.
Finally we will prove that x 0 = x for every x ∈ T . We have that Therefore, we obtain that x 0 = x.