Equivalence \`a la Mundici for commutative lattice-ordered monoids

We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered groups is equivalent to the category of MV-monoidal algebras. Roughly speaking, the structures we call unital commutative lattice-ordered groups are unital Abelian lattice-ordered groups without the unary operation $x \mapsto -x$. The primitive operations are $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example of these structures is $\mathbb{R}$, with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $x \mapsto \lnot x$. The primitive operations are $\oplus$, $\odot$, $\lor$, $\land$, $0$, $1$. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $[0, 1] \subseteq \mathbb{R}$. We obtain the original Mundici's equivalence as a corollary of our main result.


Introduction
In [16], Mundici proved that the category of unital Abelian lattice-ordered groups (unital Abelian ℓ-groups, for short) is equivalent to the category of MV-algebras. In Theorem 8.21, our main result, we establish the following generalization: The category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras.
Roughly speaking, unital commutative lattice-ordered monoids (unital commutative ℓ-monoids, for short) are unital Abelian ℓ-groups without the unary operation x → −x, whereas MV-monoidal algebras are MV-algebras without the negation x → ¬x (precise definitions will be given in Section 2). The operations of unital commutative ℓ-monoids are +, ∨, ∧, 0, 1, −1, whereas the operations of MVmonoidal algebras are ⊕, ⊙, ∨, ∧, 0, 1. A motivating example of unital commutative ℓ-monoid is R, with the obvious interpretation of the operations, whereas a motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra [0, 1]. Furthermore, for every topological space X equipped with a preorder, the set of bounded continuous order-preserving functions from X to R is an example of a unital commutative ℓ-monoid, whereas the set of continuous order-preserving functions from X to [0, 1] is an example of an MV-monoidal algebra. The author's interest for unital commutative ℓ-monoids originated from these last examples, as we now illustrate in some detail.
Given a compact Hausdorff space X, the set C(X, R) of continuous functions from X to R is a divisible Archimedean unital Abelian ℓ-group, complete in the uniform metric. In fact, we have a duality between the category CompHaus of compact Hausdorff spaces and continuous maps and the category G of divisible Archimedean metrically complete unital Abelian ℓ-groups (see [21,20]). Similarly, we may consider on the set C(X, [0, 1]) of continuous functions from X to [0, 1] pointwise-defined operations inherited from [0, 1]; for example, the operations of MV-algebras. Developing this idea, one can show that CompHaus is dually equivalent to a variety ∆ of (infinitary) algebras (see [9,11,14]). These algebras can be thought of as MV-algebras with an additional operation of countably infinite arity satisfying some additional axioms. In fact, we have an equivalence between G and ∆, which is essentially a restriction of the equivalence between unital Abelian ℓ-groups and MV-algebras.
A compact ordered space is a compact space X endowed with a partial order on X so that the set { (x, y) ∈ X × X | x y } is closed in X × X with respect to the product topology. This notion was introduced by Nachbin [17]. If we replace compact Hausdorff spaces by compact ordered spaces in the aforementioned discussion involving CompHaus, ∆ and G, then we may accordingly replace Mundici's equivalence with our Theorem 8.21. Given a compact ordered space X, let us consider the set C (X, [0, 1]) of continuous order-preserving functions from X to [0, 1]: we can endow C (X, [0, 1]) with pointwise-defined operations ⊕, ⊙, ∨, ∧, 0, 1 (which are the operations of MV-monoidal algebras). Pursuing a similar idea, in [10] it was proved that the category of compact ordered spaces and continuous order-preserving maps is dually equivalent to a quasi-variety of infinitary algebras ( [1,3] show that this quasi-variety actually is a variety). However, the operations are somewhat unwieldy, and one might want to investigate the set C (X, R) of continuous order-preserving real-valued functions, instead. In fact, C (X, R) is a unital commutative ℓ-monoid. The main motivation of this paper is to make the connection between C (X, R) (unital commutative ℓ-monoids) and C (X, [0, 1]) (MV-monoidal algebras) explicit.
There are both pros and cons in working with unital commutative ℓ-monoids or MV-monoidal algebras. On one hand, it is easier to work with the axioms of unital commutative ℓ-monoids rather than those of MV-monoidal algebras. On the other hand, the category of MV-monoidal algebras is a variety of finitary algebras axiomatized by a finite number of equations, so the tools of universal algebra apply. The equivalence established here allows to transfer the pros of one category to the other one.
Our result specializes to Mundici's equivalence between unital Abelian ℓ-groups and MV-algebras (Appendix A). We remark that, in contrast to the proof of Mundici's equivalence in [16], we do not use the axiom of choice to prove the equivalence between unital commutative ℓ-monoids and MV-monoidal algebras.
We sketch the proof of our main result, Theorem 8. 21. In order to obtain an equivalence uℓM MVM Γ Ξ between the category uℓM of unital commutative ℓ-monoids and the category MVM of MV-monoidal algebras, we show that we have two equivalences Here uℓM + is the category of 'positive-unital commutative ℓ-monoids' (Definition 4.1), which are the positive cones of unital commutative ℓ-monoids. The func-torΓ maps a unital commutative ℓ-monoid M to its 'unit interval'Γ(M ) (Section 3). We construct a quasi-inverse in two steps. As a first step, given an MV-monoidal algebra A, we define the set G(A) of 'good sequences in A' (Section 5), and we equip this set with the structure of a positive-unital commutative ℓ-monoid (Section 7). As a second step, we consider translations of the elements of G(A) by negative integers; in this way we obtain a unital commutative ℓ-monoid TG(A), where T : uℓM + → uℓM is a functor (Section 4). To show that the composition of these two steps provides a quasi-inverse ofΓ, we writeΓ as the composite of two functors (−) + and U. The functor (−) + associates to M its 'positive cone' M + ; the functor U associates to M + its unit interval. We will show that (−) + and T are quasi-inverses (Section 4), and that U and G are quasi-inverses (Section 8); from this, it follows thatΓ andΞ := TG are quasi-inverses, and hence the categories of unital commutative ℓ-monoids and MV-monoidal algebras are equivalent (Theorem 8.21). By the time of publication of the present paper, the main result (Theorem 8.21) has appeared also in the author's Ph.D. thesis [2,Chapter 4]. However, the proofs are different. In [2], the author uses Birkhoff's subdirect representation theorem, which simplifies the arguments but relies on the axiom of choice. Moreover, while in this paper we construct a quasi-inverse ofΓ as the composite of two functors, in [2] a one-step construction is adopted.
In Appendix B we show that subdirectly irreducible MV-monoidal algebras are totally ordered, and in Appendix C we show that every good sequence in a subdirectly irreducible MV-monoidal algebra is of the form (1, . . . , 1, x, 0, 0, . . . ). Even if we do not use these last two results, we have included them because they seem of interest.
For every x ∈ M there exists n ∈ N such that (−1) + · · · + (−1) n times The element 1 is called the positive unit, and the element −1 is called the negative unit.
We warn the reader that some authors do not assume the lattice to be distributive, nor that + distributes over both ∧ and ∨.
(E5) (x + y) · z = x + (y · z). (Note that Axioms E4 and E5 are equivalent, given the commutativity of + and ·.) The addition of Axiom U2 equals the addition of the following axiom.
(E5) 0 1. The addition of Axiom U3 equals the addition of the following axiom.
(E6) For every x ∈ M there exists n ∈ N such that 0 · · · · · 0 n times The class of algebras satisfying Axioms E1 to E6 is term-equivalent to the class of unital commutative ℓ-monoids. One interesting thing of Axioms E1 to E6 is that their symmetries resemble the ones in the definition of MV-monoidal algebras below. We will use Axioms E1 to E6 to explain the axioms of MV-monoidal algebras below; besides this usage, we will stick to Axioms U0 to U3 throughout the paper.
Example 2.4. For every topological space X equipped with a preorder, the set of bounded continuous order-preserving functions from X to R is a unital commutative ℓ-monoid.

MV-monoidal algebras.
In the following, we define the variety MVM of MV-monoidal algebras, which are finitary algebras axiomatised by a finite number of equations. Our main result is that the categories uℓM and MVM are equivalent. Without giving the details now, we anticipate the fact that the equivalence is given by the functorΓ : uℓM → MVM that maps a unital commutative ℓ-monoid M to the set { x ∈ M | 0 x 1 }, endowed with the operations ⊕, ⊙, ∧, ∨, 0 and 1, where ∧, ∨, 0 and 1 are defined by restriction, and ⊕ and ⊙ are defined by x ⊕ y := (x + y) ∧ 1 and x ⊙ y := (x + y − 1) ∨ 0.
Before commenting on the axioms, we remark that Axioms A4 and A5 are equivalent, given the commutativity of ⊕ and ⊙. We have included both so to make it clear that, if A; ⊕, ⊙, ∨, ∧, 0, 1 is an MV-monoidal algebra, then also the 'dual' algebra A; ⊙, ⊕, ∧, ∨, 1, 0 is an MV-monoidal algebra.
Axioms A1 to A3 coincide with Axioms E1 to E3 in Remark 2.3. So, in a sense, the difference between MV-monoidal algebras and unital commutative ℓmonoids lies in the difference bewteen the conjunction of Axioms A4 to A7 and the conjunction of Axioms E4 to E6. We mention here that 0 and 1 are bounds of the underlying lattice of an MV-monoidal algebra (Lemma 6.5); this fact is not completely obvious, given that the proof makes use of almost all the axioms of MV-monoidal algebras.
Axiom A4 is a sort of associativity, which resembles Axiom E4, i.e. (x · y) + z = x · (y + z). In particular, one verifies that the interpretation on [0, 1] of both the left-hand and right-hand side of Axiom A4 equals (1) ( Notice that the element x + y + z − 1 appearing in (1) coincides, using the definition of · from Remark 2.3, with the interpretation on R of (x · y) + z and x · (y + z). In fact, Axiom A4 is essentially the condition (x · y) + z = x · (y + z) expressed at the unital level, i.e.: Indeed, the presence of the term x · y in the left-hand side of (2) corresponds to the presence of the terms x ⊕ y and x ⊙ y in the left-hand side of Axiom A4, and the presence of the term y + z in the right-hand side of (2) corresponds to the presence of the terms y ⊕ z and y ⊙ z in the right-hand side of Axiom A4. Analogously, Axiom A5 corresponds to Axiom E5, i.e. (x + y) · z = x + (y · z).
Axiom A6 expresses how the term (x ⊙ y) ⊕ z differs from its non-truncated version (x · y) + z: essentially, the axiom can be read as Analogously, Axiom A7 can be read as We remark that MV-monoidal algebras form a variety of algebras whose primitive operations are finitely many and of finite arity, and which is axiomatised by a finite number of equations. We let MVM denote the category of MV-monoidal algebras with homomorphisms.
Remark 2.6. Bounded distributive lattices form a subvariety of the variety of MVmonoidal algebras, obtained by adding the axioms x ⊕ y = x ∨ y and x ⊙ y = x ∧ y.

The unit interval functor
In this section we define a functorΓ from the category of unital commutative ℓ-monoids to the category of MV-monoidal algebras; the main goal of the paper is to show thatΓ is an equivalence.  For all x, y, x ′ , y ′ ∈ M such that x x ′ and y y ′ , we have x + y x ′ + y ′ .
Our next goal-met in Proposition 3.6 below-is to show thatΓ(M ) is an MVmonoidal algebra. We need some lemmas. and For all x and y in a commutative ℓ-monoid we have Proof. We recall the proof, available in [6], of the two inequalities: Then Proof. We have The fact that also (x ⊙ y) ⊕ ((x ⊕ y) ⊙ z) and (x ⊕ (y ⊙ z)) ⊙ (y ⊕ z) coincide with (x + y + z − 1) ∨ 0) ∧ 1 follows from the commutativity of ⊕ and ⊙ (which is easily seen to hold) and the commutativity of +. Our main goal is to show thatΓ is an equivalence of categories.

Positive cones
In [7, Chapter 2], the authors proceed in two steps in order to prove that, for an MV-algebra A, there exists a unital Abelian ℓ-group that envelops A. First, a partially ordered monoid M A is constructed from A. Then a unital Abelian ℓ-group G A is defined (in a way which is analogous to the definition of Z from N). In this paper, we proceed analogously: the role of A is played by MV-monoidal algebras, the role of G A is played by unital commutative ℓ-monoids, and the role of M A is played by what we call positive-unital commutative ℓ-monoids. Roughly speaking, if we think of a unital commutative ℓ-monoid as the interval (−∞, ∞), then an MV-monoidal algebra is the interval [0, 1], whereas a positive-unital commutative ℓ-monoid is the interval [0, ∞).
In order to prove thatΓ is an equivalence, we show thatΓ is the composite of two equivalences uℓM uℓM + MVM, where uℓM + is the category-yet to be defined-of positive-unital commutative ℓmonoids. The idea is that, for M ∈ uℓM, we have In this section, we define the functor (−) + , and we exhibit a quasi-inverse T. We remark that one could construct a quasi-inverse functor for Γ just in one step: see the author's Ph.D. thesis [2,Chapter 4] for the employment of this approach. Given a unital commutative ℓ-monoid M , we set M + := { x ∈ M | x 0 }. With the following definition, we aim to capture the structure of M + for M a unital commutative ℓ-monoid. Definition 4.1. By a positive-unital commutative ℓ-monoid we mean an algebra M ; +, ∨, ∧, 0, 1, − ⊖ 1 (arities 2, 2, 2, 0, 0, 1) such that, for every x ∈ M , the following properties hold.
We denote with uℓM + the category of positive-unital commutative ℓ-monoids with homomorphisms. Given n ∈ N, we write n for 1 + · · · + 1 n times .
In this section, we show that uℓM and uℓM + are equivalent. Proof. The proof proceeds by induction on n ∈ N. The case n = 0 is trivial. Suppose the statement holds for n ∈ N. If x + (n + 1) = y + (n + 1), then and then, by inductive hypothesis, x = y. ⇐⇒ y + 1 = 1. This shows that the unary operation − ⊖ 1 and the constant 0 can be explicitly defined from {∨, ∧, 1, +}. Therefore, every function f : M → N between positiveunital commutative ℓ-monoids that preserves +, ∨, ∧ and 1 preserves also − ⊖ 1 and 0, and hence it is a homomorphism.
Given a morphism f : M → N of unital commutative ℓ-monoids, f restricts to a function f + from M + to N + . Moreover, f preserves +, ∨, ∧ and 1 and so, by Remark 4.3, f + is a morphism of positive-unital commutative ℓ-monoids. This establishes a functor (−) + : uℓM → uℓM + that maps M to M + , and maps a morphism f : M → N to its restriction f + : M + → N + . We will prove that (−) + is an equivalence of categories (Theorem 4.16 below). To do so, we exhibit a quasi-inverse T : uℓM + → uℓM.
Let M be a positive-unital commutative ℓ-monoid. We want to construct a unital commutative ℓ-monoid T(M ) such that, if N is a unital commutative ℓ-monoid and N + ∼ = M , then T(M ) ∼ = N . Every element of a unital commutative ℓ-monoid N can be expressed as x − n for some x ∈ N + and n ∈ N. Roughly speaking, we will obtain T(N + ) ∼ = N by translating the elements of N + by negative integers.   Proof. We have and analogously for ∧. Proof. The fact that T(M ) is a commutative monoid follows from the fact that M and N are commutative monoids. Checking that T(M ) is a distributive lattice is facilitated by Remark 4.5 and Lemma 4.6. Let us prove that + distributes over ∨: Analogously for ∧. The axioms for 1 and −1 are easily seen to hold.
For a morphism of positive-unital commutative ℓ-monoids f : M → N , we set The function Moreover, T(f ) is a morphism of unital commutative ℓ-monoids. We show only that + is preserved: One easily verifies that T : uℓM + → uℓM is a functor. For each unital commutative ℓ-monoid, we consider the function , then x + m = y + n and therefore x − n = y − m. Proof. The function ε 0 M preserves 0, 1, and −1 because For all x, y ∈ M + and n ∈ N we have Hence, ε 0 M preserves +. Moreover, for all x, y ∈ M + and n, m ∈ N, we have Proposition 4.9. ε 0 : T(−) +→ 1 uℓM is a natural transformation, i.e., for every morphism of unital commutative ℓ-monoids f : M → N , the following diagram commutes.
For every x ∈ M + and every n ∈ N we have Proof. To prove injectivity, let x, y ∈ M + , let n, m ∈ N, and suppose we have For each positive-unital commutative ℓ-monoid M , we consider the function ). Notation 4.13. Let M be a positive-unital commutative ℓ-monoid. We define, inductively on n ∈ N, a function ( · ) ⊖ n : M → M .
Lemma 4.14. Let M be a positive-unital commutative ℓ-monoid. For every x ∈ M and every n ∈ N, we have Proof. We prove the statement by induction. The case n = 0 is trivial. Suppose the statement holds for n − 1 ∈ N, and let us prove it for n. We have Then, x + 0 = y + 0, and so x = y. Second, we prove that

Good sequences
. . ) of elements of A which is eventually 0 and such that, for each n ∈ N, (x n , x n+1 ) is a good pair.
Remark 5.2. In our definition of good pair we included both the condition x 0 ⊕ x 1 = x 0 and the condition x 0 ⊙x 1 = x 1 because, in general, they are not equivalent.
As an example, one can take the MV-monoidal algebra consisting of three elements {0, a, 1}, where a ⊕ a = a, and a ⊙ a = 0.
In order to prove the equivalence between the categories of MV-algebras and unital Abelian ℓ-groups (see [16] or [7]), Mundici used the facts that subdirectly irreducible MV-algebras are totally ordered and that good sequences in totally ordered MV-algebras are of the form (1, . . . , 1, x, 0, 0, . . . ).
In this paper we do not make use of the Subdirect Representation Theorem (in fact, we do not make use of the axiom of choice) to establish the equivalence between uℓM and MVM. The reason why this is done is that, initially, the author was unable to prove that, in subdirectly irreducible MV-monoidal algebras, good sequences are of the form (1, . . . , 1, x, 0, 0, . . . ). Eventually such a proof was found, and the result is given in Corollary C.6. However, the result is not used in the present paper, for the following reasons. First, in this way, the proof that we provide for the equivalence between uℓM and MVM may be applied in similar settings, where the structure of subdirectly irreducible algebras is not known. Secondly, the proof we give does not rely on the axiom of choice. In particular, up to proving without the axiom of choice that the axioms of MV-monoidal algebras hold in any MValgebra, we obtain a proof of the equivalence between unital Abelian ℓ-groups and MV-algebras that does not make use of the axiom of choice.
For a proof of our main result that does take advantage of the Subdirect Representation Theorem, the reader is invited to consult the author's Ph.D. thesis [2, Chapter 4]. 6. Basic properties of MV-monoidal algebras Remark 6.1. Inspection of the axioms that define MV-monoidal algebras shows that, for every MV-monoidal algebra A; ⊕, ⊙, ∨, ∧, 0, 1 , also its 'dual' algebra A; ⊙, ⊕, ∧, ∨, 1, 0 is an MV-monoidal algebra.
In other words, the terms σ 1 , σ 2 , σ 3 , σ 4 in the theory of MV-monoidal algebras are all invariant under permutations of variables, and they coincide.
Proof. By commutativity of ⊕ and ⊙, in the theory of MV-monoidal algebras σ 1 is invariant under transposition of the first and the second variables, and σ 3 is invariant under transposition of the second and the third ones. Moreover, by Axiom A4, we have σ 1 (x, y, z) = σ 3 (x, y, z). Since any two distinct transpositions in the symmetric group on three elements generate the whole group, it follows that σ 1 and σ 3 are invariant under any permutation of the variables. By commutativity of ⊕ and ⊙, we have σ 1 (x, y, z) = σ 4 (z, y, x), and, by Axiom A5, we have σ 2 (x, y, z) = σ 4 (x, y, z). We conclude that σ 1 , σ 2 , σ 3 , σ 4 are invariant under permutations of variables, and they coincide.
Lemma 6.7. The following properties hold for all x, y, z, x ′ , y ′ in an MV-monoidal algebra.
(2) If x x ′ and y y ′ , then Proof. Let us call A the MV-monoidal algebra of the statement. Item 1 is guaranteed by the application of Lemma 3.1 to the commutative ℓ-monoid A; ⊕, ∨, ∧, 0 . Item 2 is dual to item 1. Item 3 holds by item 1 together with the fact that z z. Item 4 is dual to item 3. From 0 y we obtain, by item 3, x ⊕ 0 x ⊕ y. By Lemma 6.6, we have x ⊕ 0 = x. Therefore, x x ⊕ y, and so item 5 is proved. Item 6 is dual to item 5. Lemma 6.8. For all x, y, z in an MV-monoidal algebra we have Proof. Using Axioms A4 to A7, we obtain x ⊙ (y ⊕ z) = x ∧ σ(x, y, z) σ(x, y, z) σ(x, y, z) ∨ z = (x ⊙ y) ⊕ z. Lemma 6.9. Let A be an MV-monoidal algebra, let (x 0 , x 1 ) be a good pair in A, and let y ∈ A. Then and both these elements coincide with σ(x 0 , x 1 , y).

Proof.
We have For all x and y in an MV-monoidal algebra, the pair (x ⊕ y, x ⊙ y) is good.

Operations on the set of good sequences
We denote with G(A) the set of good sequences in an MV-monoidal algebra A. (In fact, G stands for 'good'.) We will endow G(A) with the structure of a positiveunital commutative ℓ-monoid. We let 0 denote the good sequence (0, 0, 0, . . . ), and we let 1 denote the good sequence (1, 0, 0, 0, . . . ). For good sequences a = (a 0 , a 1 , a 2 , . . . ) and Proposition 7.2 below asserts that a ∨ b and a ∧ b are good sequences. In order to prove it, we establish the following lemmas.
For A an MV-monoidal algebra, we have a partial order on G(A), induced by the lattice operations. Since the lattice operations are defined componentwise, we have the following.
Our first aim, reached in Lemma 7.8 below, is to show that these two ways coincide.
Lemma 7.6. Let A be an MV-monoidal algebra. For every good pair (x, y) in A and every u ∈ A, the pairs (x ⊕ u, y) and (x, y ⊙ u) are good.
Proof. The pair (x ⊕ u, y) is good because we have (x ⊕ u) ⊕ y = (x ⊕ y) ⊕ u = x ⊕ u, and, by Lemma 6.7, we have y = x⊙y (x⊕u)⊙y y, which implies (x⊕u)⊙y = y. Dually, (x, y ⊙ u) is good.
Lemma 7.7. If (x, y) and (y, z) are good pairs in an MV-monoidal algebra, then (x, z) is a good pair.
Proof. By Lemma 7.5, it is enough to show that, for i, j ∈ {0, . . . , m}, the pair (a i ⊕ b m−i , a j ⊙ b m−j ) is good. The case i = j is covered by Lemma 6.10. If i < j, then, by Lemma 7.7, the pair (a i , a j ) is good; by Lemma 7.6, the pair By Lemma 7.9, the pair (c n , c n+1 ) is good.
Proposition 7.11. Addition of good sequences is commutative.
Remark 7.12. Let A be an MV-monoidal algebra, and let a ∈ G(A). Then, a + 0 = a. Now, we show that, for all good sequences x, y, z in an MV-monoidal algebra, we have (x + y) + z = x + (y + z). A direct verification, which seems difficult in general, becomes treatable when y is of the form (y 0 , 0, 0, . . . ). In fact, Light's associativity test guarantees that this is enough to imply associativity, thanks to the fact that the elements of the form (y 0 , 0, 0, . . . ) generate G(A). In the following, we carry out the details. Lemma 7.13. For every good sequence (a 0 , . . . , a n ) in an MV-monoidal algebra we have (a 0 , . . . , a n ) = (a 0 , . . . , a n−1 ) + (a n ).
Notation 7.14. A magma X; · consists of a set X and a binary operation · on X. Given a subset T of a magma X, we define, inductively on n ∈ N >0 , the subset T n ; we set T 1 := T , and, for n ∈ N >0 , T n := { tz | t ∈ T, z ∈ T n−1 } ∪ { zt | t ∈ T, z ∈ T n−1 }. Roughly speaking, T n is the set of elements of X which can be obtained with at most n occurrences of elements of T via application of the operation ·. We say that T generates X if n∈N>0 T n = X. Proof. By induction, using Lemma 7.13.
Lemma 7.16 (Light's associativity test). Let X; · be a magma, and let T be a subset of X that generates X. Suppose that, for every x, z ∈ X and t ∈ T , we have (xt)z = x(tz). Then, the operation · is associative.
Proof. Since T generates X, we have n∈N>0 T n = X. We prove, by induction on n ∈ N >0 , that, for every y ∈ T n , and every x, y ∈ X, we have (xy)z = x(yz). The case n = 1 is ensured by hypothesis. Let n 2, and suppose that the cases 1, . . . , n − 1 hold. Then, either y = ty ′ or y = y ′ t, for some t ∈ T and y ′ ∈ T n . Suppose, for example, y = ty ′ . Then, we have The case y = y ′ t is analogous.
Lemma 7.17. Let A be an MV-monoidal algebra, let (a 0 , a 1 ) and (b 0 , b 1 ) be good pairs in A, and let x ∈ A. Then and both sides coincide with a 1 ⊕ σ(a 0 , x, b 0 ) ⊕ b 1 .
Proof. Since the pair (a 0 , a 1 ) is good, it follows from Lemma 7.6 that the pair (a 0 ⊕ (x ⊙ b 0 ) ⊕ b 1 , a 1 ) is good. Since the pair (b 0 , b 1 ) is good, it follows from Lemma 7.6 that the pair (x ⊕ b 0 , b 1 ) is good. Therefore, we have Lemma 7.18. Let A be an MV-monoidal algebra, let n ∈ N >0 , let (a 0 , . . . , a n ) and (b 0 , . . . , b n ) be good sequences in A, and let x ∈ A. Then Proof. We prove the statement by induction on n ∈ N >0 . The case n = 1 is Lemma 7.17. Now let n ∈ N \ {0, 1}, and suppose that the statement holds for n − 1. In the following chain of equalities, the second equality is obtained by an application of Lemma 7.17 with respect to the good pairs (a n−1 , a n ) and (b 0 , b 1 ), and the third equality is obtained by an application of the inductive hypothesis with respect to the good sequences (a 0 , . . . , a n−1 ) and (b 1 , . . . , b n ).
Lemma 7.19. Let A be an MV-monoidal algebra, let a and b be good sequences in A, and let x ∈ A. Then, Proof. Set d := a + (x) and e := (x) + b. For every n ∈ N, we have d n = a n ⊕ (a n−1 ⊙x) and e n = (x⊙b n−1 )⊕b n . We set f := (a+(x))+b and g := a+((x)+b). For every n ∈ N, we have Proposition 7.20. Addition of good sequences is associative.
Our next aim-reached in Proposition 7.23 below-is to show that good sequences satisfy a + (b ∨ c) = (a + b) ∨ (a + c). We need some lemmas.
Lemma 7.21. Let A be an MV-monoidal algebra, let (x 0 , x 1 ) and (y 0 , y 1 ) be good pairs in A and let z ∈ A. Then where the last equality follows from Lemma 6.9. We have Proof. We set d := (a) + (b ∨ c). Then, We set f := (a) + b, g := (a) + c and h := f ∨ g = ((a) + b) ∨ ((a) + c). We have Proof. Let us prove the first equality: the second one is analogous. Let A be the MV-monoidal algebra of the statement. SetÂ := { (x) ∈ G(A) | x ∈ A }. By Lemma 7.15,Â generates the magma G(A); + . Following Notation 7.14, for n ∈ N >0 , we letÂ n denote the set of elements of G(A) which can be obtained with at most n occurrences of elements ofÂ via application of +. We prove by induction on n ∈ N >0 that, for all a ∈Â n , and b, c ∈ G(A), we have a+(b∨c) = (a+b)∨(a+c). The case n = 1 is Lemma 7.22. Suppose that the statement holds for n ∈ N >0 , and let us prove it for n + 1. Let a ∈Â n+1 , and let b, c ∈ G(A). Then, there exists a ′ ∈Â n and x ∈Â such that a = a ′ + x or a = x + a ′ . Since addition is commutative by Proposition 7.11, these two conditions are equivalent. So, For a good sequence a = (a 0 , a 1 , a 2 , . . . ) in an MV-monoidal algebra, set a ⊖ 1 : =  (a 1 , a 2 , a 3 , . . . ). The sequence a ⊖ 1 is a good sequence.  (1, a 0 , a 1 , a 2 , . . . ). Therefore, we have Axiom P2, i.e., for all a ∈ G(A), a + 1 ⊖ 1 = a. For all a ∈ G(A), we have (a ⊖ 1) + 1 = a ∨ 1, which establishes Axiom P3. By induction, one proves n1 = (1, . . . , 1 n times , 0, 0, 0 . . . ). Since 1 is the maximum of A, we have Axiom P4, i.e., for all a ∈ G(A), there exists n ∈ N such that a n1.
Given a morphism of MV-monoidal algebras f : A → B, we set Lemma 7.25. For every morphism f of MV-monoidal algebras, the function G(f ) is a morphism of positive-unital commutative ℓ-monoids.
It is easy to see that G : MVM → uℓM + is a functor. Proof. The facts that η 1 A is a bijection and that it preserves 0, 1, ∨, ∧ are immediate. Let x, y ∈ A. Then, (x) + (y) = (x ⊕ y, x ⊙ y). Therefore Proposition 8.2. η 1 : 1 MVM→ UG is a natural transformation, i.e., for every morphism of MV-monoidal algebras f : A → B, the following diagram commutes.
8.3. The counit. For each positive-unital commutative ℓ-monoid, we consider the function Our next goal, met in Proposition 8.13, is to prove that ε 1 M is bijective; this will show that a positive-unital commutative ℓ-monoid M is in bijection with the set of good sequences in its unit interval U(M ). Lemma 8.3. Let M be a positive-unital commutative ℓ-monoid. For every x ∈ M and every n ∈ N, we have Proof. We have Since n is cancellative by Lemma 4.2, it follows that (x ∧ n) + (x ⊖ n) = x.
Lemma 8.4. Let M be a positive-unital commutative ℓ-monoid, let x ∈ M and let n ∈ N. If x n, then x ⊖ n = 0.
By Lemma 4.2, the element n is cancellative: it follows that x ⊖ n = 0.
Proof. We prove the statement by induction on m ∈ N. The case m = 0 is trivial. The case m = 1 holds by Remark 8.9. Suppose the statement holds for m ∈ N >0 , and let us prove it holds for m + 1. We have the following chain of equalities, the first of which is justified by the fact that x 0 + · · · + x m−1 + 1 1.
The case k = 0 is trivial. Let us suppose that the statement holds for a fixed k ∈ N, and let us prove that it holds for k + 1. We have x 0 + · · · + x m = y 0 + · · · + y m , then, for all i ∈ {0, . . . , m}, x i = y i .
Proposition 8.13. Let M be a positive-unital commutative ℓ-monoid, and let x ∈ M . Then, there exists exactly one good sequence (x 0 , . . . , x m ) in U(M ) such that x = x 0 + · · · + x m , given by In particular, the function ε 1 M : GU(M ) → M is bijective.
M (x 0 , . . . , x m )). 8.4. The equivalence. We are ready to prove the main result of the paper.

Further research
Some results about commutative ℓ-monoids in the literature suggest similar ones for algebras in the language {⊕, ⊙, ∨, ∧, 0, 1}. For example, in [18] it is shown that the variety generated by R; +, ∨, ∧ does not admit a finite equational basis, and a countable basis is given in the same paper. Building on these results, the content of the present paper may possibly serve to obtain a nice equational basis for the variety generated by [0, 1]; ⊕, ⊙, ∨, ∧, 0, 1 which, we conjecture, is not finitely based; in particular, we conjecture that the variety of MVM-algebras is not generated by [0, 1].
Appendix A. The equivalence restricts to lattice-ordered groups and MV-algebras In this section, we shortly hint at the fact that Mundici's equivalence follows from our main result.
Proof. Since [0, 1] generates the variety of MV-algebras [7, Theorem 2.3.5], it suffices to check that Axioms A1 to A7 hold in [0,1]. This is the case because R is easily seen to be a unital commutative ℓ-monoid and thus, by Proposition 3.6, the unit interval [0, 1] is an MV-monoidal algebra.
Using Remarks A.1 and A.3, it is not too difficult to obtain the following.
Remark A.5. To establish Theorem A.4 we used the axiom of choice. Precisely, we used the choice-based fact that [0, 1] generates the variety of MV-algebras in order to verify that every MV-algebra is an MV-monoidal algebra (Lemma A.2). If one proved without the axiom of choice that the axioms of MV-monoidal algebras are satisfied by every MV-algebra (and we suspect this to be possible), one would have a choice-free proof of Mundici's equivalence. The properties of lattices, Axiom A2, the distributivity of ⊙ over ∨ and the distributivity of ⊕ over ∧ were part of the original axiomatization of MV-algebras by Chang [5], which, as proved in [13] (see also [8,Section 2]), is equivalent to the modern one, presented here. A direct proof of Axioms A6 and A7 has been obtained with the help of Prover9, but we have not obtained proofs of the distributivity of the lattice, the distributivity of ⊕ over ∨, the distributivity of ⊙ over ∧, and Axioms A4 and A5.
Appendix B. Subdirectly irreducible MV-monoidal algebras are totally ordered In this section we prove that every subdirectly irreducible MV-monoidal algebra is totally ordered (Theorem B.3). We proceed in analogy with [19,Section 1]. Given an MV-monoidal algebra A, and a lattice congruence θ on A such that |A/θ| = 2, we set moreover, with 0(θ) and 1(θ) we denote the classes of the lattice congruence θ corresponding to smallest and greatest elements of the lattice A/θ. An MVMcongruence on an MV-monoidal algebra A is an equivalence relation on A × A that respects ⊕, ⊙, ∨, ∧, 0, 1.
Lemma B.1. Let A be an MV-monoidal algebra, and let θ be any lattice congruence on A such that |A/θ| = 2. Then, θ * is the greatest MVM-congruence contained in θ.
Proof. It is not difficult to prove that θ * ⊆ θ and that θ * contains every congruence contained in θ.
We prove that θ * is an MVM-congruence. The relation θ * is an equivalence relation because θ is so.
We denote with ∆ the identity relation { (s, s) | s ∈ A }.
Lemma B.2. If A is a subdirectly irreducible MV-monoidal algebra, then there exists a lattice congruence θ on A such that |A/θ| = 2 and θ * = ∆.
Proof. Since A is distributive as a lattice, it can be decomposed into a subdirect product of two-element lattices. Let {θ i } i∈I be the set of lattice congruences of A corresponding with such a decomposition. Then i∈I θ i = ∆. By Lemma B.1, each θ * i is an MVM-congruence, and ∆ ⊆ θ * i ⊆ θ i . Therefore we have i∈I θ * i = ∆, and the fact that A is subdirectly irreducible implies θ * j = ∆ for some j ∈ I. Theorem B.3. Every subdirectly irreducible MV-monoidal algebra is totally ordered.
Proof. Let A be a subdirectly irreducible MV-monoidal algebra. Lemma B.2 entails that there exists a lattice congruence θ on A such that |A/θ| = 2 and such that θ * = ∆, i.e., for all distinct a, b ∈ A, there exists x ∈ A such that (a ⊕ x, b ⊕ x) / ∈ θ or (a ⊙ x, b ⊙ x) / ∈ θ. Let a, b ∈ A. We shall prove that either a b or b a holds. Suppose, by way of contradiction, that this is not the case, i.e., a ∧ b = a and a ∧ b = b. Since a ∧ b = a, there exists x ∈ A such that ((a ∧ b) ⊕ x, a ⊕ x) / ∈ θ or ((a ∧ b) ⊙ x, a ⊙ x) / ∈ θ. Since a∧b = b, there exists y ∈ A such that ((a∧b)⊕ y, b ⊕ y) / ∈ θ or ((a∧b)⊙ y, b ⊙ y) / ∈ θ. We have four cases.
Appendix C. Good pairs in subdirectly irreducible MV-monoidal algebras The goal of this section-met in Corollary C.6-is to show that good sequences in a subdirectly irreducible MV-monoidal algebra are of the form (1, . . . , 1, x, 0, 0, . . . ).
Notation C.1. Let A be an MV-monoidal algebra and let x ∈ A. For a, a ′ ∈ A, set a ∼ x ⊥ a ′ if, and only if, there exist n, m ∈ N such that a ⊕ (x ⊕ · · · ⊕ x n times ) a ′ , a ′ ⊕ (x ⊕ · · · ⊕ x m times ) a.
Moreover, set a ∼ ⊤ x a ′ if, and only if, there exist n, m ∈ N such that b ⊙ (x ⊙ · · · ⊙ x n times It is not difficult to prove the following.
Lemma C.2. For every MV-monoidal algebra A and every x ∈ A, the relation ∼ x ⊥ is the smallest MVM-congruence ∼ on A such that x ∼ 0, and the relation ∼ ⊤ x is the smallest MVM-congruence ∼ on A such that x ∼ 1.
Lemma C.3. Let A be an MV-monoidal algebra, let (x 0 , x 1 ) be a good pair in A, and let a, b ∈ A be such that a b ⊕ x 1 and a ⊙ x 0 b. Then, a b.
Proof. Let us first deal with the case b a; under this hypothesis, we shall prove a = b. Since a b ⊕ x 1 , we have a ⊙ x 0 (b ⊕ x 1 ) ⊙ x 0 Lemma 6.9 = σ(b, x 0 , x 1 ).
Since a ⊙ x 0 σ(b, x 0 , x 1 ) and a ⊙ x 0 b, we have Since b a, we have b ⊙ x 0 a ⊙ x 0 . Hence, a ⊙ x 0 = b ⊙ x 0 . Analogously, a ⊕ x 1 = b ⊕ x 1 . Hence σ(a, x 0 , x 1 ) Set s := σ(a, x 0 , x 1 ) = σ(b, x 0 , x 1 ). To prove a = b it is enough to prove a ∨ s = b ∨ s and a ∧ s = b ∧ s. We have Hence, a = b. If we do not assume b a, we may replace b with a ∧ b, because a (a ⊕ x 1 ) ∧ (b ⊕ x 1 ) = (a ∧ b) ⊕ x 1 , and a ⊙ x 0 a ∧ b. Since a ∧ b a, by the previous part we have a ∧ b = a, i.e., a b.