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A finite axiomatization of positive MV-algebras

Abstract

Positive MV-algebras are the subreducts of MV-algebras with respect to the signature \(\{\oplus , \odot , \vee , \wedge , 0, 1\}\). We provide a finite quasi-equational axiomatization for the class of such algebras.

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Acknowledgements

We acknowledge the contribution of L. M. Cabrer in shaping the first ideas on positive MV-algebras at the initial stage of this research. We are grateful to the anonymous referee, who provided useful and detailed comments on the manuscript.

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Correspondence to Sara Vannucci.

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Presented by N. Galatos.

Marco Abbadini was supported by the Italian Ministry of University and Research through the PRIN project n. 20173WKCM5 Theory and applications of resource sensitive logics. Tomáš Kroupa and Sara Vannucci acknowledge the support by the project Research Center for Informatics (CZ.02.1.01/0.0/0.0/16_019/0000765).

Appendix A. Free MV-extensions

Appendix A. Free MV-extensions

By standard results in general algebra, the forgetful functor U from MV-algebras to positive MV-algebras has a left adjoint F. For every positive MV-algebra \(\mathbf {A}\), it is immediate that the component \(\iota _{\mathbf {A}} :\mathbf {A} \rightarrow UF(\mathbf {A})\) at \(\mathbf {A}\) of the unit is injective, and that the image of \(\iota _{\mathbf {A}}\) generates the MV-algebra \(F(\mathbf {A})\). We speak of the pair \((F(\mathbf {A}), \iota _{\mathbf {A}})\) (or simply of \(F(\mathbf {A})\), leaving \(\iota _{\mathbf {A}}\) understood) as the free MV-extension of \(\mathbf {A}\). In this section we prove that, given a positive subreduct \(\mathbf {A}\) of an MV-algebra \(\mathbf {B}\) such that A generates \(\mathbf {B}\), and denoting with i the inclusion of A into B, \((\mathbf {B}, i)\) is the free MV-extension of \(\mathbf {A}\); moreover, under the same conditions, for every \(x \in B\), there are \(a_1, \dots , a_n, b_1, \dots , b_n \in A\) such that \(x = \bigoplus _{j = 1}^{n} (a_j \odot \lnot b_j)\).

The following Lemma corresponds, essentially, to the fact that Mundici’s bijection between \(\mathbf {G}\) and \(\Xi (\Gamma (\mathbf {G}))\) preserves \(+\). We refer to [2, Proposition 4.67] for a proof in the setting of unital commutative distributive \(\ell \)-monoids.

Lemma A.1

Let \(\mathbf {G}\) be a unital abelian \(\ell \)-group, and let \(x, y \in G\). For each \(n \in \mathbb {Z}\), set \(x_n \,{:}{=}\,((x -n) \vee 0) \wedge 1)\) and \(y_n \,{:}{=}\,((x -n) \vee 0) \wedge 1)\). We have

$$\begin{aligned} ((x + y) \vee 0) \wedge 1 = \bigoplus _{n \in \mathbb {Z}} (x_n \odot y_{-n-1}) = \bigodot _{n \in \mathbb {Z}} (x_n \oplus y_{-n}). \end{aligned}$$

(The expression \(\bigoplus _{n \in \mathbb {Z}} (x_n \odot y_{-n-1})\) makes sense because for all but finitely many \(n \in \mathbb {Z}\) we have \(x_n \odot y_{-n-1} = 0\). Analogously, the expression \(\bigodot _{n \in \mathbb {Z}} (x_n \oplus y_{-n})\) makes sense because for all but finitely many \(n \in \mathbb {Z}\) we have \(x_n \oplus y_{-n} = 1\).)

Lemma A.2

Let \(\mathbf {G}\) be a unital abelian \(\ell \)-group. Set

$$\begin{aligned} \mathcal {M} \,\,{:}{=}\,\, \{ M \subseteq G \mid M \text { is closed under }+, \vee , \wedge , 0, 1, -1 \} \end{aligned}$$

and

$$\begin{aligned} \mathcal {A}\, \,{:}{=}\,\,\{ A \subseteq \Gamma (\mathbf {G}) \mid A \text { is closed under } \oplus , \odot , \vee , \wedge , 0, 1 \}. \end{aligned}$$
  1. (1)

    The sets \(\mathcal {M}\) and \(\mathcal {A}\) are in bijection, as witnessed by the functions

  2. (2)

    The bijection in (1) restricts to the subsets

    $$\begin{aligned} \{M \in \mathcal {M} \mid M \text { is closed under }-\} \text { and }\{A \in \mathcal {A} \mid A \text { is closed under }\lnot \}. \end{aligned}$$
  3. (3)

    For every \(M \in \mathcal {M}\), M generates the unital abelian \(\ell \)-group \(\mathbf {G}\) if and only if \(\{ x \in M \mid 0 \leqslant x \leqslant 1 \}\) generates the MV-algebra \(\Gamma (\mathbf {G})\).

Proof

The fact that the function \(f :\mathcal {M} \rightarrow \mathcal {A}\) is well-defined is immediate.

Let us prove that the function \(g :\mathcal {A} \rightarrow \mathcal {M}\) is well-defined. Let \(A \in \mathcal {A}\), and set

$$\begin{aligned} M \,\,{:}{=}\,\, \{ x \in G \mid \forall n \in \mathbb {Z}\ ((x - n) \vee 0) \wedge 1 \in A \}. \end{aligned}$$

The set M is closed under \(+\) by Lemma A.1. Moreover, it is closed under \(\vee \) because, for all \(x, y \in M\), we have \(((x \vee y) \vee 0) \wedge 1 = ((x \vee 0) \wedge 1) \vee ((x \vee 0) \wedge 1)\). Analogously, it is closed under \(\wedge \). It is not difficult to prove that \(0, 1, -1 \in M\). Therefore, \(M \in \mathcal {M}\), and thus g is well-defined.

It is easy to see that the composite \(f \circ g :\mathcal {A} \rightarrow \mathcal {A}\) is the identity on \(\mathcal {A}\). To prove that \(g \circ f\) is the identity on \(\mathcal {M}\), let \(M \in \mathcal {M}\). We shall prove that

$$\begin{aligned} M = \{x \in G \mid \forall n \in \mathbb {Z}\ (x - n) \vee 0) \wedge 1 \in M\}. \end{aligned}$$

The left-to-right inclusion is immediate. For the converse inclusion, let \(x \in G\) be such that, for every \(n \in \mathbb {Z}\), \(((x - n) \vee 0) \wedge 1 \in M\). Let \(n \in \mathbb {N}\) be such that \(-n \leqslant x \leqslant n\). Then (cf. Lemma 3.6)

$$\begin{aligned} x = -n + \sum _{i = -n}^{n-1} (((x - i) \vee 0) \wedge 1), \end{aligned}$$

and thus \(x \in M\). This proves (1).

If \(M \in \mathcal {M}\) is closed under −, f(M) is easily seen to be closed under \(\lnot \). If \(A \in \mathcal {A}\) is closed under \(\lnot \), g(A) is closed under − because, for every \(x \in g(A)\) and every \(n \in \mathbb {Z}\), we have

$$\begin{aligned} ((-x - n) \vee 0) \wedge 1 = 1 - (((x + (n+1)) \vee 0) \wedge 1) = \lnot (((x + (n+1)) \vee 0) \wedge 1). \end{aligned}$$

This proves (2).

Since the bijection in (1) preserves the inclusion order in both directions, we have (3). \(\square \)

It is a well-known fact that, if we consider a Boolean algebra \(\mathbf {B}\) generated by a bounded distributive sublattice \(\mathbf {D}\), any element of B can be written as a finite join of elements of the form \(x \wedge \lnot y\) (or, equivalently, as a finite meet of elements of the form \(x \vee \lnot y\)) for \(x,y \in D\). An analogous result holds for positive MV-algebras and MV-algebras. Whereas the result for Boolean algebras can be derived by standard applications of the distributive and De Morgan’s laws, the corresponding one for MV-algebras is not straightforward. We shall present it in the next theorem.

Theorem A.3

Let \(\mathbf {A}\) be a positive subreduct of an MV-algebra \(\mathbf {B}\), and suppose that A generates the MV-algebra \(\mathbf {B}\). For every \(x \in B\) there are \(n,m \in \mathbb {N}\) and \(s_1, \dots , s_n, t_1, \dots , t_n, u_1, \dots , u_m, v_1, \dots , v_m \in A\) such that

$$\begin{aligned} x = \bigoplus _{i = 1}^{n} s_i \odot \lnot t_i = \bigodot _{i = 1}^{m} u_i \oplus \lnot v_i. \end{aligned}$$

Proof

By Lemma A.2, the set

$$\begin{aligned} M \,{:}{=}\,\{ x \in G \mid \forall n \in \mathbb {Z}\ ((x - n) \vee 0) \wedge 1 \in A \} \end{aligned}$$

is closed under \(+\), \(\vee \), \(\wedge \), 0, 1 and \(-1\) and generates the unital abelian \(\ell \)-group \(\mathbf {G}\). Let \(z \in B\). By Lemma 4.2, there are \(x,y \in M\) such that \(z = x - y\). For every \(n \in \mathbb {N}\), we have

$$\begin{aligned} ((-y - n + 1) \vee 0) \wedge 1&= 1 - (1 - (((-y - n + 1) \vee 0) \wedge 1))\\&= 1 - (1 + (((y + n - 1) \wedge 0) \vee -1))\\&= 1 - (((y + n) \wedge 1) \vee 0)\\&= 1 - (((y + n) \vee 0) \wedge 1)\\&= \lnot (((y + n) \vee 0) \wedge 1). \end{aligned}$$

Therefore, by Lemma A.1, we have

$$\begin{aligned} z&= \bigoplus _{n \in \mathbb {Z}} \big ((((x - n) \vee 0) \wedge 1) \odot (((-y - n + 1) \vee 0) \wedge 1)\big )\\&= \bigoplus _{n \in \mathbb {Z}} \big ((((x - n) \vee 0) \wedge 1) \odot \lnot (((y + n) \vee 0) \wedge 1)\big ). \end{aligned}$$

Since \(x,y \in M\), \(((x - n) \vee 0) \wedge 1, ((y + n) \vee 0) \wedge 1 \in A\). This proves the first equality in the statement. The second one is analogous. \(\square \)

Theorem A.4

Let M be a generating subset of a unital abelian \(\ell \)-group \(\mathbf {G}\), and suppose that M is closed under \(+\), \(\vee \), \(\wedge \), 0, 1 and \(-1\). For every unital Abelian \(\ell \)-group \(\mathbf {H}\) and every function \(f :M \rightarrow H\) that preserves \(+\), \(\vee \), \(\wedge \), 0, 1 and \(-1\), there exists a unique morphism \(g :\mathbf {G} \rightarrow \mathbf {H}\) of unital abelian \(\ell \)-groups that extends f.

Proof

This follows from Lemma 4.2. For every \(z \in G\) we define the morphism g as \(g(z) = f(x) - f(y)\) where x and y are elements of M such that \(z=x-y\). The fact that g is a well-defined morphism follows from the fact that \(x - y = u - v\) is equivalent to \(x + v = u + y\). \(\square \)

The following theorem generalizes an analogous result for bounded distributive lattices, namely that the inclusion of a bounded distributive lattice into a Boolean algebra that is generated by the image of such inclusion is universal, i.e. it is a so-called Booleanization, or free Boolean extension: see [11, Theorem 4.1] for a version of this result for (not necessarily bounded) distributive lattices.

Theorem A.5

Let \(\mathbf {A}\) be a positive MV-algebra, \(\mathbf {B}\) an MV-algebra, and \(i :A \hookrightarrow B\) an injective function that preserves \(\oplus \), \(\odot \), \(\vee \), \(\wedge \), 0 and 1 and such that its image generates the MV-algebra \(\mathbf {B}\). Then \((\mathbf {B}, i)\) is the free MV-extension of \(\mathbf {A}\).

Proof

Without loss of generality, we may suppose \(A \subseteq B\) and i to be the inclusion of A into B. Let \(\mathbf {C}\) be an MV-algebra, and let \(f :A \rightarrow C\) be a function that preserves \(\oplus \), \(\odot \), \(\vee \), \(\wedge \), 0 and 1. We shall prove that there exists a unique MV-homomorphism \(g :\mathbf {B} \rightarrow \mathbf {C}\) that extends f. Uniqueness follows from the fact that A generates the MV-algebra \(\mathbf {B}\). Let us prove existence. By the equivalence in Theorem 3.4, we obtain morphisms of unital commutative distributive \(\ell \)-monoids \(\Xi (i) :\Xi (\mathbf {A}) \rightarrow \Xi (\mathbf {B})\) and \(\Xi (f) :\Xi (\mathbf {A}) \rightarrow \Xi (\mathbf {C})\). By Proposition 3.7, \(\Xi (i)\) is injective, and we may suppose \(\Xi (i)\) to be an inclusion. By Theorem A.4, there exists a unique morphism \(g' :\Xi (\mathbf {B}) \rightarrow \Xi (\mathbf {C})\) of unital abelian \(\ell \)-groups that extends \(\Xi (f)\). Then, \(g \,{:}{=}\,\Gamma (g')\) is an MV-homomorphism that extends f. \(\square \)

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Abbadini, M., Jipsen, P., Kroupa, T. et al. A finite axiomatization of positive MV-algebras. Algebra Univers. 83, 28 (2022). https://doi.org/10.1007/s00012-022-00776-3

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Keywords

  • MV-algebras
  • Positive subreducts
  • Lattice-ordered monoids
  • Quasivarieties
  • Quasi-equations
  • Axiomatization

Mathematics Subject Classification

  • 06D35
  • 06F05
  • 08C15