Abstract
Positive MValgebras are the subreducts of MValgebras with respect to the signature \(\{\oplus , \odot , \vee , \wedge , 0, 1\}\). We provide a finite quasiequational 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 MValgebras 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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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 MVextensions
Appendix A. Free MVextensions
By standard results in general algebra, the forgetful functor U from MValgebras to positive MValgebras has a left adjoint F. For every positive MValgebra \(\mathbf {A}\), 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 MValgebra \(F(\mathbf {A})\). 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 MVextension of \(\mathbf {A}\). In this section we prove that, given a positive subreduct \(\mathbf {A}\) of an MValgebra \(\mathbf {B}\) such that A generates \(\mathbf {B}\), and denoting with i the inclusion of A into B, \((\mathbf {B}, i)\) is the free MVextension 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
(The expression \(\bigoplus _{n \in \mathbb {Z}} (x_n \odot y_{n1})\) makes sense because for all but finitely many \(n \in \mathbb {Z}\) we have \(x_n \odot y_{n1} = 0\). 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
and

(1)
The sets \(\mathcal {M}\) and \(\mathcal {A}\) are in bijection, as witnessed by the functions

(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)
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 MValgebra \(\Gamma (\mathbf {G})\).
Proof
The fact that the function \(f :\mathcal {M} \rightarrow \mathcal {A}\) is welldefined is immediate.
Let us prove that the function \(g :\mathcal {A} \rightarrow \mathcal {M}\) is welldefined. Let \(A \in \mathcal {A}\), and set
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 welldefined.
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
The lefttoright 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)
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
This proves (2).
Since the bijection in (1) preserves the inclusion order in both directions, we have (3). \(\square \)
It is a wellknown 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 MValgebras and MValgebras. Whereas the result for Boolean algebras can be derived by standard applications of the distributive and De Morgan’s laws, the corresponding one for MValgebras is not straightforward. We shall present it in the next theorem.
Theorem A.3
Let \(\mathbf {A}\) be a positive subreduct of an MValgebra \(\mathbf {B}\), and suppose that A generates the MValgebra \(\mathbf {B}\). 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
Proof
By Lemma A.2, the set
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
Therefore, by Lemma A.1, we have
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 \(z=xy\). The fact that g is a welldefined 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 socalled 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 MValgebra, \(\mathbf {B}\) an MValgebra, and \(i :A \hookrightarrow B\) an injective function that preserves \(\oplus \), \(\odot \), \(\vee \), \(\wedge \), 0 and 1 and such that its image generates the MValgebra \(\mathbf {B}\). Then \((\mathbf {B}, i)\) is the free MVextension 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 MValgebra, 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 MVhomomorphism \(g :\mathbf {B} \rightarrow \mathbf {C}\) that extends f. Uniqueness follows from the fact that A generates the MValgebra \(\mathbf {B}\). 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 MVhomomorphism that extends f. \(\square \)
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Abbadini, M., Jipsen, P., Kroupa, T. et al. A finite axiomatization of positive MValgebras. Algebra Univers. 83, 28 (2022). https://doi.org/10.1007/s00012022007763
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DOI: https://doi.org/10.1007/s00012022007763