Abstract
We consider varieties of pointed lattice-ordered algebras satisfying a restricted distributivity condition and admitting a very weak implication. Examples of these varieties are ubiquitous in algebraic logic: integral or distributive residuated lattices; their \(\left\{ \cdot \right\} \)-free subreducts; their expansions (hence, in particular, Boolean algebras with operators and modal algebras); and varieties arising from quantum logic. Given any such variety \(\fancyscript{V}\), we provide an explicit equational basis (relative to \(\fancyscript{V}\)) for the semi-linear subvariety \(\fancyscript{W}\) of \(\fancyscript{V}\). In particular, we show that if \(\fancyscript{V}\) is finitely based, then so is \(\fancyscript{W}\). To attain this goal, we make extensive use of tools from the theory of quasi-subtractive varieties.
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Notes
- 1.
From now on, when we speak of the semilinear subvariety of a given variety \(\fancyscript{V}\), we invariably mean its largest semilinear subvariety. This is the variety generated by all totally ordered members of \(\fancyscript{V}\), equivalently, all totally ordered subdirectly irreducible members of \(\fancyscript{V}\).
- 2.
If \(F\subseteq A\) and \(\mathbf {A}\) has the same signature as \(\mathbf {Fm}\), \(F\) is said to be a deductive filter on \(\mathbf {A}\) of the logic \(\left( \mathbf {Fm,}\vdash \right) \) just in case \(F\) is closed with respect to all the \(\vdash \)-entailments: if \(\varGamma \vdash t\) and \(s^{\mathbf {A}}\left( \overrightarrow{a}\right) \in F\) for all \(s\in \varGamma \), then \(t^{\mathbf {A}}\left( \overrightarrow{a}\right) \in F\).
- 3.
The variety of residuated lattices is actually \(1\)-ideal determined and, in fact, in every residuated lattice the lattice of congruences is isomorphic to the lattice of ideals in the sense of Gumm-Ursini, which in turn coincide with convex normal subalgebras of such. There is a further isomorphism theorem, however (namely, between congruences and deductive filters in the sense of Galatos et al. (2007)), which does not instantiate the correspondence theorem for ideal determined varieties, but follows from Corollary 10.1.
- 4.
It should be noted that a more delicate analysis in Blount and Tsinakis (2003) demonstrates that (D) can be omitted from the hypothesis of the theorem. Such refinements of special instances of a general result are to be expected.
- 5.
Compare (Galatos et al. (2007), p. 426).
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Acknowledgments
We would like to express our appreciation to the unknown referees of this chapter for reading it with care and for offering perceptive suggestions. After submitting this chapter, we became aware of the paper Cintula et al. (2013) whose results provide a logical counterpart of some algebraic results contained in the present article. We warmly thank Petr Cintula for the pointer. The first author acknowledges the support of the Italian Ministry of Scientic Research within the FIRB project “Structures and dynamics of knowledge and cognition”, Cagliari: F21J12000140001.
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Ledda, A., Paoli, F., Tsinakis, C. (2015). Semi-linear Varieties of Lattice-Ordered Algebras. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_10
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