Abstract
Let \(\mathsf {V}\) be a variety of residuated lattices axiomatized by a set of identities in the language \(\{\vee ,\cdot ,1\}\). We characterize when \(\mathsf {V}\) has the finite embeddability property via regularity of a certain collection of languages. Several applications of this characterization are presented.
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Notes
- 1.
In fact, \(\mathcal {L}({\mathbf {B}}^{*})\) is nothing else than the dual algebra \({\mathbf {W}}^{+}\) of unital residuated frame \({\mathbf {W}}=\langle {B^{*},B^{*}\times B^{*}\times B,N,\cdot ,{\diagdown \!\!\!\!\diagdown },{\diagup \!\!\!\!\diagup },\{\varepsilon \}}\rangle \) where \(x\mathrel {N}\langle {u,v,b}\rangle \) iff \(uxv\in L_{b}\) and \(x{\diagdown \!\!\!\!\diagdown }\langle {u,v,b}\rangle =\langle {ux,v,b}\rangle \), \(\langle {u,v,b}\rangle {\diagup \!\!\!\!\diagup }x=\langle {u,xv,b}\rangle \); for details see [10].
- 2.
- 3.
Gentzen collections are closely related to Gentzen residuated frames defined in [10]. More precisely, a Gentzen collection \(\mathcal {L}=\{L_{b}\subseteq B^{*}\mid b\in B\}\) for a finite partial subalgebra \({\mathbf {B}}\) of a residuated lattice \({\mathbf {A}}\) is nothing else than the basis of the nucleus induced by the unital Gentzen residuated frame \(\langle {{\mathbf {W}},{\mathbf {B}}}\rangle \), where \({\mathbf {W}}=\langle {B^{*},B^{*}\times B^{*}\times B,N,\cdot ,{\diagdown \!\!\!\!\diagdown },{\diagup \!\!\!\!\diagup },\{\varepsilon \}}\rangle \), \(x{\diagdown \!\!\!\!\diagdown }\langle {u,v,b}\rangle =\langle {ux,v,b}\rangle \), \(\langle {u,v,b}\rangle {\diagup \!\!\!\!\diagup }x=\langle {u,xv,b}\rangle \) and \(x\mathrel {N}\langle {u,v,b}\rangle \) iff \(uxv\in L_{b}\); for details see [10].
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Acknowledgements
The author was supported by the project GBP202/12/G061 of the Czech Science Foundation and by the long-term strategic development financing of the Institute of Computer Science (RVO:67985807).
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Horčík, R. (2022). Finite Embeddability Property for Residuated Lattices via Regular Languages. In: Galatos, N., Terui, K. (eds) Hiroakira Ono on Substructural Logics. Outstanding Contributions to Logic, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-030-76920-8_7
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