Abstract
The objective of this work is to show that modules over a fixed residuated lattice \({\mathbf {R}}\)—that is, partially ordered sets acted upon by \({\mathbf {R}}\)—provide a suitable algebraic framework for extending the concept of a recognizable language as defined by Kleene. More specifically, we introduce the notion of a recognizable element of \({\mathbf {R}}\) by a finite module and provide a characterization of such an element in the spirit of Myhill’s characterization of recognizable languages. Further, we investigate the structure of the set of recognizable elements of \({\mathbf {R}}\) and also provide sufficient conditions for a recognizable element to be recognized by a Boolean module.
This work is dedicated to Hiroakira Ono in recognition of his many contributions to Mathematical Logic.
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Notes
- 1.
See Definition 3.15.
- 2.
Usually, the alphabet is taken to be finite, because of the applications of automata to Computer Science, but none of the results that we mention here depend on the finiteness of the alphabet.
- 3.
This example was suggested by N. Galatos.
- 4.
See Example 3.6.
- 5.
The reader may be familiar with the concept of a Boolean module as introduced by Brink in [6], but those are not exactly the same kind of structures, as Brink’s Boolean modules are modules over relation algebras.
- 6.
This is because if \(\gamma \) is a nucleus on a frame \({\mathbf {F}}\), then \(\gamma \) is a closure operator on \({\mathbb {F}}\), and therefore \({\mathbb {F}}_\gamma = \langle {\mathbf {F}}_\gamma ,\wedge \rangle \) is an \({\mathbf {F}}\)-module. In particular, \(\wedge \) is residuated on both coordinates on \({\mathbf {F}}_\gamma \), and therefore it distributes with respect to arbitrary joins.
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Acknowledgements
The present work is a natural sequel of Hoseung Lee’s Ph.D. dissertation [20], which includes a number of joint results of Lee with the second author. We would like to express our appreciation to the referee for the careful reading and for contributing valuable information on previous work, including Remark 5.7. The first author acknowledges the support of the grants MTM2011-25747 of the Spanish Ministry of Science and Innovation (which includes eu’s feder funds) and 2009SGR-01433 of the Catalan Government. The second author wishes to express his appreciation to Bjarni Jónsson for many stimulating discussions about Boolean modules and related topics.
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Gil-Férez, J., Tsinakis, C. (2022). Recognizability in Residuated Lattices. 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_6
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