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| Title: | Cut elimination and strong separation for substructural logics: An algebraic approach |
| Authors: | Galatos, Nikolaos
Ono, Hiroakira |
| Keywords: | substructural logics
cut elimination
strong separation
Gentzen system
Hilbert system
residuated lattice |
| Issue Date: | 2010-04-17 |
| Publisher: | Elsevier |
| Magazine name: | Annals of Pure and Applied Logic |
| Volume: | 161 |
| Number: | 9 |
| Start page: | 1097 |
| End page: | 1133 |
| DOI: | 10.1016/j.apal.2010.01.003 |
| Abstract: | We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus FL. We present a Gentzen-style sequent system GL that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of FL. Moreover, we introduce an equivalent Hilbert-style system HL and show that the logic associated with GL and HL is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics. Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of GL and various extensions, the strong separation of HL, and the finite generation of the variety of residuated lattice-ordered groupoids with unit. |
| Rights: | NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Nikolaos Galatos and Hiroakira Ono, Annals of Pure and Applied Logic, 161(9), 2010, 1097-1133, http://dx.doi.org/10.1016/j.apal.2010.01.003 |
| URI: | http://hdl.handle.net/10119/9205 |
| Material Type: | author |
| Appears in Collections: | z4-10-1. 雑誌掲載論文 (Journal Articles)
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