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Relational models of Lambek logics. (English) Zbl 1203.03029
de Swart, Harrie (ed.) et al., Theory and applications of relational structures as knowledge instruments. COST Action 274, TARSKI. Revised papers. Berlin: Springer (ISBN 3-540-20780-5/pbk). Lect. Notes Comput. Sci. 2929, 196-213 (2003).
Summary: Lambek logics are substructural logics related to the Syntactic Calculus of Lambek. In this paper we prove several representation theorems for algebras of Lambek logics (residuated semigroups, residuated monoids and others) with respect to certain algebras of binary relations. The first results of this kind were obtained by H. Andréka and S. Mikulás [“Lambek calculus and its relational semantics: Completeness and incompleteness”, J. Logic Lang. Inf. 3, No. 1, 1–37 (1994; Zbl 0808.03003)] using a method of labeled graphs. Other results were proved in [W. Buszkowski and M. Kołowska-Gawiejnowicz, “Representation of residuated semigroups in some algebras of relations (the method of canonical models)”, Ann. Soc. Math. Pol., Ser. IV, Fundam. Inf. 31, No. 1, 1–12 (1997; Zbl 0876.03036); M. Szczerba, “Relational models for the nonassociative Lambek calculus”, in: E. Orłowska et al. (eds.), Relational methods for computer science applications. Heidelberg: Physica Verlag, 149–159 (2001; Zbl 0993.03026)] using a method of labeled formulas. In the present paper, we prove these and other results using a construction of chains of partial representations. We also provide a simpler construction which works for right and left pregroups [W. Buszkowski, “Pregroups: Models and grammars”, Lect. Notes Comput. Sci. 2561, 35–49 (2002; Zbl 1027.68069)].
For the entire collection see [Zbl 1029.00017].
03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
03G25 Other algebras related to logic
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