Reports on Mathematical Logic

No. 43

Hitoshi KIHARA and Hiroakira ONO

Algebraic Characterizations of Variable Separation Properties

A b s t r a c t. This paper gives algebraic characterizations of Halld\'{e}n completeness (HC), and of Maksimova's variable separation property (MVP) and its deductive form. Though algebraic characterizations of these properties have been already studied for modal and superintuitionistic logics, e.g. in Wro\'{n}ski [12] ("Remarks on Halld\'{e}n-completeness of modal and intermediate logics", Bulletin of the Section of Logic 5, 4 (1976), pp.126--129), Maksimova [7] ("The principle of separation of variables in propositional logics", Algebra i Logika 15 (1976), pp.168--184), [9] ("n variable separation in modal and superintuitionistic logics", Studia Logica 55 (1995), pp.99--112), a deeper analysis of these properties and non-trivial modifications of these results are needed to extend them to those for substructural logics, because of the lack of some structural rules in them. The first attempt in this direction was made in the dissertation \cite{Kih06} of the first author. Results of this paper are partly announced (sometimes in their weaker form) also in Chapter 5 of the book [2] (N. Galatos, P. Jipsen, T. Kowalski and H. Ono, "Residuated Lattices: An algebraic glimpse at substructural logics",
Studies in Logic and the Foundations of Mathematics, vol.151, Elsevier, 2007).

School of Information Science, Japan Advanced Institute of Science and Technology, Asahidai, Nomi, Ishikawa, 923-1292, Japan

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