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1.
Motivated by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras we define
the functor K
• relating integral residuated lattices with 0 (IRL0) with certain involutive residuated lattices. Our work is also based on the results obtained by Cignoli about an adjunction
between Heyting and Nelson algebras, which is an enrichment of the basic adjunction between lattices and Kleene algebras.
The lifting of the functor to the category of residuated lattices leads us to study other adjunctions and equivalences. For
example, we treat the functor C whose domain is cuRL, the category of involutive residuated lattices M whose unit is fixed by the involution and has a Boolean complement c (the underlying set of C
M is the set of elements greater or equal than c). If we restrict to the full subcategory NRL of cuRL of those objects that have a nilpotent c, then C is an equivalence. In fact, C
M is isomorphic to C
e
M, and C
e is adjoint to , where assigns to an object A of IRL0 the product A × A
0 which is an object of NRL. 相似文献
2.
In this paper, a theorem on the existence of complete embedding of partially ordered monoids into complete residuated lattices is shown. From this, many interesting results on residuated lattices and substructural logics follow, including various types of completeness theorems of substructural logics. 相似文献
3.
Jeffrey S. Olson 《Studia Logica》2006,83(1-3):393-406
CRS(fc) denotes the variety of commutative residuated semilattice-ordered monoids that satisfy (x ⋀ e)k ≤ (x ⋀ e)k+1. A structural characterization of the subdi-rectly irreducible members of CRS(k) is proved, and is then used to provide a
constructive approach to the axiomatization of varieties generated by positive universal subclasses of CRS(k).
Dedicated to the memory of Willem Johannes Blok 相似文献
4.
Algebraic Aspects of Cut Elimination 总被引:2,自引:2,他引:0
We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17]. 相似文献
5.
The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of
propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class
of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed
under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological
Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators
in the above-mentioned classes.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献