Priestley Duality for Quasi-Stone Algebras

In this paper we describe the Priestley space of a quasi-Stone algebra and use it to show that the class of finite quasi-Stone algebras has the amalgamation property. We also describe the Priestley space of the free quasi-Stone algebra over a finite set. This revised version was published online in June 2006 with corrections to the Cover Date.     

Distributive Full Lambek Calculus Has the Finite Model Property

We prove the Finite Model Property (FMP) for Distributive Full Lambek Calculus (DFL) whose algebraic semantics is the class of distributive residuated lattices (DRL). The problem was left open in [8, 5]. We use the method of nuclei and quasi–embedding in the style of [10, 1]. Presented by Daniele Mundici.     

Representation of Game Algebras

We prove that every abstractly defined game algebra can be represented as an algebra of consistent pairs of monotone outcome relations over a game board. As a corollary we obtain Goranko's result that van Benthem's conjectured axiomatization for equivalent game terms is indeed complete.     

A Proof of Standard Completeness for Esteva and Godo's Logic MTL

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo's logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.     

On Some Categories of Involutive Centered Residuated Lattices

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.     

Dominions in quasivarieties of universal algebras

The dominion of a subalgebra H in an universal algebra A (in a class ) is the set of all elements such that for all homomorphisms if f, g coincide on H, then a^f = a^g. We investigate the connection between dominions and quasivarieties. We show that if a class is closed under ultraproducts, then the dominion in is equal to the dominion in a quasivariety generated by . Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.Special issue of Studia Logica: Algebraic Theory of Quasivarieties Presented by M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko