We introduce Priestley rings of upsets (of a poset) and prove that inequivalent Priestley ring representations of a bounded distributive lattice L are in 1-1 correspondence with dense subspaces of the Priestley space of L. This generalizes a 1955 result of Bauer that inequivalent reduced field representations of a Boolean algebra B are in 1-1 correspondence with dense subspaces of the Stone spa...

The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topologi...

A subresiduated lattice is a pair (A, D), where bounded distributive lattice, D sublattice of and for every $$a,b\in A$$ there $$c\in D$$ such that all $$d\in , $$d\wedge a\le b$$ if only $$d\le c$$ . This c denoted by $$a\rightarrow can be regarded as an algebra $$\left$$ type (2, 2, 0, 0) $$D=\{a\in A\mid 1\rightarrow a=a\}$$ The class lattices variety ...

It is a common trend in mathematics to study (natural) dualities for general algebraic structures and, in particular, for those arising from mathematical logic. The first step towards this direction traces back to the pioneering work by Stone for Boolean algebras [12]. Later on, Stone duality has been extended to the more general case of distributive lattices by Priestley [8], [9]. The two abov...

In 2001, an open question that was posed by Blyth, Silva and Varlet in [2] is the following: for an Ockham algebra L determine the congruences on L that are kernels of endomorphisms on L. Whereas a general solution to this is stll an open problem, there has been some progress in investigating kernels of endomorphisms in various lattice-ordered algebras. In this connection, an algebra A is said ...

In [12] the study of Positive Modal Logic (PML) is initiated using standard Kripke semantics and the positive modal algebras (a class of bounded distributive lattices with modal operators) are introduced. The minimum system of Positive Modal Logic is the (∧,∨, 2, 3,⊥,>)-fragment of the local consequence relation defined by the class of all Kripke models. It can be axiomatized by a sequent calcu...

To mark the centennial of the discovery of oxygen by Joseph Priestley (1733-1804) on August 1, 1774, H. Carrington Bolton of the Columbia College School of Mines suggested a gathering of American chemists be held to celebrate this event. A suggestion was made by Rachel Bodley of the Women’s Medical College of Pennsylvania that an appropriate place would be Northumberland, Pennsylvania, where Pr...