377 research outputs found
Representation of Nelson Algebras by Rough Sets Determined by Quasiorders
In this paper, we show that every quasiorder induces a Nelson algebra
such that the underlying rough set lattice is algebraic. We
note that is a three-valued {\L}ukasiewicz algebra if and only if
is an equivalence. Our main result says that if is a Nelson
algebra defined on an algebraic lattice, then there exists a set and a
quasiorder on such that .Comment: 16 page
Admissibility via Natural Dualities
It is shown that admissible clauses and quasi-identities of quasivarieties
generated by a single finite algebra, or equivalently, the quasiequational and
universal theories of their free algebras on countably infinitely many
generators, may be characterized using natural dualities. In particular,
axiomatizations are obtained for the admissible clauses and quasi-identities of
bounded distributive lattices, Stone algebras, Kleene algebras and lattices,
and De Morgan algebras and lattices.Comment: 22 pages; 3 figure
Stone-type representations and dualities for varieties of bisemilattices
In this article we will focus our attention on the variety of distributive
bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and
involutive bisemilattices. After extending Balbes' representation theorem to
bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn
duality and introduce the categories of 2spaces and 2spaces. The
categories of 2spaces and 2spaces will play with respect to the
categories of distributive bisemilattices and De Morgan bisemilattices,
respectively, a role analogous to the category of Stone spaces with respect to
the category of Boolean algebras. Actually, the aim of this work is to show
that these categories are, in fact, dually equivalent
Stone-Type Dualities for Separation Logics
Stone-type duality theorems, which relate algebraic and
relational/topological models, are important tools in logic because -- in
addition to elegant abstraction -- they strengthen soundness and completeness
to a categorical equivalence, yielding a framework through which both algebraic
and topological methods can be brought to bear on a logic. We give a systematic
treatment of Stone-type duality for the structures that interpret bunched
logics, starting with the weakest systems, recovering the familiar BI and
Boolean BI (BBI), and extending to both classical and intuitionistic Separation
Logic. We demonstrate the uniformity and modularity of this analysis by
additionally capturing the bunched logics obtained by extending BI and BBI with
modalities and multiplicative connectives corresponding to disjunction,
negation and falsum. This includes the logic of separating modalities (LSM), De
Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics
extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as
corollaries soundness and completeness theorems for the specific Kripke-style
models of these logics as presented in the literature: for DMBI, the
sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene
BI (connecting our work to Concurrent Separation Logic), this is the first time
soundness and completeness theorems have been proved. We thus obtain a
comprehensive semantic account of the multiplicative variants of all standard
propositional connectives in the bunched logic setting. This approach
synthesises a variety of techniques from modal, substructural and categorical
logic and contextualizes the "resource semantics" interpretation underpinning
Separation Logic amongst them
Information completeness in Nelson algebras of rough sets induced by quasiorders
In this paper, we give an algebraic completeness theorem for constructive
logic with strong negation in terms of finite rough set-based Nelson algebras
determined by quasiorders. We show how for a quasiorder , its rough
set-based Nelson algebra can be obtained by applying the well-known
construction by Sendlewski. We prove that if the set of all -closed
elements, which may be viewed as the set of completely defined objects, is
cofinal, then the rough set-based Nelson algebra determined by a quasiorder
forms an effective lattice, that is, an algebraic model of the logic ,
which is characterised by a modal operator grasping the notion of "to be
classically valid". We present a necessary and sufficient condition under which
a Nelson algebra is isomorphic to a rough set-based effective lattice
determined by a quasiorder.Comment: 15 page
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