Yiheng Wang

Over the past few years, a new cluster of abstract algebras has emerged within the general context of research in rough set theory. Among them, one is the weakly topological quasi-Boolean algebra. This paper studies the vicinity of weakly topological quasi-Boolean algebra from algebraic and logical perspectives. The weak pre-rough algebra is defined, and a cluster of intermediate algebras between it and the quasi-Boolean algebra is explored. The interrelationship and independence among them are presented. We also establish sound and complete sequential systems for these algebras. Further, we show the finite model property and construct decidable algorithms for weakly topological quasi-Boolean algebra and its weaker variants by context-free grammars. Rough set models of some of these algebras have been presented.

In this paper, we show that every normal extension of Euclidean quasi-Boolean modal logic has the finite model property. Invariance results regarding disjoint union, generated submodel, and p-morphism are established. Our results generalize and essentially echo the well-known similar result of normal extensions of classical Euclidean modal logic K5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{K5}$$\end{document} (cf. [17]).

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S5 and prove that the cut elimination holds for it. Consequently the decidability of S5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM5.

In this paper, we consider the non-normal modal logics over monotonic modal logic M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf{M}$$\end{document} with extensions of any combination of (N), (P), (T), and (4) i.e. {M,MN,…,MNPT4}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\textsf{M},\textsf{MN},\ldots,\textsf{MNPT4}\}$$\end{document}. We study the algebras corresponding to these logics and give some examples of them. We further introduce the Gentzen-style sequent calculi with soundness and completeness proved. Finally, we prove the FMP of these logics and thus decidability based on our systems by algebraic proof-theoretic methods.