Modal logics over lattices

IF 0.6 2区 数学 Q2 LOGIC
Xiaoyang Wang , Yanjing Wang
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引用次数: 0

Abstract

Lattice theory has various close connections with modal logic. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logics over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the modal languages of tense logic and polyadic modal logic to talk about lattices via standard Kripke semantics. We first obtain a series of complete axiomatizations of tense logics over lattices, (un)bounded lattices over partial orders or strict orders. In particular, we solve an axiomatization problem left open by Burgess (1984) [8]. The second half of the paper gives a series of complete axiomatizations of polyadic modal logic with nominals over lattices, distributive lattices, and modular lattices, where the binary modalities of infimum and supremum can reveal more structures behind various lattices.
格上的模态逻辑
格理论与模态逻辑有着各种密切的联系。然而,一个较少探索的方向是将格视为基于偏序的关系结构,并研究其上的模态逻辑。本文继Burgess和van Benthem在20世纪80年代的早期步骤之后,我们使用时态逻辑和多向模态逻辑的模态语言通过标准Kripke语义来讨论格。首先,我们得到了格上、(无)有界格上、偏序上和严序上的一系列张力逻辑的完全公理化。特别地,我们解决了Burgess(1984)留下的公理化问题。本文的第二部分给出了格上、分布格上和模格上的多项式多进模态逻辑的一系列完全公理化,其中上极值和上极值的二元模态可以揭示各种格背后的更多结构。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.40
自引率
12.50%
发文量
78
审稿时长
200 days
期刊介绍: The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.
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