Involutive symmetric Gödel spaces, their algebraic duals and logic

IF 0.3 4区 数学 Q1 Arts and Humanities
A. Di Nola, R. Grigolia, G. Vitale
{"title":"Involutive symmetric Gödel spaces, their algebraic duals and logic","authors":"A. Di Nola,&nbsp;R. Grigolia,&nbsp;G. Vitale","doi":"10.1007/s00153-023-00866-6","DOIUrl":null,"url":null,"abstract":"<div><p>It is introduced a new algebra <span>\\((A, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)\\)</span> called <span>\\(L_PG\\)</span>-algebra if <span>\\((A, \\otimes , \\oplus , *, 0, 1)\\)</span> is <span>\\(L_P\\)</span>-algebra (i.e. an algebra from the variety generated by perfect <i>MV</i>-algebras) and <span>\\((A,\\rightharpoonup , 0, 1)\\)</span> is a Gödel algebra (i.e. Heyting algebra satisfying the identity <span>\\((x \\rightharpoonup y ) \\vee (y \\rightharpoonup x ) =1)\\)</span>. The lattice of congruences of an <span>\\(L_PG\\)</span> -algebra <span>\\((A, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)\\)</span> is isomorphic to the lattice of Skolem filters (i.e. special type of <i>MV</i>-filters) of the <i>MV</i>-algebra <span>\\((A, \\otimes , \\oplus , *, 0, 1)\\)</span>. The variety <span>\\(\\mathbf {L_PG}\\)</span> of <span>\\(L_PG\\)</span> -algebras is generated by the algebras <span>\\((C, \\otimes , \\oplus , *, \\rightharpoonup , 0, 1)\\)</span> where <span>\\((C, \\otimes , \\oplus , *, 0, 1)\\)</span> is Chang <i>MV</i>-algebra. Any <span>\\(L_PG\\)</span> -algebra is bi-Heyting algebra. The set of theorems of the logic <span>\\(L_PG\\)</span> is recursively enumerable. Moreover, we describe finitely generated free <span>\\(L_PG\\)</span>-algebras.</p></div>","PeriodicalId":48853,"journal":{"name":"Archive for Mathematical Logic","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00153-023-00866-6.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Mathematical Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00153-023-00866-6","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Arts and Humanities","Score":null,"Total":0}
引用次数: 0

Abstract

It is introduced a new algebra \((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) called \(L_PG\)-algebra if \((A, \otimes , \oplus , *, 0, 1)\) is \(L_P\)-algebra (i.e. an algebra from the variety generated by perfect MV-algebras) and \((A,\rightharpoonup , 0, 1)\) is a Gödel algebra (i.e. Heyting algebra satisfying the identity \((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)\). The lattice of congruences of an \(L_PG\) -algebra \((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) is isomorphic to the lattice of Skolem filters (i.e. special type of MV-filters) of the MV-algebra \((A, \otimes , \oplus , *, 0, 1)\). The variety \(\mathbf {L_PG}\) of \(L_PG\) -algebras is generated by the algebras \((C, \otimes , \oplus , *, \rightharpoonup , 0, 1)\) where \((C, \otimes , \oplus , *, 0, 1)\) is Chang MV-algebra. Any \(L_PG\) -algebra is bi-Heyting algebra. The set of theorems of the logic \(L_PG\) is recursively enumerable. Moreover, we describe finitely generated free \(L_PG\)-algebras.

Abstract Image

对合对称Gödel空间及其代数对偶与逻辑
如果\((A, \otimes , \oplus , *, 0, 1)\)是\(L_P\) -代数(即由完全mv -代数生成的代数),\((A,\rightharpoonup , 0, 1)\)是Gödel代数(即满足恒等式\((x \rightharpoonup y ) \vee (y \rightharpoonup x ) =1)\)的Heyting代数),则引入一个新的代数\((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\)\(L_PG\) -代数。\(L_PG\) -代数\((A, \otimes , \oplus , *, \rightharpoonup , 0, 1)\)的同余格与mv -代数\((A, \otimes , \oplus , *, 0, 1)\)的Skolem滤波器(即特殊类型的mv -滤波器)的格同构。\(L_PG\) -代数的变体\(\mathbf {L_PG}\)是由\((C, \otimes , \oplus , *, \rightharpoonup , 0, 1)\)代数生成的,其中\((C, \otimes , \oplus , *, 0, 1)\)是Chang mv -代数。任何\(L_PG\) -代数都是bi-Heyting代数。逻辑\(L_PG\)的定理集合是递归可枚举的。此外,我们还描述了有限生成的自由\(L_PG\) -代数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Archive for Mathematical Logic
Archive for Mathematical Logic MATHEMATICS-LOGIC
CiteScore
0.80
自引率
0.00%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信