Some properties of state filters in state residuated lattices

We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: \begin{itemize} \item[(1)] $F$ is obstinate $\Leftrightarrow$ $L/F \cong\{0,1\}$; \item[(2)] $F$ is primary $\Leftrightarrow$ $L/F$ is a state local resid...

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Autor principal:	Michiro Kondo
Formato:	article
Lenguaje:	EN
Publicado:	Institute of Mathematics of the Czech Academy of Science 2021
Materias:	obstinate state filter prime state filter boolean state filter primary state filter state filter residuated lattice local residuated lattice Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/2167157825114defba317ef667f1ffaa
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id	oai:doaj.org-article:2167157825114defba317ef667f1ffaa
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spelling	oai:doaj.org-article:2167157825114defba317ef667f1ffaa2021-11-08T09:59:12ZSome properties of state filters in state residuated lattices0862-79592464-713610.21136/MB.2020.0040-19https://doaj.org/article/2167157825114defba317ef667f1ffaa2021-12-01T00:00:00Zhttp://mb.math.cas.cz/full/146/4/mb146_4_1.pdfhttps://doaj.org/toc/0862-7959https://doaj.org/toc/2464-7136We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: \begin{itemize} \item[(1)] $F$ is obstinate $\Leftrightarrow$ $L/F \cong\{0,1\}$; \item[(2)] $F$ is primary $\Leftrightarrow$ $L/F$ is a state local residuated lattice; \end{itemize} and that every g-state residuated lattice $X$ is a subdirect product of $\{X/P_{\lambda} \}$, where $P_{\lambda}$ is a prime state filter of $X$. Moreover, we show that the quotient MTL-algebra $X/P$ of a state residuated lattice $X$ by a state prime filter $P$ is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered.Michiro KondoInstitute of Mathematics of the Czech Academy of Sciencearticle obstinate state filter prime state filter boolean state filter primary state filter state filter residuated lattice local residuated latticeMathematicsQA1-939ENMathematica Bohemica, Vol 146, Iss 4, Pp 375-395 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	obstinate state filter prime state filter boolean state filter primary state filter state filter residuated lattice local residuated lattice Mathematics QA1-939
spellingShingle	obstinate state filter prime state filter boolean state filter primary state filter state filter residuated lattice local residuated lattice Mathematics QA1-939 Michiro Kondo Some properties of state filters in state residuated lattices
description	We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: \begin{itemize} \item[(1)] $F$ is obstinate $\Leftrightarrow$ $L/F \cong\{0,1\}$; \item[(2)] $F$ is primary $\Leftrightarrow$ $L/F$ is a state local residuated lattice; \end{itemize} and that every g-state residuated lattice $X$ is a subdirect product of $\{X/P_{\lambda} \}$, where $P_{\lambda}$ is a prime state filter of $X$. Moreover, we show that the quotient MTL-algebra $X/P$ of a state residuated lattice $X$ by a state prime filter $P$ is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered.
format	article
author	Michiro Kondo
author_facet	Michiro Kondo
author_sort	Michiro Kondo
title	Some properties of state filters in state residuated lattices
title_short	Some properties of state filters in state residuated lattices
title_full	Some properties of state filters in state residuated lattices
title_fullStr	Some properties of state filters in state residuated lattices
title_full_unstemmed	Some properties of state filters in state residuated lattices
title_sort	some properties of state filters in state residuated lattices
publisher	Institute of Mathematics of the Czech Academy of Science
publishDate	2021
url	https://doaj.org/article/2167157825114defba317ef667f1ffaa
work_keys_str_mv	AT michirokondo somepropertiesofstatefiltersinstateresiduatedlattices
_version_	1718442693346983936

Some properties of state filters in state residuated lattices

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