Some properties of state filters in state residuated lattices

• Published in: Mathematica bohemica Vol. 146; no. 4; pp. 375 - 395
• Main Author: Michiro Kondo
• Format: Journal Article
• Language: English
• Published: Institute of Mathematics of the Czech Academy of Science 01.12.2021
• Subjects: boolean state filter; local residuated lattice; obstinate state filter; primary state filter; prime state filter; residuated lattice; state filter
• ISSN: 0862-7959; 2464-7136
• DOI: 10.21136/MB.2020.0040-19

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.