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Keywords:
obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice
obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice
Summary:
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 $X$. \endgraf 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.
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