Some properties of state filters in state residuated lattices

Michiro Kondo

Mathematica Bohemica (2021)

  • Volume: 146, Issue: 4, page 375-395
  • ISSN: 0862-7959

Abstract

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We consider properties of state filters of state residuated lattices and prove that for every state filter F of a state residuated lattice X :

F F is obstinate ⇔ \Leftrightarrow L / F ≅ { 0 , 1 } L/F \cong \lbrace 0,1\rbrace ;

F F is primary ⇔ \Leftrightarrow L / F L/F is a state local residuated lattice;

 and that every g-state residuated lattice X is a subdirect product of { X / P λ } , where P λ 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.

How to cite

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Kondo, Michiro. "Some properties of state filters in state residuated lattices." Mathematica Bohemica 146.4 (2021): 375-395. <http://eudml.org/doc/297725>.

@article{Kondo2021,
abstract = {We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: and that every g-state residuated lattice $X$ is a subdirect product of $\lbrace X/P_\{\lambda \} \rbrace $, 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.},
author = {Kondo, Michiro},
journal = {Mathematica Bohemica},
keywords = {obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice},
language = {eng},
number = {4},
pages = {375-395},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some properties of state filters in state residuated lattices},
url = {http://eudml.org/doc/297725},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Kondo, Michiro
TI - Some properties of state filters in state residuated lattices
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 4
SP - 375
EP - 395
AB - We consider properties of state filters of state residuated lattices and prove that for every state filter $F$ of a state residuated lattice $X$: and that every g-state residuated lattice $X$ is a subdirect product of $\lbrace X/P_{\lambda } \rbrace $, 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.
LA - eng
KW - obstinate state filter; prime state filter; Boolean state filter; primary state filter; state filter; residuated lattice; local residuated lattice
UR - http://eudml.org/doc/297725
ER -

References

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