On ideals in De Morgan residuated lattices

Liviu-Constantin Holdon

Kybernetika (2018)

  • Volume: 54, Issue: 3, page 443-475
  • ISSN: 0023-5954

Abstract

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In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, ⊙ -prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.

How to cite

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Holdon, Liviu-Constantin. "On ideals in De Morgan residuated lattices." Kybernetika 54.3 (2018): 443-475. <http://eudml.org/doc/294514>.

@article{Holdon2018,
abstract = {In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, $\odot $-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.},
author = {Holdon, Liviu-Constantin},
journal = {Kybernetika},
keywords = {residuated lattice; De Morgan laws; filter; deductive system; ideal; $\cap $-prime; $\cap $-irreducible; annihilator},
language = {eng},
number = {3},
pages = {443-475},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On ideals in De Morgan residuated lattices},
url = {http://eudml.org/doc/294514},
volume = {54},
year = {2018},
}

TY - JOUR
AU - Holdon, Liviu-Constantin
TI - On ideals in De Morgan residuated lattices
JO - Kybernetika
PY - 2018
PB - Institute of Information Theory and Automation AS CR
VL - 54
IS - 3
SP - 443
EP - 475
AB - In this paper, we introduce a new class of residuated lattices called De Morgan residuated lattices, we show that the variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices. We investigate specific properties of ideals in De Morgan residuated lattices, we state the prime ideal theorem and the pseudo-complementedness of the ideal lattice, we pay attention to prime, maximal, $\odot $-prime ideals and to ideals that are meet-irreducible or meet-prime in the lattice of all ideals. We introduce the concept of an annihilator of a given subset of a De Morgan residuated lattice and we prove that annihilators are a particular kind of ideals. Also, regular annihilator and relative annihilator ideals are considered.
LA - eng
KW - residuated lattice; De Morgan laws; filter; deductive system; ideal; $\cap $-prime; $\cap $-irreducible; annihilator
UR - http://eudml.org/doc/294514
ER -

References

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