Triangle Algebras and Relative Co-annihilators
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Emile Djomgoue Nana
1
,
Ariane GABRIEL Tallee Kakeu
2
,
Blaise Bleriot Koguep Njionou
3
,
Celestin Lele
4
1 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
2 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
3 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
4 - Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon.
کلید واژه: Triangularization, Triangle algebra, Filter, Co-annihilator, Relative co-annihilator.,
چکیده مقاله :
Triangle algebras are an important variety of residuated lattices enriched with two approximation operators as well as a third angular point (different from 0 and 1). They provide a well-defined mathematical framework for formalizing the use of closed intervals derived from a bounded lattice as truth values, with a set of structured axioms. This paper introduces the concept of relative co-annihilator of a subset within the framework of triangle algebras. As filters of triangle algebras, these relative co-annihilators are explored and some of their properties and characterizations are given. A meaningful contribution of this work lies in its proof that the relative co-annihilator of a subset $T$ with respect to another subset $Y$ in a triangle algebra $\mathcal{L}$ inherits specific filter's characteristics of $Y$. More precisely, if $Y$ is a Boolean filter of the second kind, then the co-annihilator of $T$ with respect to $Y$ is also a Boolean filter of the second kind. The same statement applies when we replace the Boolean filter of the second kind with an implicative filter, pseudo complementation filter, Boolean filter, prime filter, prime filter of the third kind, pseudo-prime filter, or involution filter, respectively. Finally, we establish some conditions under which the co-annihilator of $T$ relative to $Y$ is a prime filter of the second kind.
Triangle algebras are an important variety of residuated lattices enriched with two approximation operators as well as a third angular point (different from 0 and 1). They provide a well-defined mathematical framework for formalizing the use of closed intervals derived from a bounded lattice as truth values, with a set of structured axioms. This paper introduces the concept of relative co-annihilator of a subset within the framework of triangle algebras. As filters of triangle algebras, these relative co-annihilators are explored and some of their properties and characterizations are given. A meaningful contribution of this work lies in its proof that the relative co-annihilator of a subset $T$ with respect to another subset $Y$ in a triangle algebra $\mathcal{L}$ inherits specific filter's characteristics of $Y$. More precisely, if $Y$ is a Boolean filter of the second kind, then the co-annihilator of $T$ with respect to $Y$ is also a Boolean filter of the second kind. The same statement applies when we replace the Boolean filter of the second kind with an implicative filter, pseudo complementation filter, Boolean filter, prime filter, prime filter of the third kind, pseudo-prime filter, or involution filter, respectively. Finally, we establish some conditions under which the co-annihilator of $T$ relative to $Y$ is a prime filter of the second kind.
[1] Boole G. The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning. Cambridge: Macmillan, Barclay and Macmillan; 1947.
[2] Zadeh LA. Fuzzy sets. Information and Control. 1964; 8: 338-353. DOI: https://doi.org/10.21236/ad0608981
[3] Goguen JA. L-fuzzy sets. Journal of Mathematical Analysis and Application. 1967; 18(1): 145-174. DOI: https://doi.org/10.1016/0022-247X(67)90189-8 [4] Ward M, Dilworth R. Residuated lattices. Transactions of the American Mathematical Society. 1939; 45(3): 335-354. DOI: https://doi.org/10.1090/s0002-9947-1939-1501995-3
[5] Van Gasse B, Cornelis C, Deschrijver G, Kerre EE. Triangle algebras: A formal logic approach to interval-valued residuated lattices. Fuzzy Sets and Systems. 2008; 159: 1042-1060. DOI: https://doi.org/10.1016/j.fss.2007.09.003
[6] Van Gasse B, Cornelis C, Deschrijver G, Kerre EE. Triangle algebras: Towards an axiomatization of interval-valued residuated lattices. In: Information Sciences. 2010. p.117-126. DOI: https://doi.org/10.1007/11908029_14
[7] Van Gasse B, Deschrijver G, Cornelis C, Kerre EE. Filters of residuated lattices and triangle algebras. Information Sciences. 2010; 180: 3006-3020. DOI: https://doi.org/10.1016/j.ins.2010.04.010
[8] Zahiri S, Borumand Saeid A, Eslami E. A new approach to filters in triangle algebras. Publications de l’Institut Mathematique (Belgrade). 2017; 101(115): 267-283. DOI: https://doi.org/10.2298/pim1715267z
[9] Zahiri S, Borumand Saeid A, Eslami E. On maximal filters in triangle algebras. Journal of Intelligent and Fuzzy Systems. 2016; 30(2): 1181-1193. DOI: https://doi.org/10.3233/ifs-151842
[10] Meng BL, Xin XL. Generalized co-annihilator of BL-algebras. Open Math. 2015; 13: 639-654. DOI: https://doi.org/10.1515/math-2015-0060
[11] Maroof FG, Borumand Saeid A, Eslami E. On co-annihilators in residuated lattices. Journal of Intelligent and Fuzzy Systems. 2016; 31(3): 1263-1270. DOI: https://doi.org/10.3233/ifs-162192
[12] Rasouli S. Generalized co-annihilators in residuated lattices. Annals of the University of Craiova-Mathematics and Computer Science Series. 2018; 45(2): 190-207. DOI: https://doi.org/10.52846/ami.v45i2.827
[13] Hajek P. Metamathematics of Fuzzy Logic. Kluwer Acad Publishing; 1998.
[14] Holdon CL, Borumand Saeid A. Regularity in residuated lattices. Iranian Journal of Fuzzy Systems. 2019; 16(6): 107-126. DOI: https://doi.org/10.22111/IJFS.2019.5023
[15] Van Gasse B, Cornelis C, Deschrijver G, Kerre EE. A characterization of interval-valued residuated lattices. International Journal of Approximate Reasoning. 2008; 49: 478-487. DOI: https://doi.org/10.1016/j.ijar.2008.04.006
[16] Zahiri S, Borumand Saeid A, Zahiri M. An investigation on the Co-annihilators in triangle algebras. Iranian Journal of Fuzzy Systems. 2018; 15(7): 91-102. DOI: https://doi.org/10.22111/IJFS.2018.4287
[17] Zahiri S, Borumand Saeid A, Turunen E. On local triangle algebras. Journal of Intelligent and Fuzzy Systems. 2021; 418: 126-138. DOI: https://doi.org/10.1016/j.fss.2020.07.001
[18] Zahiri S, Borumand Saeid A. An investigation on the n-fold IVRL-filters in triangle algebras. Mathematica Bohemica. 2020; 145(1): 75-91. DOI: https://doi.org/10.21136/MB.2019.0104-17
