Generalized valuations on MTL-algebras

doi:10.1007/s11464-022-1021-2

Frontiers of Mathematics in China

2022, Vol. 17

Issue (4): 521-543 https://doi.org/10.1007/s11464-022-1021-2

本期目录

Generalized valuations on MTL-algebras

Juntao WANG¹, Pengfei HE²(

)

¹. School of Science, Xi’an Shiyou University, Xi’an 710065, China
². School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710119, China

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Abstract：

In this paper, we study some kinds of generalized valuations on MTL-algebras, discuss the relationship between the cokernel of generalized valuations and types of filters on MTL-algebras. Then, we give some equivalent characterizations of positive implicative generalized valuations on MTL-algebras. Finally, we characterize the structure theory of quotient MTL algebras based on the congruence relation, which is constructed by generalized valuations. The results of this paper not only generalize related theories of generalized valuations, but also enrich the algebraic conclusion of probability measure, on algebras of triangular norm based fuzzy logic.

Key words： Fuzzy logic MTL-algebra generalized valuation filter

出版日期: 2022-12-19

Corresponding Author(s): Pengfei HE

引用本文:

. [J]. Frontiers of Mathematics in China, 2022, 17(4): 521-543.
Juntao WANG, Pengfei HE. Generalized valuations on MTL-algebras. Front. Math. China, 2022, 17(4): 521-543.

链接本文:

https://academic.hep.com.cn/fmc/CN/10.1007/s11464-022-1021-2
https://academic.hep.com.cn/fmc/CN/Y2022/V17/I4/521

1	C Busneag. Valuations on residuated lattices. An Univ Craiova Ser Mat Inform 2007; 34: 21–28
2	D Busneag. A theorem of extensions of pseudo-valuations on Hilbert algebras. An Univ Craiova Ser Mat Inform 1999; 26: 14–21
3	D Busneag. On extension of pseudo-valuations on Hilbert algebras. Discrete Math 2003; 263: 11–24
4	M I Doh, M S Kang. BCK/BCI-algebras with pseudo valuation. Honam Math J 2010; 32: 217–226
5	F Esteva, L Godo. Monoidal t-norm based logic: towards a logic for left continuous t-norms. Fuzzy Sets and Systems 2001; 124(3): 271–288
6	G Ghorbani. Quotient BCI-algebras induced by pseudo-valuations. Iran J Math Sci Inform 2010; 5(2): 13–14
7	G Hhorbani. Involutive monoidal t-norm based algebras with internal states. J Mult-Valued Logic Soft Comput 2018; 31(5-6): 531–565
8	L Z Liu. On the existence of states on MTL-algebras. Inform Sci 2013; 220: 559–567
9	D Mundici. Averaging the truth-value in Łukasiewicz sentential logic. Studia Logica 1995; 55: 113–127
10	H B Wu. Basis R0-algebra and basis L* system. Adv Math (China) 2003; 32(5): 565–576
11	J T Wang, S A Borumand, P F He. Similarity MTL-algebras and their corresponding logics. J Mult-Valued Logic Soft Comput 2019; 32(5-6): 607–628
12	J T Wang, X L Xin, P F He. Monadic bounded hoops. Soft Comput 2018; 22: 1749–1762
13	J T Wang, P F He, J Yang, M Wang, X L He. Monadic NM-algebras: An algebraic approach to monadic predicate nilpotent minimum logic. J Logic Comput 2022; 32: 741–766
14	J T Wang, X L Xin. Monadic algebras of an involutive monoidal t-norm based logic. Iran J Fuzzy Syst 2022; 19(3): 187–202
15	M Wang, X L Xin, J T Wang. Implicative pseudo valuations on hoops. Chinese Quart J Math 2018; 33(1): 51–60
16	F Woumfo, B B Koguep, E R Temgoula. et al.. Ideals and Basbach states on residuated lattices. New Math Nat Comput 2020; 16(3): 551–557
17	Y W Yang, L L Lu, Q L Wang. Pseudo valuation on hoop-algebra respect to filters. Int J Math Stat 2017; 11: 12–19
18	Y W Yang, X L Xin. EQ-algebras with pseudo pre-valuation. Ital J Pure Appl Math 2017; 36: 29–48
19	Y W Yang, Y Zhu. MV-algebras with pseudo MV-valuation. J Intell Fuzzy Syst 2019; 37(2): 2621–2636
20	L A Zadeh. Fuzzy sets. Information and Control 1965; 8(3): 338–353
21	J M Zhan, Y B Jun. (Implicative) Pseudo-valuations on R0-algebras. Univ Bucharest Sci Bull Ser 2013; 75: 101–112
22	X H Zhang. Fuzzy logics and their algebraic analysis. Beijing: Science Press, 2008 (in Chinese)
23	X H Zhang. Non-commutative fuzzy logic system PUL* based on left continuous pseudo t-norms. Adv Math (China) 2007; 36(3): 295–308
24	Y Q Zhu, X W Cao. On PFI-algebras and residuated lattices. Adv Math(China) 2006; 35(2): 223–231
25	Y Q Zhu, Y Xu. On filter theory of residuated lattices. Inform Sci 2010; 180(19): 3614–3632

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