Generalized valuations on MTL-algebras

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.

Keywords

Fuzzy logic / MTL-algebra / generalized valuation / filter

References

[1] Busneag C. Valuations on residuated lattices. An Univ Craiova Ser Mat Inform 2007; 34: 21–28

[2] Busneag D. A theorem of extensions of pseudo-valuations on Hilbert algebras. An Univ Craiova Ser Mat Inform 1999; 26: 14–21

[3] Busneag D. On extension of pseudo-valuations on Hilbert algebras. Discrete Math 2003; 263: 11–24

[4] Doh M I, Kang M S. BCK/BCI-algebras with pseudo valuation. Honam Math J 2010; 32: 217–226

[5] Esteva F, Godo L. Monoidal t-norm based logic: towards a logic for left continuous t-norms. Fuzzy Sets and Systems 2001; 124(3): 271–288

[6] Ghorbani G. Quotient BCI-algebras induced by pseudo-valuations. Iran J Math Sci Inform 2010; 5(2): 13–14

[7] Hhorbani G. Involutive monoidal t-norm based algebras with internal states. J Mult-Valued Logic Soft Comput 2018; 31(5-6): 531–565

[8] Liu L Z. On the existence of states on MTL-algebras. Inform Sci 2013; 220: 559–567

[9] Mundici D. Averaging the truth-value in Łukasiewicz sentential logic. Studia Logica 1995; 55: 113–127

[10] Wu H B. Basis R₀-algebra and basis L^* system. Adv Math (China) 2003; 32(5): 565–576

[11] Wang J T, Borumand S A, He P F. Similarity MTL-algebras and their corresponding logics. J Mult-Valued Logic Soft Comput 2019; 32(5-6): 607–628

[12] Wang J T, Xin X L, He P F. Monadic bounded hoops. Soft Comput 2018; 22: 1749–1762

[13] Wang J T, He P F, Yang J, Wang M, He X L. Monadic NM-algebras: An algebraic approach to monadic predicate nilpotent minimum logic. J Logic Comput 2022; 32: 741–766

[14] Wang J T, Xin X L. Monadic algebras of an involutive monoidal t-norm based logic. Iran J Fuzzy Syst 2022; 19(3): 187–202

[15] Wang M, Xin X L, Wang J T. Implicative pseudo valuations on hoops. Chinese Quart J Math 2018; 33(1): 51–60

[16] Woumfo F, Koguep B B, Temgoula E R. . Ideals and Basbach states on residuated lattices. New Math Nat Comput 2020; 16(3): 551–557

[17] Yang Y W, Lu L L, Wang Q L. Pseudo valuation on hoop-algebra respect to filters. Int J Math Stat 2017; 11: 12–19

[18] Yang Y W, Xin X L. EQ-algebras with pseudo pre-valuation. Ital J Pure Appl Math 2017; 36: 29–48

[19] Yang Y W, Zhu Y. MV-algebras with pseudo MV-valuation. J Intell Fuzzy Syst 2019; 37(2): 2621–2636

[20] Zadeh L A. Fuzzy sets. Information and Control 1965; 8(3): 338–353

[21] Zhan J M, Jun Y B. (Implicative) Pseudo-valuations on R₀-algebras. Univ Bucharest Sci Bull Ser 2013; 75: 101–112

[22] ZhangX H. Fuzzy logics and their algebraic analysis. Beijing: Science Press, 2008 (in Chinese)

[23] Zhang X H. Non-commutative fuzzy logic system PUL^* based on left continuous pseudo t-norms. Adv Math (China) 2007; 36(3): 295–308

[24] Zhu Y Q, Cao X W. On PFI-algebras and residuated lattices. Adv Math(China) 2006; 35(2): 223–231

[25] Zhu Y Q, Xu Y. On filter theory of residuated lattices. Inform Sci 2010; 180(19): 3614–3632

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12001423, 12171294, 61976244, 11961016), Natural Science Foundation of Shaanxi Province (Grant Nos. 2020JQ-762, 2021JQ-580, 2021JQ-579), Natural Science Foundation of Education Committee of Shannxi Province (Grant No. 19JK0626), and Fundamental Research Funds for the Central Universities (Grant Nos. GK202003003, GK200101009).

RIGHTS & PERMISSIONS

2022 Higher Education Press 2022