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.

Key words： Fuzzy logic; MTL-algebra; generalized valuation; filter

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).

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