<p>The study of the migrativity of fuzzy logic connectives is an important topic in fuzzy logic theory research. The <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity between conjunction connectives (t-norms, overlap functions, uninorms, and nullnorms) has been extensively studied. Recently, Baczyński et al. (Inf Sci 531:87-96, 2020) proposed the concept of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of fuzzy implications, providing a new perspective for the study of migrativity of fuzzy logic connectives. Meanwhile, grouping functions, as an important class of non-necessarily associative fuzzy logic connectives, have attracted much attention. This paper aims to investigate the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions over fuzzy implications. Firstly, the concept of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions is introduced. It is shown that the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions is the standard dual of the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of overlap functions over 1-product overlap functions, and that the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions can be defined using fuzzy implications. Secondly, the concepts of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity and generalized <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions over general fuzzy implications are proposed, and the equivalent characterization of the <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions over fuzzy implications is given by the ordinal sum of grouping functions. Finally, the <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of grouping functions over several specific fuzzy implications is characterized by the additive generator pair and ordinal sum of grouping functions.</p>

Migrativity of grouping functions over fuzzy implications

期刊论文
发表日期：

2025年10月15日

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Peng Yu,
Huxiong Song

摘要

The study of the migrativity of fuzzy logic connectives is an important topic in fuzzy logic theory research. The $\alpha $ $α$ -migrativity between conjunction connectives (t-norms, overlap functions, uninorms, and nullnorms) has been extensively studied. Recently, Baczyński et al. (Inf Sci 531:87-96, 2020) proposed the concept of $\alpha $ $α$ -migrativity of fuzzy implications, providing a new perspective for the study of migrativity of fuzzy logic connectives. Meanwhile, grouping functions, as an important class of non-necessarily associative fuzzy logic connectives, have attracted much attention. This paper aims to investigate the $\alpha $ $α$ -migrativity of grouping functions over fuzzy implications. Firstly, the concept of $\alpha $ $α$ -migrativity of grouping functions is introduced. It is shown that the $\alpha $ $α$ -migrativity of grouping functions is the standard dual of the $\alpha $ $α$ -migrativity of overlap functions over 1-product overlap functions, and that the $\alpha $ $α$ -migrativity of grouping functions can be defined using fuzzy implications. Secondly, the concepts of $\alpha $ $α$ -migrativity and generalized $\alpha $ $α$ -migrativity of grouping functions over general fuzzy implications are proposed, and the equivalent characterization of the $\alpha $ $α$ -migrativity of grouping functions over fuzzy implications is given by the ordinal sum of grouping functions. Finally, the $\alpha $ $α$ -migrativity of grouping functions over several specific fuzzy implications is characterized by the additive generator pair and ordinal sum of grouping functions.