<p>The property of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity between binary operations is both interesting and mathematically challenging, and is of particular importance in the study of binary aggregation functions, both from a theoretical and a practical point of view. Here, we continue the investigation of this research direction by focusing on S-uninorms, general overlap and general grouping functions, where S-uninorms can be seen as a significant generalization of nullnorms and conjunctive uninorms. First, we discuss the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of S-uninorms over general overlap or general grouping functions and identify the necessary and sufficient conditions for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity to hold. Next, we investigate the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of overlap and grouping functions over S-uninorms and obtain that for overlap functions it cannot hold when <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha \in \,]0,1]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mspace width="0.166667em" /> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, whereas for grouping functions it cannot hold when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha \in [0,\varrho ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>ϱ</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\varrho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϱ</mi> </math></EquationSource> </InlineEquation> is the IFC element of the S-uninorm. Finally, we study the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-migrativity of general overlap or general grouping functions over S-uninorms and obtain full characterizations.</p>

On the migrativity between S-uninorms and general overlap or general grouping functions

期刊论文
发表日期：

2026年3月3日

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Jieqiong Shi,
Bin Zhao,
Bernard De Baets

摘要

The property of $\alpha $ $α$ -migrativity between binary operations is both interesting and mathematically challenging, and is of particular importance in the study of binary aggregation functions, both from a theoretical and a practical point of view. Here, we continue the investigation of this research direction by focusing on S-uninorms, general overlap and general grouping functions, where S-uninorms can be seen as a significant generalization of nullnorms and conjunctive uninorms. First, we discuss the $\alpha $ $α$ -migrativity of S-uninorms over general overlap or general grouping functions and identify the necessary and sufficient conditions for $\alpha $ $α$ -migrativity to hold. Next, we investigate the $\alpha $ $α$ -migrativity of overlap and grouping functions over S-uninorms and obtain that for overlap functions it cannot hold when $\alpha \in \,]0,1]$ $α \in] 0, 1]$ , whereas for grouping functions it cannot hold when $\alpha \in [0,\varrho ]$ $α \in [0, ϱ]$ , where $\varrho $ $ϱ$ is the IFC element of the S-uninorm. Finally, we study the $\alpha $ $α$ -migrativity of general overlap or general grouping functions over S-uninorms and obtain full characterizations.