Uninorms, as a significant class of fuzzy connectives, have extensive applications in numerous fields. However, a comprehensive survey of the existing literature indicates that there are currently no well-developed methods for generating uninorms starting from two given uninorms. Recently, Gupta and Vemuri, leveraging the approaches for generating continuous t-norms from two such ones, proposed certain generating methods for several restricted families of uninorms, such as pseudo-continuous uninorms, uninorms with continuous Archimedean underlying operators, etc. Building upon the prior work, the present study initiates its exploration from two uninorms and conducts an in-depth investigation into the generating methods and algebraic structures of uninorms with continuous underlying operators that are locally internal in A(e). For convenience, this family of uninorm is hereinafter abbreviated as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathfrak {U}}_{linc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">U</mi> <mrow> <mi mathvariant="italic">linc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. By defining two binary operations <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\ddot{\sqcup }_{as}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mo>⊔</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">$\ddot{\sqcap }_{as}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mo>⊓</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> and a binary relation <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\sqsubseteq _{as}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mo>⊑</mo> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> on the set <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {U}^{e}_{linc}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="fraktur">U</mi> </mrow> <mrow> <mi mathvariant="italic">linc</mi> </mrow> <mi>e</mi> </msubsup> </math></EquationSource> </InlineEquation>, which is a subset of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {U}_{linc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">U</mi> <mrow> <mi mathvariant="italic">linc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, composed of elements with neutral element e in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathfrak {U}_{linc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="fraktur">U</mi> <mrow> <mi mathvariant="italic">linc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, our findings reveal that the system <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq8.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$({\mathfrak {U}}^{e}_{linc},\sqsubseteq _{as},\ddot{\sqcup }_{as})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="fraktur">U</mi> </mrow> <mrow> <mi mathvariant="italic">linc</mi> </mrow> <mi>e</mi> </msubsup> <mo>,</mo> <msub> <mo>⊑</mo> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo>,</mo> <msub> <mover accent="true"> <mo>⊔</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> (or <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$({\mathfrak {U}}^{e}_{linc},\sqsubseteq _{as},\ddot{\sqcap }_{as})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msubsup> <mrow> <mi mathvariant="fraktur">U</mi> </mrow> <mrow> <mi mathvariant="italic">linc</mi> </mrow> <mi>e</mi> </msubsup> <mo>,</mo> <msub> <mo>⊑</mo> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo>,</mo> <msub> <mover accent="true"> <mo>⊓</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) forms a join-semilattice (or meet-semilattice). Furthermore, by introducing an equivalence relation <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq10.gif" Format="GIF" Height="6" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\sim $</EquationSource> <EquationSource Format="MATHML"><math> <mo>∼</mo> </math></EquationSource> </InlineEquation> on the set <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">${{\mathfrak {U}}^{e}}_{linc}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <msup> <mrow> <mi mathvariant="fraktur">U</mi> </mrow> <mi>e</mi> </msup> </mrow> <mrow> <mi mathvariant="italic">linc</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>, it shows that for any equivalence class Z, the set <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3331_Article_IEq12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$(Z,\sqsubseteq _{as},\ddot{\sqcup }_{as}, \ddot{\sqcap }_{as})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>,</mo> <msub> <mo>⊑</mo> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo>,</mo> <msub> <mover accent="true"> <mo>⊔</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo>,</mo> <msub> <mover accent="true"> <mo>⊓</mo> <mo>¨</mo> </mover> <mrow> <mi mathvariant="italic">as</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> constitutes a distributive lattice.

Generating methods and algebraic structures of a class of uninorms

期刊论文
发表日期：

2025年7月22日

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Wen-Huang Li,
Hui-Zhen Fan,
Mian Xv,
Feng Qin

摘要

Uninorms, as a significant class of fuzzy connectives, have extensive applications in numerous fields. However, a comprehensive survey of the existing literature indicates that there are currently no well-developed methods for generating uninorms starting from two given uninorms. Recently, Gupta and Vemuri, leveraging the approaches for generating continuous t-norms from two such ones, proposed certain generating methods for several restricted families of uninorms, such as pseudo-continuous uninorms, uninorms with continuous Archimedean underlying operators, etc. Building upon the prior work, the present study initiates its exploration from two uninorms and conducts an in-depth investigation into the generating methods and algebraic structures of uninorms with continuous underlying operators that are locally internal in A(e). For convenience, this family of uninorm is hereinafter abbreviated as ${\mathfrak {U}}_{linc}$ $U_{linc}$ . By defining two binary operations $\ddot{\sqcup }_{as}$ ${\ddot{⊔}}_{as}$ , $\ddot{\sqcap }_{as}$ ${\ddot{⊓}}_{as}$ and a binary relation $\sqsubseteq _{as}$ $⊑_{as}$ on the set $\mathfrak {U}^{e}_{linc}$ $U_{linc}^{e}$ , which is a subset of $\mathfrak {U}_{linc}$ $U_{linc}$ , composed of elements with neutral element e in $\mathfrak {U}_{linc}$ $U_{linc}$ , our findings reveal that the system $({\mathfrak {U}}^{e}_{linc},\sqsubseteq _{as},\ddot{\sqcup }_{as})$ $(U_{linc}^{e}, ⊑_{as}, {\ddot{⊔}}_{as})$ (or $({\mathfrak {U}}^{e}_{linc},\sqsubseteq _{as},\ddot{\sqcap }_{as})$ $(U_{linc}^{e}, ⊑_{as}, {\ddot{⊓}}_{as})$ ) forms a join-semilattice (or meet-semilattice). Furthermore, by introducing an equivalence relation $\sim $ $\sim$ on the set ${{\mathfrak {U}}^{e}}_{linc}$ ${U^{e}}_{linc}$ , it shows that for any equivalence class Z, the set $(Z,\sqsubseteq _{as},\ddot{\sqcup }_{as}, \ddot{\sqcap }_{as})$ $(Z, ⊑_{as}, {\ddot{⊔}}_{as}, {\ddot{⊓}}_{as})$ constitutes a distributive lattice.