In the framework of completely distributive lattice L, we propose a new type of fuzzy coarse structure which is imposed with some algebraic flavor. We start by introducing a kind of L-fuzzy sets on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$X\times 2^X_{fin}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>×</mo> <msubsup> <mn>2</mn> <mrow> <mi mathvariant="italic">fin</mi> </mrow> <mi>X</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> with an algebraic characteristic. At the same time, we propose some of their operation laws such as supremum, infimum, composition and inversion, and discuss some of their basic properties. Based on this, we introduce notions of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic entourage and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic (resp. quasi-, semi-) coarse structure, and present some mappings among them. We provide several examples of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic coarse structures constructed by L-fuzzifying restricted hull operator, L-metric, L-coarse structure and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic relation. Further, we introduce the notion of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic ball structure by which we characterize <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic coarse structure. Finally, we study some categorical relationships among <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic entourage space, <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic semi-coarse space, <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic quasi-coarse space and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$L$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation>-algebraic coarse space.

Lattice-valued algebraic coarse structures

期刊论文
发表日期：

2026年5月2日

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Xiu-Yun Wu,
Zi-Han Zhao

摘要

In the framework of completely distributive lattice L, we propose a new type of fuzzy coarse structure which is imposed with some algebraic flavor. We start by introducing a kind of L-fuzzy sets on $X\times 2^X_{fin}$ $X \times 2_{fin}^{X}$ with an algebraic characteristic. At the same time, we propose some of their operation laws such as supremum, infimum, composition and inversion, and discuss some of their basic properties. Based on this, we introduce notions of $L$ $L$ -algebraic entourage and $L$ $L$ -algebraic (resp. quasi-, semi-) coarse structure, and present some mappings among them. We provide several examples of $L$ $L$ -algebraic coarse structures constructed by L-fuzzifying restricted hull operator, L-metric, L-coarse structure and $L$ $L$ -algebraic relation. Further, we introduce the notion of $L$ $L$ -algebraic ball structure by which we characterize $L$ $L$ -algebraic coarse structure. Finally, we study some categorical relationships among $L$ $L$ -algebraic entourage space, $L$ $L$ -algebraic semi-coarse space, $L$ $L$ -algebraic quasi-coarse space and $L$ $L$ -algebraic coarse space.