Lattice representation of $L$-quasi-convex spaces

Abstract

Based on a complete Heyting algebra $L$, we first propose the concept of $L$-quasi-convex spaces and construct an adjunction between the category of  $L$-$S_0$-quasi-convex spaces and the opposite category of complete $L$-ordered sets. Then we present the concept of weakly fuzzy algebraic lattices and prove that an $L$-quasi-convex structure endowed with the fuzzy inclusion order is precisely a weakly fuzzy algebraic lattice. Secondly, we introduce the notion of sobriety in $L$-quasi-convex spaces from the perspective of categorical equivalence, showing that the category of sober $L$-quasi-convex spaces is dually equivalent to that of weakly fuzzy algebraic lattices. Finally, we construct a monad on the category of $L$-$S_0$-quasi-convex spaces and obtain that the Eilenberg–Moore algebras of this monad are precisely sober $L$-quasi-convex spaces.

Keywords

• fuzzy convex structure
• fuzzy algebraic lattice
• adjunction
• dual equivalence
• Eilenberg–Moore algebra

Project Number

the Natural Science Foundation of China (Nos. 12471428, 12071033, 12271036)

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