{"title":"Bounded complete J-algebraic lattices","authors":"Shengwei Han, Yu Xue","doi":"10.1007/s10485-025-09801-7","DOIUrl":null,"url":null,"abstract":"<div><p>The present article aims to develop a categorical duality for the category of bounded complete <i>J</i>-algebraic lattices. In terms of the lattice of weak ideals, we first construct a left adjoint to the forgetful functor <b>Sup</b><span>\\(\\rightarrow \\)</span> <span>\\({\\textbf {Pos}}_\\vee \\)</span>, where <b>Sup</b> is the category of complete lattices and join-preserving maps and <span>\\({\\textbf {Pos}}_\\vee \\)</span> is the category of posets and maps that preserve existing binary joins. Based on which, we propose the concept of <i>W</i>-structures over posets and give a <i>W</i>-structure representation for bounded complete <i>J</i>-algebraic posets, which generalizes the representation of algebraic lattices. Finally, we show that the category of join-semilattice <i>WS</i>-structures and homomorphisms is dually equivalent to the category of bounded complete <i>J</i>-algebraic lattices and homomorphisms.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"33 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Categorical Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10485-025-09801-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The present article aims to develop a categorical duality for the category of bounded complete J-algebraic lattices. In terms of the lattice of weak ideals, we first construct a left adjoint to the forgetful functor Sup\(\rightarrow \)\({\textbf {Pos}}_\vee \), where Sup is the category of complete lattices and join-preserving maps and \({\textbf {Pos}}_\vee \) is the category of posets and maps that preserve existing binary joins. Based on which, we propose the concept of W-structures over posets and give a W-structure representation for bounded complete J-algebraic posets, which generalizes the representation of algebraic lattices. Finally, we show that the category of join-semilattice WS-structures and homomorphisms is dually equivalent to the category of bounded complete J-algebraic lattices and homomorphisms.
期刊介绍:
Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant.
Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.