{"title":"Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon","authors":"W. Dziobiak, M. V. Schwidefsky","doi":"10.1007/s10469-025-09776-3","DOIUrl":null,"url":null,"abstract":"<p>According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"114 - 140"},"PeriodicalIF":0.4000,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-025-09776-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 0
Abstract
According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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