Finite-valued algebraic lattices
type: Item
instance of: scholarly article
title: Finite-valued algebraic lattices
zbMATH Open document ID: 0806.06011
DOI: 10.1007/BF01190439
author: J. T. Snodgrass
author: Constantine Tsinakis
published in: Algebra Universalis
publication date: 1993-12-20T00:00:00Z
review text: Let \(L\) be a distributive algebraic lattice. Then \(L\) is said to be finite-valued if for every compact element \(c\in L\), the poset of meet- irreducible elements not exceeding \(c\) has finitely many maximal elements. It is a classical result that a bialgebraic (= algebraic and dually algebraic) distributive lattice is finite-valued. The converse does not hold in general [see \textit{J. Martinez}, Proc. Am. Math. Soc. 33, 213-220 (1972; Zbl 0241.06007)]. The principal result of the paper under review says that the converse of the above classical result is possible under some additional assumptions. More precisely, let \(L\) be an algebraic distributive lattice whose compact elements form a sublattice of whose meet-irreducible elements form a root-system. (A root-system is a poset for which the principal order-filter \([p)=\{x\in P\): \(p\leq x\}\) is a chain for all \(p\in P\).) Let \(L\) be finite-valued. Then \(L\) is bialgebraic. As a corollary of this result one can derive the following known theorem [see \textit{P. Conrad}, Czech. Math. J. 15, 101-123 (1965; Zbl 0135.063)]: The congruence lattice \(\text{Con}(L)\) of an \(l\)-group \(L\) is finite-valued if and only if each positive element of \(L\) is a finite product of pairwise orthogonal elements.
Mathematics Subject Classification ID: 06D20
Mathematics Subject Classification ID: 06F15
zbMATH DE Number: 446952
zbMATH Keywords: relatively normal lattice
zbMATH Keywords: bialgebraic lattice
zbMATH Keywords: algebraic lattice
zbMATH Keywords: meet- irreducible elements
zbMATH Keywords: distributive lattice
zbMATH Keywords: root-system
reviewed by: Tibor Katriňák
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 446952