On lattices determined up to isomorphisms by their graphs. (English) Zbl 0557.06004

Jakubík, Ján

Czech. Math. J. 34(109), 305-314 (1984).

The graph of a lattice L is the undirected graph whose vertices are the elements of L, and whose edges are the pairs a,b such that a covers b or b covers a. A graph isomorphism of the graph of \(L_ 1\) onto the graph of \(L_ 2\) is a bijection \(h:L_ 1\to L_ 2\) that preserves edges of the graphs. The paper deals with the problem of characterizing those lattices whose graph determines them up to isomorphism. Let \(C_ 0\) be the class of these lattices. The author defines \(C_ 1\) to be the class of lattices L such that when h is an isomorphism of the graph of L onto the graph of \(L_ 1\), then it turns out to be either an isomorphism or a dual isomorphism of L onto \(L_ 1.\)
Reviewer’s note: The author’s intention seems to be that for L to be in \(C_ 1\), there must actually exist instances where h is an isomorphism as well as instances where it is a dual isomorphism. The main results are: (i) Every lattice may be embedded into a lattice belonging to \(C_ 1\). (ii) Every bounded lattice may be embedded into a lattice belonging to \(C_ 0\cap C_ 1\). (iii) Each bounded modular (distributive) lattice may be embedded as a convex sublattice into a bounded modular (distributive) lattice belonging to \(C_ 0\cap C_ 1\). Examples are provided to show that \(C_ 0\) and \(C_ 1\) are incomparable.

Reviewer: M.F.Janowitz

MSC:

06B05	Structure theory of lattices
06B15	Representation theory of lattices
05C99	Graph theory

Keywords:

graph isomorphism; lattices; dual isomorphism; bounded lattice; convex sublattice

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