Geodesic
Prime ideals in the lattice of additive induced-hereditary graph properties
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 117-127

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An additive induced-hereditary property of graphs is any class of finite simple graphs which is closed under isomorphisms, disjoint unions and induced subgraphs. The set of all additive induced-hereditary properties of graphs, partially ordered by set inclusion, forms a completely distributive lattice. We introduce the notion of the join-decomposability number of a property and then we prove that the prime ideals of the lattice of all additive induced-hereditary properties are divided into two groups, determined either by a set of excluded join-irreducible properties or determined by a set of excluded properties with infinite join-decomposability number. We provide non-trivial examples of each type.
Keywords: hereditary graph property, prime ideal, distributive lattice, induced subgraphs 
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Berger, Amelie; Mihók, Peter. Prime ideals in the lattice of additive induced-hereditary graph properties. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 117-127. http://geodesic.mathdoc.fr/item/DMGT_2003_23_1_a7/