In this paper, we give a consistent presentation of dimension properties and properties of bases for vector spaces over distributive lattices. The bases consisting of join irreducible vectors are studied and their uniqueness is proved. Criteria for the following are proved: the set of join irreducible vectors is a generating set for a vector space; this set is a vector space basis; all bases contain the same number of vectors. A criterion of uniqueness for the basis is proved. The basis containing the greatest number of vectors is found. We give a description for all standard bases of a vector space. We prove a theorem allowing one to calculate the space dimension and to find the basis of the smallest number of vectors by known algorithms. These results are applied to vector spaces over chains: we prove that there exists a standard basis, that the basis of join irreducible vectors is the standard basis, that a standard basis is unique. We calculate the dimension of the arithmetic space and describe all bases containing the smallest number of vectors. It is proved that all such bases are standard.
Найдено выражение числа пересечений графа через минимальное число полных подграфов, образующих покрытие графа. Указанный результат позволяет единым методом получить известные свойства числа пересечений графа. Выделен класс графов, для которых число пересечений равно наименьшему числу клик, покрывающих граф. Доказано, что число пересечений полного r-дольного графа r x K 2 равно наименьшему n такому, что r 6 n 1 OEn=2 1. Ранее была известна асимптотика для числа пересечений графа r x K 2. Доказано, что число пересечений графа r x K 2 C K m равно наименьшему n такому, что m C r 6 2 n 1 , r 6 n 1 OEn=2 1. Найдены формулы для числа пересечений графов rC 4 , r Chain.3/, r.C 4 C K m /, rW 5 .
We find an expression of the intersection number of a graph in terms of the minimum number of complete subgraphs that form a covering of the graph. This provides us with a uniform approach to studying properties of the intersection number of a graph. We distinguish the class of graphs for which the intersection number is equal to the least number of cliques covering the graph. It is proved that the intersection number of a complete r-partite graph r x K 2 is equal to the least n such that r n 1 OEn=2 1. It is proved that the intersection number of the graph r x K 2 C K m is equal to the least n such that m C r 2 n 1 , r n 1 OEn=2 1 . Formulas for the intersection numbers of the graphs rC 4 , r Chain.3/, r.C 4 C K m /, rW 5 are obtained.
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