In this paper the equivalence Q U on a semigroup S in terms of a set U of idempotents in S is defined. A semigroup S is called a U-liberal semigroup with U as the set of projections and denoted by S(U ) if every Q U -class in it contains an element in U . A class of U-liberal semigroups is characterized and some special cases are considered.
It is of great use to determine whether a graph is planar in both information technology and engineering areas. Although there are some known algorithms, they are quite difficult to understand and to implement. This paper proposes a new method to determine the planarity of a graph by adjacency matrix, which is very easy to implement. Especially, we give the degree conditions for the planarity of 6-vertex undirected simple graphs.
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