A problem of Bosák concerning the graphs of semigroups
Y.-F. Lin · Proceedings of the American Mathematical Society · 1969
Let S be a semigroup, 8 the set of all proper subsemigroups of S. By the graph G(3) of S, we mean the nondirected graph whose set of vertices is 8, in which vertices A and B are adjacent (that is, are joined by an edge) if and only if A F B and AnB 0. Recall that a graph is connected provided there is a path between every pair of its vertices. In his paper [1], Bosak proved: If S is a nondenumerable semigroup or a periodic semigroup with more than two elements, then its graph G(S) is connected. Bosak then raised the following open problem. Bosak's Problem [1, p. 1221:1 Does there exist a semigroup with more than two elements whose graph is disconnected? The purpose of this article is to answer this problem, in the negative, by proving the following theorem.