On the semigroup whose elements are subgraphs of a complete graph (Q6156683)

Summary: Let \(K_n\) be a complete graph on \(n\) vertices. Denote by \(S K_n\) the set of all subgraphs of \(K_n\). For each \(G, H \in S K_n\), the ring sum of \(G\) and \(H\) is a graph whose vertex set is \(V(G) \cup V(H)\) and whose edges are that of either \(G\) or \(H\), but not of both. Then \(S K_n\) is a semigroup under the ring sum. In this paper, we study Green's relations on \(S K_n\) and characterize ideals, minimal ideals, maximal ideals, and principal ideals of \(S K_n\). Moreover, maximal subsemigroups and a class of maximal congruences are investigated. Furthermore, we prescribe the natural order on \(S K_n\) and consider minimal elements, maximal elements and covering elements of \(S K_n\) under this order.