PRINCIPAL IDEAL GRAPHS OF RECTANGULAR BANDS

Abstract

Let  be a finite regular semigroup. We define the principal left ideal graph of  as the graph  with  and two vertices  and  () are adjacent in  if and only if . The principal right ideal graph is defined accordingly and is denoted by . In this paper we describe the structures of principal ideal graphs of rectangular bands. We see that if  is a rectangular band, then  is a disconnected graph with  components in which each component is complete with  vertices, while  is a disconnected graph with  components in which each component is complete with  vertices. We also describe the number of edges in  and  when  is a rectangular band.

Keywords: Rectangualar bands, principal ideal graphs, connected graphs, complete graphs.