Complemented graphs and blow-ups of Boolean graphs, with applications to co-maximal ideal graphs

Abstract

For a set X, let 2X be the power set of X. Let BX be the Boolean graph, which is defined on the vertex set 2X \ {X, ?}, with M adjacent to N if M ? N = ?. In this paper, several purely graph-theoretic characterizations are provided for blow-ups of a finite or an infinite Boolean graph (respectively, a preatomic graph). Then the characterizations are used to study co-maximal ideal graphs that are blow-ups of Boolean graphs (pre-atomic graphs, respectively).