On the sandwich semigroups of circulant Boolean matrices

Abstract

Let C n be the semigroup of all the n × n ( n ⩾ 2) circulant Boolean matrices, and R a nonzero element in C n . The sandwich semigroup of C n with the sandwich element R is denoted by C n ( R). The purpose of this paper is to discuss the Green's classes, idempotent elements, regular elements, and maximal subgroups in C n ( R). In Section 4, a necessary and sufficient condition on the Green's class L R ( A) is given, where L R ( A) is the Green's class in the sandwich semigroup C n ( R) containing A, and A is an arbitrary circulant Boolean matrix in C n ( R). In Sections 5 and 6, the idempotent elements and the maximal subgroups containing an idempotent element in C n ( R) are discussed. Some necessary and sufficient conditions which characterize the idempotent elements and maximal subgroups are obtained. In Section 7, we use the results of Sections 5 and 6 to obtain a necessary and sufficient condition which characterizes the regular elements in C n ( R). Otherwise, some counting theorems about the idempotent elements, the regular elements, and the maximal subgroups are given.