The Sandwich Semigroup Of Generalized Circulant Boolean Matrices

LetB={0,1} be the binary Boolean algebra and n a positive integer . The matrices which we consider in this paper are n X n matrices over B, called Boolean matrices. Let rbe a nonnegative integer. An n X nr-circulant (generalized circulant) Boolean matrix is an n X nmatrix over B in which each row, except the first, is obtained from the preceding row by shifting the elements cyclically r columns to the right, i.e.,a_i,j=a_i-1,j-r,i,j=0,1,2,...n-1, where the indices are reduces to their least nonnegative remainder modulo n. Let C_n(r) be the set of all n X n r-circulant matrices over the Boolean algebra B={0,1},G_n. For any fixed C in G_n,We can define an operation"*" in G_n as follows: A*B=ACB for any A,B in G_n, where ACB is the usual product of Boolean matrics. Then (G_n*) is a semigroup. We denote this semigroup by G_n(C) and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix C. The purpose of this paper is to characterize the idempotent elements , maximal subgroups and regular elements in G_n(C) . In chapter 2, a necessary and sufficient condition for an element in G_n(C) to be idempotent is obtained. And an algorithm to find all the idempotents in G_n(C) is given. In chapter 3, a necessary and sufficient condition for an element in G_n(C) to be in M(F) is obtained, and an algorithm to find all the elements of M(F) is given, where F is an idempotent element in G_n(C) and M(F) is the maximal subgroup in G_n(C) containing F. In final chapter, necessary and sufficient conditions for an element in G_n(C) to have a generalized inverse in G_n(C) is obtained. Algorithms to find all the g-inverses for a regular element and all the group inverses for a completely regular element in the semigroup G_n(C) are given.