On the sandwich semigroups of circulant Boolean matrices
Denote by \(C_ n\) the semigroup of all \(n \times n\) \((n \geq 2)\) circulant Boolean matrices. Fix \(R \in C_ n\) and for any \(A,B \in C_ n\), define the product \(A * B\) by \(A * B = ARB\) where \(ARB\) denotes the usual product of the Boolean matrices \(A\), \(B\) and \(C\). Then \((C_ n,*)\) is a commutative semigroup. It is referred to as a sandwich semigroup of \(C_ n\) with sandwich matrix \(R\) and it is denoted by \(C_ n(R)\). Let \(E = [e_{ij}]\) be the matrix where \(e_{ii} = 1\) and \(e_{ij} = 0\) for \(i \neq j\). Let \(P = [p_{ij}]\) be the circulant Boolean matrix where \(p_{n1} = 1\), \(p_{i,i+1} = 1\) for \(1 \leq i \leq n-1\) and \(p_{ij} = 0\) otherwise. Let \(R = P^{k_ 1} + P^{k_ 2} + \cdots + P^{k_ l}\) where \(0 \leq k_ 1 < k_ 2 < \cdots < k_ l < n\). Then \(\{k_ 1,k_ 2,\dots,k_ l\}\) is referred to as the set of support numbers of \(R\) and is denoted by \(\text{SN}(R)\). Let \(\mathcal L\) denote Green's \({\mathcal L}\)-equivalence and for \(A \in C_ n(R)\), denote by \({\mathcal L}(A)\) the \({\mathcal L}\)-equivalence class containing \(A\). It is shown that either \({\mathcal L}(A) = \{A\}\) or \({\mathcal L}(A) = \{A,PA,PA^ 2,\dots,P^{n- 1}A\}\). It is also shown that if \(0 \in \text{SN}(R)\) and \(d\) is a positive common divisor of \(k_ 1,k_ 2,\dots,k_ l,n\), then \(I = E + P^ d + P^{2d} + \cdots + P^{(t-1)d}\) where \(t = n/d\), is a nonzero idempotent of \(C_ n(R)\) and all nonzero idempotents of \(C_ n(R)\) are of this form. The author goes on to show that \(A \in C_ n(R)\) is regular if and only if there exists an integer \(m\) and an idempotent \(I \in C_ n(R)\) such that \(A = P^ mI\). In addition to all this, he describes the maximal subgroups of \(C_ n(R)\).
scientific article; zbMATH DE number 166252