# On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices

• Volume: 56, Issue: 4, page 1117-1129
• ISSN: 0011-4642

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## Abstract

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Let $n$ be a positive integer, and ${C}_{n}\left(r\right)$ the set of all $n×n$$r$-circulant matrices over the Boolean algebra $B=\left\{0,1\right\}$, ${G}_{n}={\bigcup }_{r=0}^{n-1}{C}_{n}\left(r\right)$. For any fixed $r$-circulant matrix $C$ ($C\ne 0$) in ${G}_{n}$, we 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 matrices. Then $\left({G}_{n},*\right)$ is a semigroup. We denote this semigroup by ${G}_{n}\left(C\right)$ and call it the sandwich semigroup of generalized circulant Boolean matrices with sandwich matrix $C$. Let $F$ be an idempotent element in ${G}_{n}\left(C\right)$ and $M\left(F\right)$ the maximal subgroup in ${G}_{n}\left(C\right)$ containing the idempotent element $F$. In this paper, the elements in $M\left(F\right)$ are characterized and an algorithm to determine all the elements in $M\left(F\right)$ is given.

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