# Article

Full entry | PDF   (0.3 MB)
Keywords:
generalized ciculant Boolean matrix; sandwich semigroup; idempotent element; maximal subgroup
Summary:
Let $n$ be a positive integer, and $C_{n} (r)$ the set of all $n\times n$ $r$-circulant matrices over the Boolean algebra $B=\lbrace 0,1\rbrace$, $G_{n}=\bigcup _{r=0}^{n-1}C_{n}(r)$. For any fixed $r$-circulant matrix $C$ ($C\ne 0$) in $G_{n}$, we define an operation “$\ast$” in $G_{n}$ as follows: $A\ast B=ACB$ for any $A,B$ in $G_{n}$, where $ACB$ is the usual product of Boolean matrices. Then $(G_{n},\ast )$ 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 $C$. Let $F$ be an idempotent element in $G_{n}(C)$ and $M(F)$ the maximal subgroup in $G_{n}(C)$ containing the idempotent element $F$. In this paper, the elements in $M(F)$ are characterized and an algorithm to determine all the elements in $M(F)$ is given.
References:
[1] C.-Y.  Chao, M.-C.  Zhang: On generalized circulants over a Boolean algebra. Linear Algebra Appl. 62 (1984), 195–206. DOI 10.1016/0024-3795(84)90095-8 | MR 0761067
[2] W.-C.  Huang: On the sandwich semigroups of circulant Boolean matrices. Linear Algebra Appl. 179 (1993), 135–160. MR 1200148 | Zbl 0768.20031
[3] Mou-Chen Zhang: On the maximal subgroup of the semigroup of generalized circulant Boolean matrices. Linear Algebra Appl. 151 (1991), 229–243. MR 1102151
[4] A. H. Clifford, G. B. Preston: The Algebra Theory of Semigroups, Vol.  1. Amer. Math. Soc., Providence, 1961. MR 0132791
[5] J. S. Montague, R. J. Plemmons: Maximal subgroup of semigroup of relations. J. Algebra 13 (1969), 575–587. MR 0252539
[6 K.-H. Kim, S. Schwarz] The semigroup of circulant Boolean matrices. Czechoslovak Math. J. 26(101) (1976), 632–635. MR 0430121 | Zbl 0347.20037
[7] J.-S. Chen, Y.-J.  Tan: The idempotent elements in the sandwich semigroup of generalized elements Boolean mareices. J.  Fuzhou Univ. Nat. Sci. 31 (2003), 505–509. MR 2023977
[8] K. Ireland, M.  Rosen: A Classical Introduction to Modern Number Theory. Springer-Verlag, New York, 1982. MR 0661047

Partner of