Geodesic
On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices
Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1117-1129 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 06F30, 15A33, 15A36
Keywords: generalized ciculant Boolean matrix; sandwich semigroup; idempotent element; maximal subgroup 
@article{CMJ_2006_56_4_a3,
     author = {Chen, Jinsong and Tan, Yijia},
     title = {On the maximal subgroup of the sandwich semigroup of generalized circulant {Boolean} matrices},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1117--1129},
     year = {2006},
     volume = {56},
     number = {4},
     mrnumber = {2280798},
     zbl = {1164.15323},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a3/}
}
TY  - JOUR
AU  - Chen, Jinsong
AU  - Tan, Yijia
TI  - On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices
JO  - Czechoslovak Mathematical Journal
PY  - 2006
SP  - 1117
EP  - 1129
VL  - 56
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a3/
LA  - en
ID  - CMJ_2006_56_4_a3
ER  - 
%0 Journal Article
%A Chen, Jinsong
%A Tan, Yijia
%T On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices
%J Czechoslovak Mathematical Journal
%D 2006
%P 1117-1129
%V 56
%N 4
%U http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a3/
%G en
%F CMJ_2006_56_4_a3
Chen, Jinsong; Tan, Yijia. On the maximal subgroup of the sandwich semigroup of generalized circulant Boolean matrices. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1117-1129. http://geodesic.mathdoc.fr/item/CMJ_2006_56_4_a3/

[1] C.-Y.  Chao, M.-C.  Zhang: On generalized circulants over a Boolean algebra. Linear Algebra Appl. 62 (1984), 195–206. | DOI | MR

[2] W.-C.  Huang: On the sandwich semigroups of circulant Boolean matrices. Linear Algebra Appl. 179 (1993), 135–160. | MR | Zbl

[3] Mou-Chen Zhang: On the maximal subgroup of the semigroup of generalized circulant Boolean matrices. Linear Algebra Appl. 151 (1991), 229–243. | MR

[4] A. H. Clifford, G. B. Preston: The Algebra Theory of Semigroups, Vol.  1. Amer. Math. Soc., Providence, 1961. | MR

[5] J. S. Montague, R. J. Plemmons: Maximal subgroup of semigroup of relations. J. Algebra 13 (1969), 575–587. | MR

[6 K.-H. Kim, S. Schwarz] The semigroup of circulant Boolean matrices. Czechoslovak Math. J. 26(101) (1976), 632–635. | MR | Zbl

[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

[8] K. Ireland, M.  Rosen: A Classical Introduction to Modern Number Theory. Springer-Verlag, New York, 1982. | MR