布尔代数上的线性群

ON LINEAR GROUPS OVER BOOLEAN ALGEBRA

布尔代数上的线性群罗铸楷，刘任任（湘潭大学计算机科学系，湘潭４１１１０５）ＯＮＬＩＮＥＡＲＧＲＯＵＰＳＯＶＥＲＢＯＯＬＥＡＮＡＬＧＥＢＲＡ￥ＬＵＯＺＨＵＫＡＩ（ＬｏＣＺＵＫＡＩ）；ＬＩＵＲＥＮＲＥＮ（ＤｅｐａｒｔｍｅｎｔｏｆＣｏｍｐｕｔｅｒＳｃｉｅｎ...

The set GL(n,B) of all invertible n × n matrices over the Boolean algebra B is a group, and is called the n-level linear group. In this paper, we prove that the following rules hold: 1) GL(n,B) = G1…Gn, Gi…Gk ∩ Gk+ = E, k = 1,…,n-1,where Gk is an Abelian group. 2) Let k = and p1,…,pt be prime numbers. Then Gk = H1 ×…× Ht (Direct product), where , i = 1,…,t. 3) Let B be a complete atomic Boolean algebra, and denote by S the set of all atoms in B. Then Hi=a∈S(Aa)(Diect product),i=1,…, t, where Aa is a cyclic matrix defineil by atom a and|(Aa)| = 4) If|S| = m, then|GL(n, B)| = (n!)m.