Transformation semigroups and matrix semigroups having proper dense subsemigroups
Patcharee Hirunmastsywan, Yupaporn Kemprasit · 1989
For any set X, let Gₓ, Mₓ, Oₓ, CPₓ and CTₓ denote the symmetric group on X, the transformation semigroup of all 1-1 transformations of X, the transformation seigroup of all onto transformations of X, the transformation semigroup of all constant transformations of X, respectively. For any field F and any positive integer n, lt Mn(F), Gn(F), Un(F), Ln(F) and Dn(F) dnote the matrix semigroup of all n x n matrices over F, the matrix group of all n x n nonsingular matrices over F, the matrix semigroup of all n x n upper triangular matrices over F, the matrix semigroup of all n x n lower triangular matrices over F and the matrix semigroup of all n x n diagonal matrices over F, respectively. The main results of this research are Theorem 1. Let X be a set. (1) If S = Gₓ, Mₓ or Oₓ, then S has a proper dense subsemigroup if and only if X is infinite. (2) CPₓ has a proper dense subsemigroup if and only if |X| > 1. (3) CTₓ has no proper dense subsemigroup. Theorem 2. Let F = (F,+,•) be a field n a positive integer and let S = Mn(F), Gn(F), Un(F), Ln(F) or Dn(F). (1) If (F,•) has a proper dense subsemigroup, then so does S. (2) If S has a proper dense subsemigroup, then F is infinite.