Transformation semigroups and matrix semigroups having proper dense subsemigroups

Abstract

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.