Certain Regular Semigroups of Infinite Matrices

Let F be a field and N the set of natural numbers. It is known that the multiplicative semigroup of all bounded N x N matrices over F is a regular semigroup. Our purpose is to consider the multiplicative semigroup U* (F) of all column bounded upper triangular N x N matrices A over F with for some k E N, A(ii) not equal 0 for i is an element of{1,..., k} and A(ij) = 0 for i > k and all j is an element of N. In this paper, we show that U* (F) is a regular semigroup which is a disjoint union of right simple regular semigroups, and its idempotents are also determined.