Let F be a ¯eld and N the set of natural numbers. It is known that
the multiplicative semigroup of all bounded N £ 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 £ N matrices A over F with for some k 2 N,
Aii 6= 0 for i 2 f1; : : : ; kg and Aij = 0 for i > k and all j 2 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.