  We consider wreath product decompositions for semigroups of triangular
matrices. We exhibit an explicit wreath product decomposition for the semigroup
of all n-by-n upper triangular matrices over a given field k, in terms of
aperiodic semigroups and affine groups over k. In the case that k is finite
this decomposition is optimal, in the sense that the number of group terms is
equal to the group complexity of the semigroup. We also obtain some
decompositions for semigroups of triangular matrices over more general rings
and semirings.
