  This is a survey paper on algorithms that have been developed during the last
25 years for the explicit computation of the structure of an associative
algebra of finite dimension over either a finite field or an algebraic number
field. This constructive approach was initiated in 1985 by Friedl and Ronyai
and has since been developed by Cohen, de Graaf, Eberly, Giesbrecht, Ivanyos,
Kuronya and Wales. I illustrate these algorithms with the case n = 2 of the
rational semigroup algebra of the partial transformation semigroup PT_n on n
elements; this generalizes the full transformation semigroup and the symmetric
inverse semigroup, and these generalize the symmetric group S_n.
