Let S be a semigroup and
T be a semigroup with zero (T = T0). An ideal extension of S by T is a
semigroup V containing S as an ideal and such that the Rees quotient V∕S is
isomorphic to T. A mapping α from T∗= T −{0} into S is said to be a partial
homomorphism, if t1,t2∈ T∗,t1t2≠0 implies (f1,t2)α = (t1α)(t2α). Every partial
homomorphism from τ∗ into S gives rise to an ideal extension of )S by T. Further, in
certain cases every ideal extension of S by T is obtained in this way. In this
paper a characterization is given for all partial homomorphisms from τ∗ into
S.
