If X and Y are sets, we let P(X, Y) denote the set of all partial transformations from X into Y (that is, all mappings whose domain and range are subsets of X and Y, respectively). We define an operation * on P(X, Y) by choosing 0 E P(Y, X) and writing. alpha * beta = alpha circle theta circle beta, for each alpha, beta epsilon P(X, Y). Then (P(X, Y), *) is a semigroup, and some authors have determined when this is regular (Magill and Subbiah, 1975), when it contains a "proper dense subsemigroup" (Wasanawichit and Kemprasit, 2002) and when it is factorisable (Saengsura, 2001). In this paper, we extend the latter work to certain subsemigroups of (P(X, Y), *). We also consider the corresponding idea for partial linear transformations from one vector space into another. In this way, we generalise known results for total transformations and for injective partial transformations between sets, and we establish new results for linear transformations between vector spaces.