On the rank of generalized order-preserving transformation semigroups

For any two non-empty (disjoint) chains X and Y, and for a fixed order-preserving transformation 0 : Y-+ X, let GO(X, Y ; 0) be the generalized order-preserving transformation semigroup. Let O(Z) be the order -preserving transformation semigroup on the set Z = X UY with a defined order. In this paper, we show that GO(X, Y ; 0) can be embedded in O(Z, Y ) = (alpha E O(Z) : Z alpha C Y }, the semigroup of order-preserving transformations with restricted range. If 0 E GO(Y, X) is one-to-one, then we show that GO(X, Y ; 0) and O(X, im(0)) are isomorphic semigroups. If we suppose that |X| = m, |Y | = n, and |im(0)| = r where m, n, r E N, then we find the rank of GO(X, Y ; 0) when 0 is one-to-one but not onto. Moreover, we find lower bounds for rank(GO(X, Y ; 0)) when 0 is neither one-to-one nor onto and when 0 is onto but not one-to-one.