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 θ : Y → X , let GO ( X, Y ; θ ) be the generalized order-preserving transformation semigroup. Let O ( Z ) be the order-preserving transformation semigroup on the set Z = X ∪ Y with a defined order. In this paper, we show that GO ( X, Y ; θ ) can be embedded in O ( Z, Y ) = { α ∈ O ( Z ) : Zα ⊆ Y } , the semigroup of order-preserving transformations with restricted range. If θ ∈ GO ( Y, X ) is one-to-one, then we show that GO ( X, Y ; θ ) and O ( X, im ( θ )) are isomorphic semigroups. If we suppose that | X | = m , | Y | = n , and | im ( θ ) | = r where m, n, r ∈ N , then we find the rank of GO ( X, Y ; θ ) when θ is one-to-one but not onto. Moreover, we find lower bounds for rank ( GO ( X, Y ; θ )) when θ is neither one-to-one nor onto and when θ is onto but not one-to-one.