A careful study is made of embeddings of posets which have a convex range. We observe that such embeddings share nice properties with the homomorphisms of more restrictive categories; for example, we show that every order embedding between two lattices with convex range is a continuous lattice homomorphism. A number of posets are considered; for one of the simplest examples, we prove that every product order embedding σ : ℕℕ → ℕℕ with convex range is of the form 1 σ ( x ) ( n ) = ( x ◦ g σ ) + y σ ( n ) if n ∈ K σ , $$ \sigma(x)(n)=\left( (x\circ g_{\sigma})+y_{\sigma}\right)(n) ~~~~\text{if}~ n\in K_{\sigma}, $$ and σ(x)(n) = y σ (n) otherwise, for all x ∈ ℕℕ, where K σ ⊆ ℕ, g σ : K σ → ℕ is a bijection and y σ ∈ ℕℕ. The most complex poset examined here is the quotient of the lattice of Baire measurable functions, with codomain of the form ℕ I for some index set I, modulo equality on a comeager subset of the domain, with its ‘natural’ ordering.