surjectyw

Surjectyw is a term used in some mathematical discussions to denote a specific type of surjective function that admits a right-inverse, i.e., a function s: Y -> X with f ∘ s = id_Y. In category-theoretic language, such a map f: X -> Y is a split epimorphism. Equivalently, Y is a retract of X via the pair (f, s).

Definition and basic properties:

Let f: X -> Y be a function. f is surjectyw if it is surjective and there exists a section s: Y -> X satisfying f(s(y)) = y for all y in Y. This implies that Y is a retract of X, and that there is a subset A ⊆ X and a bijection h: A -> Y with f|_A = h. The inverse h^{-1}: Y -> A then provides a canonical choice of preimage for each y, yielding a right-inverse s = inclusion ∘ h^{-1}.

Examples:

A standard example is the projection map p_Y: X × Y -> Y, which is surjective and has a right-inverse s(y) = (x0, y) for any fixed x0 ∈ X. More generally, any surjective function that comes equipped with a section (a choice of preimage for every y) is surjectyw.

Relationships and scope:

In ZF set theory, the existence of a right-inverse for every surjection is equivalent to a form of the axiom of choice; without AC, some surjections may not be surjectyw. Thus surjectyw can be viewed as the surjection-plus-section condition, i.e., split surjections.

See also:

split epimorphism, retract, right-inverse, section, axiom of choice.