Another instance of a partition-based transversal occurs when one considers the equivalence relation known as the (set-theoretic) kernel of a function, defined for a function formula_4 with domain "X" as the partition of the domain formula_5. which partitions the domain of "f" into equivalence classes such that all elements in a class map via "f" to the same value. If "f" is injective, there is only one transversal of formula_6. For a not-necessarily-injective "f", fixing a transversal "T" of formula_6 induces a one-to-one correspondence between "T" and the image of "f", henceforth denoted by formula_8. Consequently, a function formula_9 is well defined by the property that for all "z" in formula_10 where "x" is the unique element in "T" such that formula_11; furthermore, "g" can be extended (not necessarily in a unique manner) so that it is defined on the whole codomain of "f" by picking arbitrary values for "g(z)" when "z" is outside the image of "f". It is a simple calculation to verify that "g" thus defined has the property that formula_12, which is the proof (when the domain and codomain of "f" are the same set) that the full transformation semigroup is a regular semigroup. formula_13 acts as a (not necessarily unique) quasi-inverse for "f"; within semigroup theory this is simply called an inverse. Note however that for an arbitrary "g" with the aforementioned property the "dual" equation formula_14 may not hold. However if we denote by formula_15, then "f" is a quasi-inverse of "h", i.e. formula_16.