are the elements, and the hyperedges are the sets. Then, a transversal (defined as a system of not-necessarily-distinct representatives) is a vertex cover in a hypergraph. Examples In group theory, given a subgroup H of a group G, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. In this case, the "sets" (cosets) are mutually disjoint, i.e. the cosets form a partition of the group. As a particular case of the previous example, given a direct product of groups , then H is a transversal for the cosets of K. In general, since any equivalence relation on an arbitrary set gives rise to a partition, picking any representative from each equivalence class results in a transversal. 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 with domain X as the partition of the domain . 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 . For a not-necessarily-injective f, fixing a transversal T of induces a one-to-one correspondence between T and the image of f, henceforth denoted by . Consequently, a function is well defined by the property that for all z in where x is the unique element in T such that ; 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 , which is the proof (when the domain and codomain of f are the same set) that the full transformation semigroup is a regular semigroup. 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 may not hold. However if we denote by , then f is a quasi-inverse of h, i.e. . Common Transversals A common transversal of the collections A and B (where ) is a set that is a transversal of both A and B. The collections A and