The aim of this paper is to generalize several basic results from transversal theory, primarily the theorem of Edmonds and Fulkerson.

Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number ${\gamma }_{\times k}\left(G\right)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $|N_G\left[v\right]\cap S|\ge k$. Also the total k-domination number ${\gamma }_{\times k,t}\left(G\right)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $|N_G\left(v\right)\cap S|\ge k$. The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for any graph...

The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma$ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and there exists...

The theorem of Edmonds and Fulkerson states that the partial transversals of a finite family of sets form a matroid. The aim of this paper is to present a symmetrized and continuous generalization of this theorem.