Isthmus Structure in a Summary Matroid

A summary matroid of the form $M^k=M+\dots+M$ ($k$ times) is considered, where $M$ is an arbitrary matroid on finite set $E$. Subset $F\subset E$ on which equality is attained in the Nash-Williams formula for determining the rank of matroid-sum $M^k$ is called a $k$-extremal subsets. The article investigates the class of $k$-extremal sets of matroid $M$.