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Source. Section 2 and the opening paragraphs of Sections 3–4, printed pp. 1323–1325 (published PDF).

Let $S$ be a finite set and let $\mathcal A=(A_i)_{i\in I}$ be a finite indexed family of subsets of $S$. Equal subsets at different indices remain different family members. For $J\subseteq I$, write

$$A(J)=\bigcup_{i\in J}A_i.$$

Let $p=(p_i)_{i\in I}$ have nonnegative integer coordinates and put

$$p(J)=\sum_{i\in J}p_i, \qquad N=p(I).$$

A $p$-transversal of $\mathcal A$ is a subset $X\subseteq S$ for which there are pairwise disjoint sets $X_i\subseteq A_i$ such that

$$X=\bigsqcup_{i\in I}X_i, \qquad |X_i|=p_i.$$

To represent the multiplicities, define the replicated index set and family

$$I^p=\{(i,h):i\in I, 1\le h\le p_i\}, \qquad A^p_{i,h}=A_i.$$

Thus $p$-transversals are exactly the ranges of full transversals of the replicated family $\mathcal A^p$. Coordinates with $p_i=0$ create no copy. If $I=\varnothing$, the empty set is the unique $p$-transversal.

For an integer $k\ge0$, a $k$-transversal is the support

$$X=\{y_i:i\in I\}$$

of an indexed assignment $i\mapsto y_i\in A_i$ for which every fiber has size at most $k$:

$$1\le |\{i:y_i=x\}|\le k \qquad(x\in X).$$

The assignment, rather than its support, records repeated representatives. This makes precise the source's set $Y=\{Y_i,i\in I\}$ “of not necessarily distinct elements” (p. 1325). For $k=0$, such an assignment exists exactly when $I$ is empty.

When $k\ge1$, put

$$S^k=S\times[k], \qquad \pi:S^k\longrightarrow S, \qquad \pi(x,h)=x.$$

An indexed assignment with multiplicity at most $k$ lifts to distinct labeled copies in $S^k$: for each $x$, label its at most $k$ occurrences injectively by $1,\ldots,k$. Conversely, a transversal in the copied sets $A_i\times[k]$ projects to a $k$-transversal.

All matroids in this source unit are finite. Their rank functions are denoted by $r$. A base is a maximal independent set; all bases have the rank of the ground set. The empty set has rank zero.