Opposite Elements in Clutters
研究了有限集上杂乱集合中的对立元素，发现识别对立元素可保持理想性、最大流最小割性质和包装性质，并给出反例和工具，与生成树、Steiner树、拟阵等概念关联。
Let E be a finite set of elements, and let L be a clutter over ground set E. We say distinct elements e, f are opposite if every member and every minimal cover of L contains at most one of e, f. In this paper, we investigate opposite elements and reveal a rich theory underlying such a seemingly simple restriction. The clutter C obtained from L after identifying some opposite elements is called an identification of L; inversely, L is called a split of C. We will show that splitting preserves three clutter properties, i.e., idealness, the max-flow min-cut property, and the packing property. We will also display several natural examples in which a clutter does not have these properties but a split of them does. We will develop tools for recognizing when splitting is not a useful operation, and as well, we will characterize when identification preserves the three mentioned properties. We will also make connections to spanning arborescences, Steiner trees, comparability graphs, degenerate projective planes, binary clutters, matroids, as well as the results of Menger, Ford and Fulkerson, the Replication Conjecture, and a conjecture on ideal, minimally nonpacking clutters.