Abstract. The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer polymatroids. We focus on characterizations of integer polymatroids using their bases, their circuits, and their cyclic flats along with the rank of each cyclic flat and each element; we offer some new characterizations and insights into known characterizations.

Abstract. If [Formula: see text] is a minor-closed class of matroids, the class [Formula: see text] of integer polymatroids whose natural matroids are in [Formula: see text] is also minor closed, as is the class [Formula: see text] of [Formula: see text]-polymatroids in [Formula: see text]. We find the excluded minors for [Formula: see text] when [Formula: see text] is (i) the class of binary matroids, (ii) the class of matroids with no [Formula: see text]-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class [Formula: see text] has finitely many excluded minors, but that is true of [Formula: see text] only in case (ii). We also introduce the [Formula: see text]-natural matroid, a variant of the natural matroid for a [Formula: see text]-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality.

An intertwine of a pair of matroids is a matroid such that it, but none of its proper minors, has minors that are isomorphic to each matroid in the pair. For pairs for which neither matroid can be obtained, up to isomorphism, from the other by taking free extensions, free coextensions, and minors, we construct a family of rank-k intertwines for each sufficiently large integer k. We also treat some properties of these intertwines.

One definition of an interval order is as an order isomorphic to that of a family of nontrivial intervals of a linearly ordered set with [a,b] < [c,d] if b ⩽ c. Fishburn's theorem states that an order is an interval order if and only if it has no four-element restriction isomorphic to the ordered set (shown in Fig. 1) “2 + 2”. We show that an order is isomorphic to a family of nontrivial intervals of a weak order, ordered as above, if and only if it has no restriction to one of the four ordered sets (shown in Fig. 2) “3 + 2”, “2 + N”, a six-element crown or a six-element fence.