Results on infinite matroids, ideals, ultrafilters and local bases, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Closure basics — Lemma 1.2. Closure is monotone, bases span, and a maximal independent subset of Y spans Y.
• Direct sums of infinite matroids — Proposition 3.1. A direct sum of matroids is a matroid; bases and closures are componentwise; self-duality is inherited.
• Self-dual reduction (OpenAI) — Lemma 4.2. For self-dual matroids whose spanning sets contain bases, a packing/covering partition gives an independent covering.
• Independent columns — Lemma 2.1. Continuum many subsets of D whose finite membership patterns have exact proportion 2^-r in late blocks.
• Ultrafilter extension — Lemma 3.1. Every family with the finite intersection property lies in an ultrafilter (by transfinite recursion).
• An ultrafilter avoiding the larger ideal (OpenAI) — Lemma 3.2. An ultrafilter disjoint from K, containing every G_t, each member K-contained in only finitely many G_t.
• Raising the ordinal rank (OpenAI) — Lemma 4.2. Between a small set and a large set there are large sets of arbitrarily high ordinal rank.
• Simultaneous reservations (OpenAI) — Lemma 5.1. For fewer than continuum many positive sets, one index gives two disjoint K-small sets meeting all of them positively.
• Local bases: antichains and exchanges — Lemma 1.3. The local basis families are antichains closed under balanced finite exchanges.
• Interpolation (OpenAI) — Lemma 2.2. Between a set with lower limit at most -1 and one with positive lower limit lies a local basis.
• Interval alternatives (OpenAI) — Proposition 2.3. For intervals near the class, some local basis lies below, between, or above the endpoints.
• The basis family (OpenAI) — Proposition 3.1. A complement-closed antichain of admissible sets, closed under balanced exchanges, meeting every interval with infinite gap.
• A self-dual matroid with an ordinal inequality (OpenAI) — Theorem 4.1. A self-dual matroid Q on a countable set whose bases satisfy rho(B_L) > rho(D minus B_R) when B_L is large.
• Uniformity of Q — Proposition 4.2. Every subset is independent or contains a basis; finite sets are independent; spanning sets of restrictions contain maximal independent sets.
• Spanning property of the pair — Lemma 1.1. In the two direct sums along the double ray, spanning sets of restrictions contain maximal independent subsets.
• Packing/covering fails for infinite matroids (OpenAI) — Theorem 2.1. Two self-dual matroids on a countable set have no independent covering and no packing/covering partition.
• Self-dual uniform matroids in ZFC (OpenAI) — Corollary 3.1. A countable self-dual matroid with every subset independent or spanning and infinite bases and cobases exists in ZFC.
• Controlled basis extension — Lemma 2.2. Independent A and basis D extend to a basis contained in A union D.
• Dual matroid and double dual — Theorem 2.3. All independence axioms, including maximal extension on arbitrary subsets; no finitary hypothesis.
• Complementary maximality — Lemma 2.4. Maximality of the inside part of a basis is equivalent to dual maximality of its outside complement.
• Restriction and contraction matroids — Theorem 2.5. Both operations yield matroids on every selected subset.
• Contraction bases and independent gluing — Lemma 2.6. Basis extension through contraction and gluing to every independent set on the complementary ground set.
• Closure equality and spanning bases — Lemma 2.7. A basis of a subset has its closure; every spanning set contains a basis; restriction closure identity.
• Spanning/independence duality — Corollary 2.8. Complement identities for restrictions and contractions with the ground set explicit.
• Intersection and packing/covering — Proposition 4.3. Explicit witnesses in both directions for arbitrary infinite matroids M,N: intersection(M,N) iff packing/covering(M,N*).
• Closure and minor ground sets — Lemma 4.4. Idempotent closure, enlarged contraction independence, restriction/contraction identity and spanning gluing.
• Greatest wave support — Lemma 4.5. Arbitrary set-indexed families; transfinite closure argument, not a union of independent sets.
• Family-to-pair encoding — Proposition 4.6. Explicit Cartesian coordinates, row matroids and both witness constructions, including nonexhaustive packings and overlapping covers.
• Five universal equivalences — Theorem 4.7. Pairwise intersection, pairwise and family packing/covering, separate covering and packing conditions; all implications, no finite or countability restriction.

Reading order

Read the lessons in order; each uses the notation and results of the previous ones.