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Concept

Hasse diagram

A drawing of an ordering in which each element sits directly above those it covers, with the implied steps left out.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

124361251020153060× 2 →× 3 ↑2^2 × 3 × 5 — 3 × 2 × 2 = 12 divisors

The shape of a number's divisors

Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

number · unique factorisation better for side one, worse for side twoM1 A4 B2 C1 D3side one's ranks 2, 2, 2, 2M2 A3 B2 C1 D4side one's ranks 3, 2, 2, 3M3 A4 B1 C2 D3side one's ranks 2, 4, 3, 2M4 A3 B1 C2 D4side one's ranks 3, 4, 3, 3join and meet, on every pairpairjoinmeetM1, M2M1M2M1, M3M1M3M1, M4M1M4M2, M3M1M4M2, M4M2M4M3, M4M3M44 of the 24 matchings are stable, drawn with the best for side one at the topevery one of the 16 ordered pairs was joined and met, and all 32 results were themselves stablethe top A4 B2 C1 D3 is what side one proposing returns; the bottom A3 B1 C2 D4 is what side two proposing returns

The side that proposes wins

An instance usually has several stable matchings, and the set of them is not a heap — it is a lattice, closed under taking the better partner and under taking the worse. The two ends of that lattice are exactly what deferred acceptance returns from the two sides, so whoever proposes decides which end the instance lands on.

applied · stable matching

Named alongside it

The objects these essays reach for when they reach for this one.

Order latticeBlocking pairCounting two waysDeferred acceptanceDivisor functionDivisor sumGeometric seriesLatticeMultiplicative functionPartial orderPerfect numberPreference profile

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