Any Interval of a Geometric Lattice Is Again a Geometric Lattice

In the mathematics of matroids and lattices, a geometric lattice is a finite atomistic semimodular lattice, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, course the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this mode.

Definition [edit]

A lattice is a poset in which whatsoever 2 elements ${\displaystyle x}$ and ${\displaystyle y}$ take both a supremum, denoted past ${\displaystyle x\vee y}$ , and an infimum, denoted by ${\displaystyle x\wedge y}$ .

The following definitions apply to posets in general, non simply lattices, except where otherwise stated.

When a graded poset has a lesser element, one may assume, without loss of generality, that its rank is aught. In this case, the atoms are the elements with rank one.

${\displaystyle r(x)+r(y)\geq r(x\wedge y)+r(ten\vee y).\,}$

• A matroid lattice is a lattice that is both atomistic and semimodular.^{[2] [iii]} A geometric lattice is a finite matroid lattice.^[4]

Some authors consider simply finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.^[5]

Cryptomorphism [edit]

The geometric lattices are cryptomorphic to (finite, simple) matroids, and the matroid lattices are cryptomorphic to simple matroids without the assumption of finiteness.

Similar geometric lattices, matroids are endowed with rank functions, but these functions map sets of elements to numbers rather than taking private elements as arguments. The rank function of a matroid must be monotonic (adding an element to a gear up tin can never decrease its rank) and they must exist submodular functions, meaning that they obey an inequality like to the one for semimodular lattices:

${\displaystyle r(10)+r(Y)\geq r(10\cap Y)+r(X\loving cup Y).\,}$

The maximal sets of a given rank are called flats. The intersection of two flats is over again a flat, defining a greatest lower bound operation on pairs of flats; one tin likewise define a least upper leap of a pair of flats to be the (unique) maximal superset of their union that has the aforementioned rank as their spousal relationship. In this way, the flats of a matroid form a matroid lattice, or (if the matroid is finite) a geometric lattice.^[4]

Conversely, if ${\displaystyle L}$ is a matroid lattice, one may ascertain a rank function on sets of its atoms, past defining the rank of a set of atoms to be the lattice rank of the greatest lower jump of the fix. This rank function is necessarily monotonic and submodular, so it defines a matroid. This matroid is necessarily simple, meaning that every two-element set has rank two.^[4]

These ii constructions, of a simple matroid from a lattice and of a lattice from a matroid, are inverse to each other: starting from a geometric lattice or a simple matroid, and performing both constructions one after the other, gives a lattice or matroid that is isomorphic to the original one.^[4]

Duality [edit]

At that place are ii different natural notions of duality for a geometric lattice ${\displaystyle L}$ : the dual matroid, which has every bit its ground sets the complements of the bases of the matroid corresponding to ${\displaystyle L}$ , and the dual lattice, the lattice that has the same elements equally ${\displaystyle L}$ in the reverse order. They are not the same, and indeed the dual lattice is generally not itself a geometric lattice: the belongings of being atomistic is not preserved by social club-reversal. Cheung (1974) defines the adjoint of a geometric lattice ${\displaystyle Fifty}$ (or of the matroid defined from information technology) to be a minimal geometric lattice into which the dual lattice of ${\displaystyle Fifty}$ is order-embedded. Some matroids do not have adjoints; an instance is the Vámos matroid.^[vi]

Additional backdrop [edit]

Every interval of a geometric lattice (the subset of the lattice between given lower and upper jump elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a minor of the associated matroid. Geometric lattices are complemented, and because of the interval property they are too relatively complemented.^[seven]

Every finite lattice is a sublattice of a geometric lattice.^[viii]

References [edit]

1. ^ Birkhoff (1995), Theorem xv, p. 40. More precisely, Birkhoff'south definition reads "Nosotros shall call P (upper) semimodular when it satisfies: If a≠b both cover c, so at that place exists a d∈P which covers both a and b" (p.39). Theorem 15 states: "A graded lattice of finite length is semimodular if and only if r(ten)+r(y)≥r(10∧y)+r(x∨y)".
2. ^ Maeda, F.; Maeda, South. (1970), Theory of Symmetric Lattices, Die Grundlehren der mathematischen Wissenschaften, Ring 173, New York: Springer-Verlag, MR 0282889 .
3. ^ Welsh, D. J. A. (2010), Matroid Theory, Courier Dover Publications, p. 388, ISBN9780486474397 .
4. ^ ^{a b c d} Welsh (2010), p. 51.
5. ^ Birkhoff, Garrett (1995), Lattice Theory, Colloquium Publications, vol. 25 (tertiary ed.), American Mathematical Lodge, p. 80, ISBN9780821810255 .
6. ^ Cheung, Alan L. C. (1974), "Adjoints of a geometry", Canadian Mathematical Bulletin, 17 (three): 363–365, correction, ibid. 17 (1974), no. 4, 623, doi:10.4153/CMB-1974-066-5, MR 0373976 .
7. ^ Welsh (2010), pp. 55, 65–67.
8. ^ Welsh (2010), p. 58; Welsh credits this event to Robert P. Dilworth, who proved it in 1941–1942, but does non give a specific citation for its original proof.

External links [edit]

• "Geometric lattice". PlanetMath.
• OEIS sequence A281574 (Number of unlabeled geometric lattices with northward elements)

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Source: https://en.wikipedia.org/wiki/Geometric_lattice