geometric lattice

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In the mathematics of
matroid In combinatorics, a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemati ...
s and lattices, a geometric lattice is a
finite Finite is the opposite of Infinity, infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected ...
atomistic
semimodular lattice Image:Centred hexagon lattice D2.svg, The centred hexagon lattice ''S''7, also known as ''D''2, is semimodular but not modular. In the branch of mathematics known as order theory, a semimodular lattice, is a lattice (order), lattice that satisfies t ...
, and a matroid lattice is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of Matroid#Flats, flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.

# Definition

A lattice (order), lattice is a partially ordered set, poset in which any two elements $x$ and $y$ have both a supremum, denoted by $x\vee y$, and an infimum, denoted by $x\wedge y$. : The following definitions apply to posets in general, not just lattices, except where otherwise stated. * For a minimal element $x$, there is no element $y$ such that $y < x$. * An element $x$ covering relation, covers another element $y$ (written as $x :> y$ or $y <: x$) if $x > y$ and there is no element $z$ distinct from both $x$ and $y$ so that $x > z > y$. * A cover of a minimal element is called an Atom (order theory), atom. * A lattice is atomistic (order theory), atomistic if every element is the supremum of some set of atoms. * A poset is Graded poset, graded when it can be given a rank function $r\left(x\right)$ mapping its elements to integers, such that $r\left(x\right)>r\left(y\right)$ whenever $x>y$, and also $r\left(x\right)=r\left(y\right)+1$ whenever $x :> y$. : When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one. * A graded lattice is semimodular lattice, semimodular if, for every $x$ and $y$, its rank function obeys the identity :: $r\left(x\right)+r\left(y\right)\ge r\left(x\wedge y\right)+r\left(x\vee y\right). \,$ * A matroid lattice is a lattice that is both atomistic and semimodular. A geometric lattice is a ''finite'' matroid lattice., p. 51. : Some authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.

# Cryptomorphism

The geometric lattices are cryptomorphism, cryptomorphic to (finite, simple) matroids, and the matroid lattices are cryptomorphic to simple matroids without the assumption of finiteness. Like geometric lattices, matroids are endowed with matroid rank, rank functions, but these functions map sets of elements to numbers rather than taking individual elements as arguments. The rank function of a matroid must be monotonic (adding an element to a set can never decrease its rank) and they must be submodular functions, meaning that they obey an inequality similar to the one for semimodular lattices: :$r\left(X\right)+r\left(Y\right)\ge r\left(X\cap Y\right)+r\left(X\cup Y\right). \,$ The maximal element, maximal sets of a given rank are called flats. The intersection of two flats is again a flat, defining a greatest lower bound operation on pairs of flats; one can also define a least upper bound of a pair of flats to be the (unique) maximal superset of their union that has the same rank as their union. In this way, the flats of a matroid form a matroid lattice, or (if the matroid is finite) a geometric lattice. Conversely, if $L$ is a matroid lattice, one may define a rank function on sets of its atoms, by defining the rank of a set of atoms to be the lattice rank of the greatest lower bound of the set. 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. These two 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.

# Duality

There are two different natural notions of duality for a geometric lattice $L$: the dual matroid, which has as its basis sets the complement (set theory), complements of the bases of the matroid corresponding to $L$, and the Duality (order theory), dual lattice, the lattice that has the same elements as $L$ in the reverse order. They are not the same, and indeed the dual lattice is generally not itself a geometric lattice: the property of being atomistic is not preserved by order-reversal. defines the adjoint of a geometric lattice $L$ (or of the matroid defined from it) to be a minimal geometric lattice into which the dual lattice of $L$ is order embedding, order-embedded. Some matroids do not have adjoints; an example is the Vámos matroid.