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In the mathematics of [[matroid]]s and [[lattice (order)|lattices]], a '''geometric lattice''' is a [[finite set|finite]] [[Atom (order theory)|atomistic]] [[semimodular lattice]], and a '''matroid lattice''' is an atomistic semimodular lattice without the assumptions of finiteness. Geometric lattices and matroid lattices, respectively, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.


==Definition==
Recall that a '''[[lattice (order)|lattice]]''' is a [[partially ordered set]] in which any two elements <math>x</math> and <math>y</math> have a least upper bound <math>x\vee y</math> and a greatest lower bound <math>x\wedge y</math>. In a lattice, or more generally a partially ordered set, an element <math>x</math> '''covers''' another element <math>y</math> (written as <math>x :> y</math> or <math> y <: x</math>) if <math>x > y</math> and there is no third element <math>z</math> between <math>x</math> and <math>y</math>. A lattice or partially ordered set is '''[[Graded poset|graded]]''' when it can be given a rank function <math>r(x)</math> mapping its elements to integers, such that <math>r(x)>r(y)</math> whenever <math>x>y</math> and in particular <math>r(x)=r(y)+1</math> whenever even <math>x :> y</math>.


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In a lattice having a bottom element, one may assume without loss of generality that its rank is zero. The '''[[Atom (order theory)|atoms]]''' of such a lattice are the elements with rank one, and the lattice is '''atomistic''' if every element is the least upper bound of some set of atoms.  
 
A graded lattice is '''[[semimodular lattice|semimodular]]''' if, for every <math>x</math> and <math>y</math>, its rank function obeys the identity<ref>{{harvtxt|Birkhoff|1995}}, Theorem 15, p.&nbsp;40. More precisely, Birkhoff's definition reads "We shall call P (upper) semimodular when it satisfies: If ''a''≠''b'' both cover ''c'', then there 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''(''x'')+''r''(''y'')≥''r''(''x''∧''y'')+''r''(''x''∨''y'')".</ref>
 
:<math>r(x)+r(y)\ge r(x\wedge y)+r(x\vee y). \, </math>
 
A '''matroid lattice''' is a lattice that is both atomistic and semimodular,<ref>{{citation
  | last1 = Maeda | first1 = F.
| last2 = Maeda | first2 = S.
| location = New York
| mr = 0282889
| publisher = Springer-Verlag
| series = Die Grundlehren der mathematischen Wissenschaften, Band 173
| title = Theory of Symmetric Lattices
| year = 1970}}.</ref><ref>{{citation
| last = Welsh | first = D. J. A.
| isbn = 9780486474397
| page = 388
| publisher = Courier Dover Publications
| title = Matroid Theory
| year = 2010}}.</ref> and a '''geometric lattice''' is a matroid lattice with finitely many elements.<ref name="w10-51">{{harvtxt|Welsh|2010}}, p.&nbsp;51.</ref>
 
Some authors consider only finite matroid lattices, and use the terms "geometric lattice" and "matroid lattice" interchangeably for both.<ref>{{citation|title=Lattice Theory|volume=25|series=Colloquium Publications|publisher=American Mathematical Society|first=Garrett|last=Birkhoff|authorlink=Garrett Birkhoff|edition=3rd|year=1995|isbn=9780821810255|page=80|url=http://books.google.com/books?id=0Y8d-MdtVwkC&pg=PA80}}.</ref>
 
==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 function]]s, meaning that they obey an identity similar to the one for semimodular lattices:
 
:<math>r(X)+r(Y)\ge r(X\cup Y)+r(X\cap Y). \,</math>
 
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.<ref name="w10-51"/>
 
Conversely, if <math>L</math> 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.<ref name="w10-51"/>
 
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.<ref name="w10-51"/>
 
==Duality==
There are two different natural notions of duality for a geometric lattice <math>L</math>: the dual matroid, which has as its basis sets the [[complement (set theory)|complements]] of the bases of the matroid corresponding to <math>L</math>, and the [[order dual|dual lattice]], the lattice that has the same elements as <math>L</math> 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. {{harvtxt|Cheung|1974}} defines the [[adjoint]] of a geometric lattice <math>L</math> (or of the matroid defined from it) to be a minimal geometric lattice into which the dual lattice of <math>L</math> is [[order embedding|order-embedded]]. Some matroids do not have adjoints; an example is the [[Vámos matroid]].<ref>{{citation
| last = Cheung | first = Alan L. C.
| doi = 10.4153/CMB-1974-066-5
| issue = 3
| journal = [[Canadian Mathematical Bulletin]]
| mr = 0373976
| pages = 363–365; correction, ibid. 17 (1974), no. 4, 623
| title = Adjoints of a geometry
| volume = 17
| year = 1974}}.</ref>
 
==Additional properties==
Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound elements) is itself geometric; taking an interval of a geometric lattice corresponds to forming a [[matroid minor|minor]] of the associated matroid. Geometric lattices are [[complemented lattice|complemented]], and because of the interval property they are also relatively complemented.<ref>{{harvtxt|Welsh|2010}}, pp. 55, 65–67.</ref>
 
Every finite lattice is a sublattice of a geometric lattice.<ref>{{harvtxt|Welsh|2010}}, p. 58; Welsh credits this result to [[Robert P. Dilworth]], who proved it in 1941–1942, but does not give a specific citation for its original proof.</ref>
 
==References==
{{reflist}}
 
==External links==
* {{planetmath reference|id=7972|title=Geometric lattice}}
 
[[Category:Lattice theory]]
[[Category:Matroid theory]]

Revision as of 10:10, 7 January 2014

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, form the lattices of flats of finite and infinite matroids, and every geometric or matroid lattice comes from a matroid in this way.

Definition

Recall that a lattice is a partially ordered set in which any two elements x and y have a least upper bound x∨y and a greatest lower bound x∧y. In a lattice, or more generally a partially ordered set, an element x covers another element y (written as x:>y or y<:x) if x>y and there is no third element z between x and y. A lattice or partially ordered set is graded when it can be given a rank function r(x) mapping its elements to integers, such that r(x)>r(y) whenever x>y and in particular r(x)=r(y)+1 whenever even x:>y.

In a lattice having a bottom element, one may assume without loss of generality that its rank is zero. The atoms of such a lattice are the elements with rank one, and the lattice is atomistic if every element is the least upper bound of some set of atoms.

A graded lattice is semimodular if, for every x and y, its rank function obeys the identity[1]

r(x)+r(y)≥r(x∧y)+r(x∨y).

A matroid lattice is a lattice that is both atomistic and semimodular,[2][3] and a geometric lattice is a matroid lattice with finitely many elements.[4]

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

Cryptomorphism

The geometric lattices are 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 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 identity similar to the one for semimodular lattices:

r(X)+r(Y)≥r(X∪Y)+r(X∩Y).

The 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.[4]

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.[4]

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.[4]

Duality

There are two different natural notions of duality for a geometric lattice L: the dual matroid, which has as its basis sets the complements of the bases of the matroid corresponding to L, and the 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. Template:Harvtxt 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-embedded. Some matroids do not have adjoints; an example is the Vámos matroid.[6]

Additional properties

Every interval of a geometric lattice (the subset of the lattice between given lower and upper bound 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 also relatively complemented.[7]

Every finite lattice is a sublattice of a geometric lattice.[8]

References

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External links

  1. ↑ Template:Harvtxt, Theorem 15, p. 40. More precisely, Birkhoff's definition reads "We shall call P (upper) semimodular when it satisfies: If a≠b both cover c, then there 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(x)+r(y)≥r(x∧y)+r(x∨y)".
  2. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  3. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  4. ↑ 4.0 4.1 4.2 4.3 Template:Harvtxt, p. 51.
  5. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  6. ↑ Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  7. ↑ Template:Harvtxt, pp. 55, 65–67.
  8. ↑ Template:Harvtxt, p. 58; Welsh credits this result to Robert P. Dilworth, who proved it in 1941–1942, but does not give a specific citation for its original proof.
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