Class Note for MATH 796 at KU
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Date Created: 02/06/15
Friday 2108 Modular and Semimodular Lattices De nition 1 A lattice L is modular if for every x y z E L with x S 2 l xVy2xyz It is upper semimodular if for every x y E L 2 xylty gtxltxy Last time we showed that modular gt semimodular Lemma 1 Suppose L is semimodular and let x y 2 E L fag lt y then either sz sz or 95 V2 lt sz Proof Let in as 2 Note that x S w 3 y Therefore either 11 x or w y o lfwythenx22y Soszyxzyz o lfwx then szyxlty Therefore szltxzyyz D Theorem 2 L is semimodular if and only if it is ranked with a rank function 7 satisfying 3 we v y m M s me my my e L Proof Suppose that L is a ranked lattice with rank function 7 satisfying If x y lt y then x y gt as otherwise as 2 y and x y On the other hand Ty Tx y 1 so by 3 T06 V y T06 S My N96 A y 1 which implies that in fact as y gt x The hard direction is showing that a semimodular lattice has such a rank function First observe that if L is semimodular then 4 xyltxygtxyltxy Denote by CL the maximum length of a chain in L We will show that L is ranked by induction on CL Base case If CL 0 or CL 1 then this is trivial Inductive step Suppose that CL n 2 2 Assume by induction that every semimodular lattice with no chain of length CL has a rank function satisfying First we show that L is ranked LetOacoltaclltlt9cn1lt9cn 1beachainofmaximumlength Let0y0lty1ltltym1ltym 1 be any maximalT chain in L We wish to show that m n Let L 951 and L 31 By induction these sublattices are both ranked Moreover cL n 7 1 If an yl then we are done by induction since the interval L 951 is a lattice and cL n 7 1 On the other hand if x1 y yl then let 21 x1 yl By 4 21 covers both 951 and y1 Let 2122 1 be a maximal chain in L thus in L O L Remember that the length of a chain is the number of minimal relations in it which is one less than its cardinality as a subset of L So for example C n n not n 1 lThe terms maximum and maximal are not synonymous Maximum means of greatest possible cardinality while maximal means not contained in any other such object In general maximum is a stronger condition than maximal Since L is ranked and 2 gt 951 the chain 21 i i i T has length n7 2 So the chain y121u T has length n71 On the other handAL is ranked and y1y2 i i l is a maximal chain so it also has length n 7 1 Therefore the chain 0y1 i i 1 has length n as desired Second we show that the rank function r of L satis es Let x y E L and take a maximal chain as y c0 lt c1 lt lt 7171 lt cn 95 Note that n Mac 7 rx Then we have a chain ycoVySC1Vy SCnVyxVy By Lemma 1 each 3 in this chain is either an equality or a covering relation Therefore the distinct elements cl y form a maximal chain from y to x y whose length must be S n Hence rx y 7 ry S n Mac 7 rx y and so N96 V y N96 A y S n T06 My D The same argument shows that L is lower semimodular if and only if it is ranked with a rank function satisfying the reverse inequality of 3 Theorem 3 L is modular if and only if it is ranked with a rank function r satisfying 5 we v y was Ag 7 me my my e L Proof If L is modular then it is both upper and lower semimodular so the conclusion follows by Theorem 2 On the other hand suppose that L has rank function r satisfying Let x S 2 E L We already know that x y 2 S x y 2 On the other hand rx rx ryz7rxyz me my To 7 my v z 7 was A y A z 2 me my m2 7 we v yv 2 7 we Ag xVyrzrIV2Vz TWVWM implying D 7 Geometric Lattices Recall that a lattice is atomic if every element is the join of atoms De nition 2 A lattice is geometric if it is upper semimodular and atomic The term geometric comes from the following construction Let E be a nite set of nonzero vectors in a vector space V Let LE W0 E l W Q V is a vector subspace which is a poset under inclusion In fact LE is a geometric lattice homework problem lts atoms are the singleton sets l s E E and its rank function is 7 Z dimZgt where Z denotes the linear span of the vectors in Z A closely related construction is the lattice La E W O E l W Q V is an af ne subspace An af ne subspace of V is a translate of a vector subspace for example a line or plane not necessarily containing the origin In fact any lattice of the form La E can be expressed in the form LE where E is a certain point set constructed from E homework problem However the rank of Z E La E is one more than the dimension of its af ne span making it more convenient to picture geometric lattices of rank 3 Example 1 Let E be the point con guration on the left below Then La E is the lattice on the right which in this case is modular abcd 039 abc ad bd cd 39 W a b c d o o o a b c l
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