Generating ALl Modular Lattices Of A Given Size
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Transcript of Generating ALl Modular Lattices Of A Given Size
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8/12/2019 Generating ALl Modular Lattices Of A Given Size
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2n3
n
n= 11 n= 12
n= 24
n= 22
n= 18
n= 18 n= 19
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Algebra univers.
L
a c a (b c) = (a b) c a,b,c L.
a b a b
L a, b L
a b a, b a, b a b.
L a, b L
a, b a b a b a,b.
L L
C P CD P D
P C= D
L
C D |C|= |D|
L A L \ {0}
a b {0} A a, b A A= {b| b a a A}
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n {0, 1, 2, . . . , n 1}
0 1
A n L LA n+ 1
n L A
LA
A L \{0} n L
L (n+ 1) LA n
A
w(L) = (w2(L), . . . , wn1(L)) n L wi(L) =
ij2j
n L M w(L)
w(M) i n 1 wi(L) k = |Db|.
Da Db M
n L A L
a, b A LA
LA n n a n b a a b
b a b LA
M LA a, b M
L n k
a, b levk2(L) L
a, b a, b
a
b a
b
a a
b
b
a, b
a, b
levk(L) levk1(L) a, b levk2(L) M L
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Algebra univers.
levk1(L)
L
A
A levk1(L) A levk(L) A levk(L) levk1(L) x, y A x y
L
A
A levk(L) levk1(L) x, y levk1(L), x , y x y x y x, y
n n 1
A levk1(L) A= levk(L)
n L A levk(L)
n+ 1 LA
A levk(L) b levk(L) bA a A
a, b levk(L) Ca Cb a b
k
LA n a n b Da :={0, n} Ca Db := {0} Cb
|Da|= 2 +k >1 + k= |Db|,
LA
LA
O(n!)
L
g L {g(a)| a A}= B
A, B
L A LA
L
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n L cL {0, . . . , n 1}
n M L=M
{(cL(x), cL(y))| x y
L}= {(cM(x), cM(y))| x y
M},
(n+ 1) LA L A
cLA LA
n := c1LA(n) n m(LA) = {LA, a | f(a) = n, f Aut(LA)} LA, a a LA
L= LA, n L m(LA) LA
n
L
LA
m m L
m < n
LAC:={A| A L
O Aut(L) LAC
A O
c:= LA
n := c1(n) f(n) = n fAut(LA)
m+ 1 LA
m= n L
L
L
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Algebra univers.
L
mv(n)
m(n) =n
k=2
mv(k) m(n k+ 1), n 2
levk(L) L n LA
LA
|levk(L)|= 1
n
L
n 1
L{n1}
levk(LA) = {n} levk1(LA) = {n 1}
n n 1 {n} {n 1}
M LA
m M m LA m = 0 m n 1
a L \ {0} n 1 a M
levk1(L)
mn n
n 2n3 mn
L3
n L
L := LA
A= {x L | x 0}
L :=L{b}
b b n 1
n L
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n = 8
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Algebra univers.
n
L3
L L
L L
L
n n1 L b
b n n 1
n
L L L
L b
n 1
L3 M2
0 n n 1 b
k Mk+1 0 b
n k n
b 0
L
L L
L L
L
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Mk
M L M L=M
L = M L M
L M L=M
L =M k +1
k
k
L M
L = M
n= 3 233 = 1 L3 2n3
n
2n+13 n+1
ln dn
n
(22/4)(n2)
3/2+o((n2)3/2) < ln < 6.11343(n2)3/2
1.81n4 < dn < 2.46n1.
n >7
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Algebra univers.
2
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