Revisions to Existence of a non-Eulerian atomistic lattice with this property on the Möbius function

data update (form 12 to 13 nodes)

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edited Oct 14, 2017 at 4:23

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Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

Remark: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any Eulerian lattice.
Proof: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \mapsto |a|$ the rank function. The result is immediate. $\square$

Question 2: Is there a non-Eulerian lattice with the above property on the Möbius function?
Yes, see the answer of John Machacek.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 12$$|L| \le 13$, as checked by the following Sage program (using these lists of Martin Malandro):

from itertools import product
def relationtest(L,n):
    for l in L:
        P=Poset((range(n),l))
        b = P.bottom()
        t = P.top()
        if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
            L=LatticePoset(P)
            if L.is_atomic():
                if not L.is_graded():
                    print(P.cover_relations())
                if L.is_graded():
                    for x, y in product(P, P):
                        if P.compare_elements(x,y)==-1:
                            if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
                                print(P.cover_relations())
                                break

Are the small atomistic lattices listed somewhere?

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

Remark: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any Eulerian lattice.
Proof: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \mapsto |a|$ the rank function. The result is immediate. $\square$

Question 2: Is there a non-Eulerian lattice with the above property on the Möbius function?
Yes, see the answer of John Machacek.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 12$, as checked by the following Sage program (using these lists of Martin Malandro):

from itertools import product
def relationtest(L,n):
    for l in L:
        P=Poset((range(n),l))
        b = P.bottom()
        t = P.top()
        if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
            L=LatticePoset(P)
            if L.is_atomic():
                if not L.is_graded():
                    print(P.cover_relations())
                if L.is_graded():
                    for x, y in product(P, P):
                        if P.compare_elements(x,y)==-1:
                            if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
                                print(P.cover_relations())
                                break

Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.

Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},a)=\mu(\hat{0},\hat{1})\mu(a,\hat{1}), \ \forall a \in L$$ It follows that $\mu(\hat{0},\hat{1}) = \pm1$.

Remark: It is satisfied by any boolean lattice, more generally by the face lattice of any convex polytope (as suggested by John Shareshian), and more generally by any Eulerian lattice.
Proof: An Eulerian lattice is a graded lattice $L$ such that for any $a,b \in L$ with $a \le b$, we have $\mu(a,b)= (-1)^{|b|-|a|} $, with $a \mapsto |a|$ the rank function. The result is immediate. $\square$

Question 2: Is there a non-Eulerian lattice with the above property on the Möbius function?
Yes, see the answer of John Machacek.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 13$, as checked by the following Sage program (using these lists of Martin Malandro):

from itertools import product
def relationtest(L,n):
    for l in L:
        P=Poset((range(n),l))
        b = P.bottom()
        t = P.top()
        if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
            L=LatticePoset(P)
            if L.is_atomic():
                if not L.is_graded():
                    print(P.cover_relations())
                if L.is_graded():
                    for x, y in product(P, P):
                        if P.compare_elements(x,y)==-1:
                            if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
                                print(P.cover_relations())
                                break

Are the small atomistic lattices listed somewhere?

data update (form 9 to 12 nodes) + link

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edited Oct 13, 2017 at 3:14

Sebastien Palcoux

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186

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 9$$|L| \le 12$, as checked by the following Sage program (using these lists of Martin Malandro):

sage: n=0
sage: forfrom Pitertools inimport Posets():product
....:    def N=P.cardinalityrelationtest(L,n)
....:     if N>n:
....:      for l in n=N
....L:  
        printP=Poset((range(n)
....:     if P.is_lattice(,l):)
....:         b = P.bottom()
....:         t = P.top()
....:         if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
....:             L=LatticePoset(P)
....:             if L.is_atomic():
....:                 if not L.is_graded():
....:                     print(P.cover_relations())
....:                 if L.is_graded():
....:                     for x in P:
....:                         for, y in product(P, P):
....:                             if P.compare_elements(x,y)==-1:
....:                                 if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
....:                                     print(P.cover_relations())
                                break

Remark: This program is slow because it computes on the finite posets instead of the finite lattices. Any improvement are welcome.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 9$, as checked by the following Sage program:

sage: n=0
sage: for P in Posets():
....:     N=P.cardinality()
....:     if N>n:
....:         n=N
....:         print(n)
....:     if P.is_lattice():
....:         b = P.bottom()
....:         t = P.top()
....:         if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
....:             L=LatticePoset(P)
....:             if L.is_atomic():
....:                 if not L.is_graded():
....:                     print(P.cover_relations())
....:                 if L.is_graded():
....:                     for x in P:
....:                         for y in P:
....:                             if P.compare_elements(x,y)==-1:
....:                                 if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
....:                                     print(P.cover_relations())

Remark: This program is slow because it computes on the finite posets instead of the finite lattices. Any improvement are welcome.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 12$, as checked by the following Sage program (using these lists of Martin Malandro):

from itertools import product
def relationtest(L,n):
    for l in L: 
        P=Poset((range(n),l))
        b = P.bottom()
        t = P.top()
        if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
            L=LatticePoset(P)
            if L.is_atomic():
                if not L.is_graded():
                    print(P.cover_relations())
                if L.is_graded():
                    for x, y in product(P, P):
                        if P.compare_elements(x,y)==-1:
                            if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
                                print(P.cover_relations())
                                break

Sage checking

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edited Oct 11, 2017 at 19:37

Sebastien Palcoux

27k
5
74
186

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 9$, as checked by the following Sage program:

sage: n=0
sage: for P in Posets():
....:     N=P.cardinality()
....:     if N>n:
....:         n=N
....:         print(n)
....:     if P.is_lattice():
....:         b = P.bottom()
....:         t = P.top()
....:         if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
....:             L=LatticePoset(P)
....:             if L.is_atomic():
....:                 if not L.is_graded():
....:                     print(P.cover_relations())
....:                 if L.is_graded():
....:                     for x in P:
....:                         for y in P:
....:                             if P.compare_elements(x,y)==-1:
....:                                 if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
....:                                     print(P.cover_relations())

Remark: This program is slow because it computes on the finite posets instead of the finite lattices. Any improvement are welcome.

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?

As suggested by Sam Hopkins:
Question 3: Is there a non-Eulerian atomistic lattice with the above property on the Möbius function?
No for $|L| \le 9$, as checked by the following Sage program:

sage: n=0
sage: for P in Posets():
....:     N=P.cardinality()
....:     if N>n:
....:         n=N
....:         print(n)
....:     if P.is_lattice():
....:         b = P.bottom()
....:         t = P.top()
....:         if all(P.moebius_function(b,x) == P.moebius_function(b,t)*P.moebius_function(x,t) for x in P):
....:             L=LatticePoset(P)
....:             if L.is_atomic():
....:                 if not L.is_graded():
....:                     print(P.cover_relations())
....:                 if L.is_graded():
....:                     for x in P:
....:                         for y in P:
....:                             if P.compare_elements(x,y)==-1:
....:                                 if not P.moebius_function(x,y)==(-1)^(P.rank(y)-P.rank(x)):
....:                                     print(P.cover_relations())

Remark: This program is slow because it computes on the finite posets instead of the finite lattices. Any improvement are welcome.

minor edit

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edited Sep 29, 2017 at 10:37

Sebastien Palcoux

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title edit

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edited Sep 14, 2017 at 22:46

Sebastien Palcoux

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New sub-question with the atomistic assumption, as suggested by Sam Hopkins.

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edited Sep 12, 2017 at 20:31

Sebastien Palcoux

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Eulerian OK

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edited Sep 12, 2017 at 15:01

Sebastien Palcoux

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title edit

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edited Sep 12, 2017 at 2:00

Sebastien Palcoux

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improvement

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edited Sep 12, 2017 at 0:25

Sebastien Palcoux

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asked Sep 11, 2017 at 22:40

Sebastien Palcoux

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