Dear Professor Cohen,
Thank you for your reply. Okay you are right. Sage tests if a poset is a
meet semi-lattice by the following commands
>>P4=[p for p in Posets(4) if p.is_meet_semilattice()]
>>for p in P4: show(p.plot())
I understood this and thank you.
My aim is different from my example but the fault belongs to me. I could
not explain my problem exactly. I mean what almost Professor Thiery wrote.
Also thank Professor Thiery for his comments.
Anyway, I will try to explain the problem:
We have exactly one meet semi-lattice with 2 elements: say S_2. It is just
a chain. Adding another element to S2 we have two non-isomorphic meet
semi-lattices: say S_3={1,2,3}. It is clear that S_3 woud be a chain OR
1-<2 and 1-<3 in S_3. Here a-<b denotes that a is covered by b. By the same
method we can obtain S_4.
Can I do this recursively? After doing this, can I obtain the Möbius
funciton of S_n from S_{n-1} recursively? Naturally I am not sure that this
is possible.
I want to use the method that I am looking for in the divisibility problem
of gcd and lcm matrices because Divisibility of a lcm matrix by a gcd
matrix on a gcd closed set S_n depends on the structure of S_n, which is a
meet semi-lattice. We have obtained some results on the divisiility problem
and we want to check them with the aid of SAGE.
To be more clear I can give a recent reference:
Jianrong Zhao, Divisibility of power LCM matrices by power GCD matrices on
gcd-closed sets, Linear and Multilinear Algebra. Volume 62, Issue 6, 2014
http://www.tandfonline.com/doi/full/10.1080/03081087.2013.786717#abstract
DOI:10.1080/03081087.2013.786717
best regards
Ercan Altinisik
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