I discovered experimentally that a certain finite poset (sorry, I cannot give its definition here) seems to be in fact a (non-distributive, non-graded) lattice. The covering relations are reasonably simple, but it seems not so easy to find out whether one element is smaller than the other, let alone find the meet (or join) of two elements. However, it is (relatively) easy to see that the poset has a minimum and a maximum.


I wonder whether there are standard techniques for proving that a poset is a lattice, that do not need knowledge about how the meet of two elements looks like. (In fact, any example would be very helpful.)


Some more hints:


1. I don't see a way to embed the poset in a larger lattice...
2. the poset is (in general) not self-dual, but the dual poset is itself a member of the set of posets I am looking at.
3. to get an idea, [here](http://service.ifam.uni-hannover.de/%7Erubey/poset.pdf) ([Wayback Machine](https://web.archive.org/web/20150512115725/http://service.ifam.uni-hannover.de/%7Erubey/poset.pdf)) is a picture of one example (produced by sage-combinat and dot2tex)



Proving that a poset is a lattice
I have often found the following lemma of Björner, Eidelman and Ziegler to be useful:



> 
> Let $P$ be a bounded poset of finite rank such that, for any $x$ and $y$ in $P$, if $x$ and $y$ both cover an element $z$, then the join $x \vee y$ exists. Then $P$ is a lattice.
> 
> 
> 


See Lemma 2.1 "[Hyperplane arrangements with a lattice of regions](https://mathscinet.ams.org/mathscinet-getitem?mr=1036875)", *Discrete and Computational Geometry*, Volume 5, Number 1, 263--288.


The hypothesis that $P$ is bounded means that $P$ has a minimal and maximal element. In fact, this hypothesis is slightly stronger than necessary: One can show that if $P$ has a minimal element and the other hypotheses hold then $P$ has a maximal element.


