Lattice: Frequently asked questions
Question 1. What is a minimal big lattice?
Answer. A lattice is called big if it is a maximal sublattice of an infinite lattice; it is a minimal big lattice if it is big and has no proper big sublattice. In the paper "Lattices with large minimal extensions" by R. Freese, J. Jezek and J. B. Nation, published in Algebra Universalis 45 (2001), 221-309, a complete list of all minimal big lattices has been found. It contains 145 items. The description can be reduced to a list of 81 lattices, as the remaining 64 ones are their duals. It has been also proved in that paper that a lattice is big if and only if it contains a sublattice isomorphic to one of the 145 lattices.
Question 2. When I input an ordered set, it is redrawn each time I hit a key. Why is the picture so ugly?
Answer. Admittedly, there are much better lattice drawing programs; you can find one due to R. Freese at LatDraw. Here I chose to use just a very simple algorithm, which does not give the best results.
Question 3. When I input the lattice 0a1 0bc1 and want to drag the element c lower to the bottom and the element b higher, I can never get c under b.
Answer. It is for purpose. If you were permitted to drag c under b, it would look like that b is less than c, contrary to your definition.
Question 4. The posets are numbered P0, P1, etc. But the first poset should be P1, the second P2, etc.
Answer. No, the first poset should be P0, the second should be P1, etc.
Question 5. You didn't do a nice job when drawing those 81 minimal big lattices. They could be drawn much better.
Answer. Please redraw them and send me the file at jezek@karlin.mff.cuni.cz.