On 07/04/2025 22:34, Kurt Pagani wrote:
I assume you have read https://mathoverflow.net/questions/8970/number- of-valid-topologies-on-a-finite-set-of-n-elements There are many unsolved problems, e.g. https://staff.fnwi.uva.nl/ j.vanmill/papers/papers1990/opit.pdf (digital).So the sensible thing to do here is give up. Not necessarily. From a computational point of view there is certainly a lack of a concise package that covers the essentials, like https://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf (you surely know it).
I have done a quick search online and skimmed some of the papers. Since no one has found a reason why this cannot work I will take some more time and read them more carefully, the paper you mention above look especially useful so thank you for that. My initial thought (which I have not fully thought through) was to handle finite, monster lattices in the same way that FriCAS already handles finite monster groups. So groups involve: 1) start with a set 2) define permutations on the set (generators) 3) compose generators to get all elements 4) represent as strong generators so that we can scale up. So for finite lattices can we: 1) start with a set 2) define generators as maps which remove one element (instead of permutations). 3) compose generators to get lattice (subset of power set) 4) Find a way to represent this more efficiently than sets of sets (is there anything like stabilisers and orbits in this case). I know that Birkhoff's representation theorem says that finite distributive lattices can always be represented by sets of sets but I guess there could be more efficient representations? The papers about enumeration of finite topologies seem to use different representations. For example this one uses transitive digraphs: https://dl.acm.org/doi/pdf/10.1145/363282.363311 and this one uses semi-tensor products of matrices: https://www.mdpi.com/2227-7390/10/7/1143 Its very hard for me to judge which of these approaches would scale up best on FriCAS but my intuition suggests that what works best for groups might work best for lattices. Martin -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To unsubscribe from this group and stop receiving emails from it, send an email to fricas-devel+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/fricas-devel/79a3f39b-92f4-4526-b4bc-bb2c602d07f9%40martinb.com.
