Dmitry I. Ignatov presented his report at NTR company webinar
On November 15, 2022 Dmitry Ignatov - laboratory Head at Laboratory for Models and Methods of Computational Pragmatics, associate Professor at School of Data Analysis and Artificial Intelligence took part in NTR company webinar.
In the talk, the speaker explained how Data Mining and Formal Concept Analysis can help to solve combinatorial problems from Lattice Theory and establish connections between seemingly unrelated algebraic objects.
On November 15, 2022 a webinar by NTR was held where Dmitry Ignatov presented his paper "On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom".
This talk presented how we count set closure systems (also known as Moore families) for the case when all single element sets are closed. In particular, we give the numbers of such strict (empty set included) and non-strict families for the base set of size n=6. The problem of generation of such families on object-attribute or transactional data is well-known in Data Mining as frequent (closed) itemset mining. We also provide the number of such inequivalent Moore families with respect to all permutations of the base set up to n=6. The search in OEIS and existing literature revealed the coincidence of the found numbers with the entry for D. M. Davis' set union lattice (OEIS sequence A235604, up to n=5) and |L_n|, the number of atomic lattices on n atoms, obtained by S. Mapes (up to n=6), respectively. Thus we study all those cases, establish one-to-one correspondences between them via Galois adjunctions and Formal Concept Analysis, and provide our listeners with two enumerative algorithms as well as with the results of these algorithms used for additional tests. Other results include the largest size of intersection free families forn=6 plus our conjecture for n=7, an upper bound for the number of atomic lattices L_n, and some structural properties of L_n based on the theory of extremal lattices.
This material was also presented at DAMDID 2022 conference during an invited talk session.
Original paper
Prof. Davis' paper
FCA Tutorial
Contributions to OEIS:
https://oeis.org/A334254
https://oeis.org/A334255
https://oeis.org/A235604
https://oeis.org/A355517
Presentation