ClosureSeparation

Computation of OEIS sequences A334254 and A334255 for n=6 and checking their values for n=3,4,5.

• A334254 (https://oeis.org/A334254) represents the number of closure operators on a set of n elements which satisfy the T_1 separation axiom, i.e. all singleton sets {x} are closed. For n>1, this property guarantees that the empty set is closed, i.e. the closure operators are strict.

• A334255 (https://oeis.org/A334255) represents Number of strict closure operators on a set of n elements which satisfy the T_1 separation axiom.

Outline of A334254_and_A334255.ipynb

1. The main code with the recursive procedure to generate all closure systems represented as formal contexts and to select those satisfying the T_1 separation axiom.

2. Counting A334254 for n=6 with multiprocessing parallelisation.

3. Checking the values A334254 for n=3, 4, 5 with the code from Section 1.

Outline of IneqA334254.ipynb

1. The main code with the recursive procedure to generate all closure systems represented as formal contexts and to select those satisfying the T_1 separation axiom.

2. Counting A334254 for n=2,3,4,5.

3. Counting nonisomorphic closure systems from those of A334254 for n=2,3,4,5.

References

• Bernhard Ganter, Rudolf Wille: Formal Concept Analysis - Mathematical Foundations. Springer 1999, ISBN 978-3-540-62771-5, pp. I-X, 1-284

• Dmitry I. Ignatov: On the Cryptomorphism between Davis' Subset Lattices, Atomic Lattices, and Closure Systems under T1 Separation Axiom, https://arxiv.org/abs/2209.12256, 2022

For any usage the acknowledgement of this repository and the paper is the necessary condition.