Cameron–Fon-Der-Flaass IBIS theorem

In mathematics, Cameron–Fon-Der-Flaass IBIS theorem arises in the dynamical algebraic combinatorics. The theorem was discovered in 1995 by two mathematicians Peter Cameron and Dima Von-Der-Flaass.{{Cite journal |last1=Patrias |first1=Rebecca |last2=Pechenik |first2=Oliver |date=2020 |title=Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture |url=https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/dynamics-of-plane-partitions-proof-of-the-cameronfonderflaass-conjecture/96E578BF1DCB2C4FECEABDE6336772AA |journal=Forum of Mathematics, Sigma |language=en |volume=8 |pages=62 |doi=10.1017/fms.2020.61 |issn=2050-5094|arxiv=2003.13152 }} The theorem is considered to be a link between group theory and graph theory as it studies redundancy of a group.{{Cite journal |last1=Cameron |first1=P. J |last2=Fon-Der-Flaass |first2=D. G |date=1995-11-01 |title=Bases for permutation groups and matroids |url=https://dx.doi.org/10.1016/0195-6698%2895%2990035-7 |journal=European Journal of Combinatorics |language=en |volume=16 |issue=6 |pages=537–544 |doi=10.1016/0195-6698(95)90035-7 |issn=0195-6698|url-access=subscription }}

Statement

Let $P$ be a permutational group of $\Omega$ ,{{clarify|what’s Omega?|date=July 2024}} then the following are equivalent:

Irredundant bases of $P$ are stored by re-ordering.
The bases of matroid are formed due to the irredundant bases of $P$ .
Every irredundant base of $P$ got the same size.

Cameron–Fon-Der-Flaass IBIS theorem

Statement

References

Further reading