Combinatorics

Beneš Conjecture ★★★

Author(s): Beneš

Let $ E $ be a non-empty finite set. Given a partition $ \bf h $ of $ E $, the stabilizer of $ \bf h $, denoted $ S(\bf h) $, is the group formed by all permutations of $ E $ preserving each block of $ \mathbf h $.

Problem  ($ \star $)   Find a sufficient condition for a sequence of partitions $ {\bf h}_1, \dots, {\bf h}_\ell $ of $ E $ to be complete, i.e. such that the product of their stabilizers $ S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell) $ is equal to the whole symmetric group $ \frak S(E) $ on $ E $. In particular, what about completeness of the sequence $ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $, given a partition $ \bf h $ of $ E $ and a permutation $ \delta $ of $ E $?
Conjecture  (Beneš)   Let $ \bf u $ be a uniform partition of $ E $ and $ \varphi $ be a permutation of $ E $ such that $ \bf u\wedge\varphi(\bf u)=\bf 0 $. Suppose that the set $ \big(\varphi S({\bf u})\big)^{n} $ is transitive, for some integer $ n\ge2 $. Then $$ \frak S(E) = \big(\varphi S({\bf u})\big)^{2n-1}. $$

Keywords:

Posted by Vadim Lioubimov
updated January 3rd, 2010
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Chain-meet-closed sets ★★

Author(s): Porton

Let $ \mathfrak{A} $ is a complete lattice. I will call a filter base a nonempty subset $ T $ of $ \mathfrak{A} $ such that $ \forall a,b\in T\exists c\in T: (c\le a\wedge c\le b) $.

Definition   A subset $ S $ of a complete lattice $ \mathfrak{A} $ is chain-meet-closed iff for every non-empty chain $ T\in\mathscr{P}S $ we have $ \bigcap T\in S $.
Conjecture   A subset $ S $ of a complete lattice $ \mathfrak{A} $ is chain-meet-closed iff for every filter base $ T\in\mathscr{P}S $ we have $ \bigcap T\in S $.

Keywords: chain; complete lattice; filter bases; filters; linear order; total order

Posted by porton
updated December 12th, 2009
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