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Tag Archives: semigroups
All kinds of mathematics …
Please reserve the dates 2427 July 2017 in your diary! Next year, I will turn 70. Some good friends (notably João Araújo) are arranging a conference in Lisbon to mark the occasion, and many other good friends have agreed to … Continue reading
Rainbows in the plane
As usual in Lisbon, I have been working with João Araújo on semigroups. But sometimes research has unusual spinoffs, such as the following curious fact: Fact If the points of a projective plane are coloured with four colours (all of … Continue reading
Partition homogeneity
This post discusses a paper by Jorge André, João Araújo and me, which has just been accepted for the Journal of Algebra. There is a slight tangle to the history of this paper, which I want to describe. First a … Continue reading
Posted in exposition, history
Tagged homogeneity, partitions, semigroups, symmetric groups, transitivity
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12160
12160 is an interesting number; but I didn’t know that until last night. As part of the work on semigroups, we are looking at the following problem. Given n and k, with n ≥ 2k, suppose that G is a khomogeneous subgroup … Continue reading
Posted in doing mathematics, exposition
Tagged 4homogeneous, permutation groups, semigroups
3 Comments
Generating singular maps
Last week, Max Gadouleau and Alonso CastilloRamirez visited St Andrews. With James Mitchell, we worked on a problem about generating the semigroup of singular maps on a set (the full transformation semigroup minus the symmetric group). As usual in this … Continue reading
Chains of semigroups
I have written here about the lovely formula for the length of the longest chain of subgroups in the symmetric group Sn: take n, increase it by 50% (rounding up if necessary), subtract the number of ones in the base … Continue reading
Subsets and partitions
There are several packing and covering problems for subsets of a set, which have been worked over by many people. For example, given t, k and n, how many ksubsets of an nset can we pack so that no tsubset … Continue reading
Posted in mathematics, open problems
Tagged primitivity, sections, semigroups, transversals
1 Comment
Algebra in Novi Sad
I have just spent a very wet week in Novi Sad at the fourth Novi Sad Algebraic Conference. Wet both in the sense of the large amount of rain that fell (most afternoons brought lightning, thunder and a heavy downpour) … Continue reading
Posted in events, exposition
Tagged 42, clones, hyperplane arrangements, polymorphisms, semigroups, Sierpinski rank
4 Comments
A permutation group challenge, 2
The result in the preceding post can be formulated as follows: A permutation group of degree n = 2k which is transitive on partitions of shape (k,k) but not on ordered partitions of this shape, has a fixed point and is (k−1)homogeneous … Continue reading
A permutation group challenge
Long ago, in the distant past before the Classification of Finite Simple Groups, Peter Neumann, Jan Saxl and I investigated the class of permutation groups acting on sets of even cardinality n = 2k, with the following interchange property: Any subset of … Continue reading
Posted in exposition, open problems
Tagged CFSG, homogeneity, permutation groups, semigroups
1 Comment