Tag Archives: semigroups

All kinds of mathematics …

Posted on 11/12/2016 by

Please reserve the dates 24-27 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 →

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Rainbows in the plane

Posted on 24/11/2016 by

As usual in Lisbon, I have been working with João Araújo on semigroups. But sometimes research has unusual spin-offs, such as the following curious fact: Fact If the points of a projective plane are coloured with four colours (all of … Continue reading →

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Partition homogeneity

Posted on 11/01/2016 by

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 →

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12160

Posted on 27/10/2015 by

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 k-homogeneous subgroup … Continue reading →

Posted in doing mathematics, exposition | Tagged , , | 3 Comments

Generating singular maps

Posted on 12/04/2015 by

Last week, Max Gadouleau and Alonso Castillo-Ramirez 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 →

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Chains of semigroups

Posted on 27/01/2015 by

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 →

Posted in exposition, open problems, symmetric group | Tagged , , , , , , | 7 Comments

Subsets and partitions

Posted on 24/02/2014 by

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 k-subsets of an n-set can we pack so that no t-subset … Continue reading →

Posted in mathematics, open problems | Tagged , , , | 1 Comment

Algebra in Novi Sad

Posted on 09/06/2013 by

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 →

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A permutation group challenge, 2

Posted on 26/12/2012 by

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 →

Posted in open problems | Tagged , , | 1 Comment

A permutation group challenge

Posted on 19/12/2012 by

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 , , , | 1 Comment