Pseudofields, quasifields, near-domains

Here is a piece of evidence which those who think that mathematics is invented rather than discovered will like.

This concerns definitions. I wrote last year about the definitions of a group, a matroid, the real numbers, and primitivity. In each case, there is a concept which is so important that it has shown up in several different contexts and can be defined in several different ways, but it is not too hard to show that the definitions are all equivalent. My story here is of a case where we are not quite so fortunate.

A permutation group G on a set X is sharply 2-transitive if, given two pairs (a1,a2) and (b1,b2) of distinct elements of X, there is a unique element of G which maps the first pair to the second. (Without the word “unique”, this would define 2-transitivity.)

Sharply 2-transitive groups arise in various contexts including model theory, universal algebras (independence algebras), and geometry (projective planes). There is no argument about the definition I have given.

However, three different mathematicians in around 1960 decided to encode a sharply 2-transitive group into an algebraic structure. Jacques Tits in 1952 used a gadget which he called a pseudofield (pseudo-corps in French); George Grätzer in 1963 used a “quasifield” (I use quotes because, in the meantime, the word “quasifield” has acquired a completely different meaning); and Helmut Karzel in 1965 used a near-domain (Fastbereich in German).

To oversimplify slightly, each of these gadgets has two operations, called addition and multiplication, and has some of the properties of a field; the elements of the group are then the affine maps of shape x → ax+b, where a is non-zero. (This is not quite correct; Grätzer used subtraction instead of addition.)

You might naively think that there should be only one natural way to do this, so the structures should all be the same. Perhaps there might be some differences; the maps might act on the left instead of the right, you might have to reverse the order of multiplication or the order of addition (or even subtraction); but basically they should be more or less the same.

With a shock, we discovered yesterday that they are not.

We asked Prover 9 whether the subtraction in a “quasifield” was a loop (in other words, the operation has left and right inverses) – this property is one of the axioms for addition in a near-domain. “No”, it said, and provided an example of a “quasifield” with three elements in which columns for the subtraction table have repeated elements. However, the maps axb turned out to be all the six permutations of the domain, as indeed they should be.

Subsequently Michael Kinyon found a 1972 paper by F. W. Wilke in the Bulletin of the Australian Mathematical Society showing that addition in a pseudofield need not be a loop, but that one could replace “pseudofield” by “strong pseudofield” in which addition is a loop.

We searched the web for anyone who might have looked at this and established rules for translating between the other pairs of structures – in the end without success. We asked (by email) some experienced universal algebraists whether they knew anything about this, and again drew a blank.

There must be a correspondence. Indeed, if you take an element ax+b in one structure, encode it as a permutation, and decode this as a’x+b’ in a different structure, it should be possible to express a’ and b’ in terms of a and b. Perhaps, if this were done, one could establish term equivalence, or some highbrow notion from universal algebra, of the different structures.

If you know how this is done, or if you know of a paper which does this, we would very much like to know!

While I am on the subject, I will mention a piece of fairly recent news which had escaped me; others might be interested in this as well.

Any finite sharply 2-transitive group possesses an abelian regular normal subgroup consisting of the identity and the fixed-point-free elements. (If you are thinking “Oh yes, I see how to prove that from Frobenius’ Theorem”, it is actually much easier than that; it was proved by Jordan and involves little more than simple counting.) In algebraic terms, this means that the near-domain associated with the group is a near-field.

In the case of a sharply 2-transitive group with an abelian regular normal subgroup, it is obvious, at least to a group theorist, how we should proceed. We know that any set on which a group G acts regularly can be identified with the group G, uniquely after the choice of a base point (up to an annoying glitch about left or right action.) So, if G is sharply 2-transitive on X with an abelian regular normal subgroup N, we select an element of X to be 0, then identify X with N as additive group, and the set of non-zero elements of X with the stabiliser of 0 (which acts regularly on the remaining points) as multiplicative group. The near-field axioms now more or less write themselves: addition is an abelian group, multiplication is a group with zero, and the right distributive law comes from the fact that G0 acts as automorphisms of N. In my view, it is slightly perverse to coordinatise the symmetric group of degree 3 by a structure in which the additive group is not the cyclic group of order 3.

It was an open problem for half a century whether the existence of an abelian regular normal subgroup was also true for infinite sharply 2-transitive groups and/or near-domains. This year, Rips, Segev and Tent posted a paper on the arXiv giving a counterexample (indeed, a large family of counterexamples).

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Counting coauthors, 2

I have decided to move one step towards practising what I preach.

The arXiv is now the de facto place of publication of many mathematics papers; Google Scholar recognises it, as do various other sites such as ResearchGate. So shouldn’t co-authors of papers on the arXiv be on my list of coauthors?

The step I have taken is to include a separate section in my list including people with whom I have a joint paper on the arXiv (or other similar repositories) which has not yet appeared in a more traditional form.

This adds 16 to my previous total of 152 coauthors: I have reached the order of the second non-abelian finite simple group

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Evoa

Lapwings

A very different kind of excursion last Sunday, to Evoa, a nature reserve near the Tejo estuary.

Quite a drive from Lisboa: along the highway to Vila Franca, and the iron bridge which was the last bridge across the Tejo before the two new bridges in the city were built; over the bridge; then 12.5km along the kind of road you find in the Australian outback (gravel, potholed and corrugated) to the reserve. Along the road we saw several creatures which we took at first to be huge scorpions; we learned later that they were Louisiana freshwater lobsters, introduced by the Spanish (who apparently liked to eat them) – now, it seems, storks like to eat them.

Lobster

The land here is dead flat, in contrast to the hills on the other side of the river. Farmers grow rice and raise fine black horses and bulls for the bullring. Among the horses we saw many white egrets, with whom they have a symbiotic relationship.

The weather had changed after the cool damp days of my visit so far, and it was quite hot as we went for about a 5km walk around the several lakes of the nature reserve, along tracks with tall reeds towering overhead. From the reeds came a lovely warbling sound, and indeed a reed warbler was sitting on a sign to welcome us to the first hide. (Their Portuguese name means “nightingale”; I have never heard a nightingale, so I don’t know how the song compares.)

We stopped in several hides and saw a variety of waterbirds. Among them were gulls, terns, grey herons, white egrets, mallard and teal ducks, coots, avocets, stilts, and for me the highlight: a spoonbill, holding its bill just under the water and waving its head from side to side to catch small aquatic creatures.

After some delay when the car refused to start, we left along the same track, seeing many more herons and egrets and a couple of hawks.

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Tomar

Tomar

The Portuguese city of Tomar was founded, on the site of a Roman town, by the Knights Templar in the twelfth century. (Many businesses in the town still use the Templar name or their logo, which is also found in the calçada pavements throught the town.) The town lies on the Nabão River (the name means “turnip”, I don’t know why).

It was saved from the wreck of the Templars (they had grown too rich and powerful and were brought down by an alliance of the Pope and the King of France who wanted to get his hands on their treasure) by King Dinis of Portugal, who had the town and its assets transferred to a newly created Order of Christ.

The famous Portuguese prince Henry the Navigator became head of the Order. It may be that some of its wealth funded the journeys of exploration and discovery made by the Portuguese during his reign.

The focus of the town, standing on top of the highest hill, is the castle and convent of the Order of Christ, with a 12th century round chapel said to be modelled on the Temple of Jerusalem. Later, additional building works were carried on in the Manueline period, showing the characteristic armillary sphere and naval ropes in the stonework. This destroyed the austere simplicity of the chapel, but contains a window which has become the symbol of Tomar. Water was supplied by a 6-kilometre aqueduct. The hilltop is now a Unesco World Heritage Site.

Manueline window

We went there last Saturday, primarily to visit the Convent. After wandering down the long dormitory corridors, and through the gardens with sweet-smelling lavender, we went down to the town in search of refreshment. Despite some difficulty finding a parking place, we ended up in a small café.

Every Portuguese town with a convent has at least one speciality sweet cake, and Tomar is no exception: I had a cake which looked like a large slice of mango but tasted like the food of angels. (I recently found out the reason for this. The convents consumed large numbers of eggs – they used the whites to stiffen the nuns’ wimples – and they had to find a use for the yolks.)

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Comics, literacy and self-help

Neill recently spent a week pouring out his passion and commitment to children’s comics on his blog. I thought it made remarkable reading. It is indexed on a side panel, or you can get the series from Readlists (a site I didn’t know about) or as an ePublication from Amazon.

In short, his argument is that a serious decline in children’s literacy has coincided with a catastrophic decline in the availablity of cheap children’s comics; everyone says we should do something about literacy, and comics have a big part to play in any such campaign. But rather than just bewailing the fact that publishers no longer produce, and corner stores and newsagents no longer stock, comics for kids, there is something else we can do: encourage children to produce their own comics, and help them with alternative distribution channels, which might be on the Internet, or comics clubs in schools or local libraries, or whatever.

His own observations of doing workshops for children (and living with one) make it clear that children love reading comics, even (or especially) comics by other children, and with a bit of encouragement they can become engaged in both the creative and the entrepreneurial side of producing them. Moreover, there are books about how to do it that can be put into their hands, and talented people around who run workshops.

Why am I mentioning this here? I think there are some surprising similarities, as well as some differences, with mathematical publishing. If you look at what I wrote, and substitute “mathematicians” and “theorems” for “children” and “comics”, a lot of it makes sense. We love reading theorems, especially if the presentation as well as the content are creatively done; our job is proving new theorems and crafting presentations of them; and some mathematicians at least can become engaged in the distribution side. In particular, I’d like to pay tribute to Herb Wilf and Neil Calkin, who founded the Electronic Journal of Combinatorics.

What are the differences? With us it is not that publishers will not produce the stuff and put it into our hands; but what they ask (for subscription or page charges) is rather higher than “prices realistically within the realm of pocket money while still maybe even leaving enough change for a bag of Skips”, as Neill puts it. So we can’t get them ourselves, and depend on our employers to buy them for us, or to pay the steep entry fee for us to publish in them (which means putting power over what we read and where we publish into the hands of the bureaucrats).

Of course, what keeps us locked in is the insistence of bureaucrats that we publish only in “approved” journals (approved by whom? by them, of course, not by us) or our publication won’t count in evaluations of our research (evaluations by whom?).

Here I would echo Neill and say, we can wring our hands and bewail the situation, or take what steps we can to remedy it. Expertise in running freely available journals is available; we should use it. If those of us who are no longer subject to these stupid bureaucratic rules support this enterprise, eventually “they” will not be able to ignore these outlets.

So I wish the comics creators every success, and at the same time I wish every success to the committed and creative people who are trying to provide us with outlets for our best work. Support them! Help them if you can, and send them good papers.

As a final note, I am delighted when Neill’s world and mine link up in this way.

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9,21,27,45,81,153,…

This is the sequence of degrees of primitive groups which don’t synchronize a map of rank 3, equivalently graphs with clique number and chromatic number 3 having primitive automorphism groups. You could argue that the sequence should start with 3, but this is a trivial case.

The numbers must all be multiples of 3, since the colouring has all colour classes of equal size. But all numbers above are odd, and all except 21 are multiples of 9; I don’t know why this is.

How does the sequence go on? It contains 243, 441, 729, … but I don’t know if there are other terms before these.

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Butterflies

I am in Lisbon working with João Araújo and Wolfram Bentz on synchronization.

We say that a permutation group G on the set {1,…n} synchronizes a non-permutation f from this set to itself if the semigroup generated by G and f contains a constant map. Also, the kernel classes of a map f are the inverse images of the points in the image of f; and f is uniform if all the kernel classes have the same size, and non-uniform otherwise.

A group that synchronizes every non-permutation must be primitive. It is not true that a primitive group synchronizes every non-permutation; there are some very interesting exceptions, and deciding the question is very difficult. The current “big conjecture” on synchronization for primitive permutation groups is due to João, and says:

Conjecture: A primitive group synchronizes every non-uniform map.

I have spoken about this conjecture in several places. We had proved it for maps of rank at most 4 (where the rank is the cardinality of the image) or at least n−2, in a paper published this year in the Journal of Combinatorial Theory Series B (doi: 10.1016/j.jctb.2014.01.006). We are currently hard at work improving the upper bound, and hope to get it down to n−4 or even n−5.

This morning, we discovered that the conjecture is false.

Here is a brief description of the counterexample.

The graph in question is the line graph of the Tutte–Coxeter graph, also known as Tutte’s 8-cage. The Tutte–Coxeter graph is trivalent; its line graph has valency 4, and any edge lies in a unique triangle (the triangles corresponding to the vertices in the original graph), so the closed neighbourhood of a vertex is a butterfly, consisting of two triangles with a common vertex.

Butterfly

As in dynamics, it is the butterfly that causes chaos with a flap of its wings …

Our graph has automorphism group G = PΓL(2,9), and has chromatic number 3; this means that it has a homomorphism onto one of its triangles, this being a (uniform) map of rank 3 not synchronized by the primitive group G.

We used GAP and its share package GRAPE to determine all the independent sets of size 15 in the graph (up to the action of the group G, there are just two of these), and to examine the induced subgraph on the complement of each such set. In both cases, this induced subgraph turns out to be a disjoint union of cycles of even length; so each independent set of size 15 is a colour class in a 3-colouring. For one of these independent sets, it occurs that the induced subgraph on the complement has two components, of sizes 10 and 20. In this case, the original independent set A and the bipartite blocks B and C of the component of size 10 and D and E of the other component are all independent sets, with edges between A and the others, and between B and C and between D and E, and no further edges.

Thus the edges between these five sets can be mapped homomorphically to the butterfly, with A mapping to the central vertex. This gives us a non-uniform map of rank 5 (with kernel classes of sizes 15, 5, 5, 10, 10) which is an endomorphism of the graph, and hence not synchronized by the group G.

Indeed, this butterfly resembles Lorenz’s attractor in one respect. If you wander round the graph and follow your image under the homomorphism, it will move around one wing of the butterfly and then (apparently randomly) switch to the other wing, and so on. Actually, since the wings are of unequal sizes, you will spend most of your time on the larger wing.

Is this an isolated example, or the first butterfly of the summer? (It is not completely isolated; the line graph of the Biggs–Smith graph gives another example with degree 153, where the kernel type is (6,6,45,45,51).)

Butterfly

As with any good counterexample, it opens various new questions. Is it the smallest counterexample? What can we say about the gap between the ranks of the smallest map and the smallest non-uniform map not synchronized by a primitive group? (It can’t be 1; how large can it be?) Does a primitive group of degree n synchronize every map of rank greater than n/2? Can one determine all the primitive groups which fail to synchronize some map of rank 3? And so on …

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