Definitions

Time to generalise from the preceding examples.

There are several good reasons why a choice of definitions is a good thing. First, as several points in the discussion of graphs suggested, different definitions may be adapted for different kinds of uses. There is no doubt that if you study covering projections of graphs you will incline to Tomaz Pisanski’s definition, while if you study maps the permutation definition may be more suitable. The partition definition is a specialisation of the definition of a chamber system, and extends naturally in that direction.

If you are interested in the line graph of a graph, whose vertices are the edges of the original graph, two vertices joined if the corresponding edges intersect, then you probably want a definition where the edges are actual objects, rather than say the “relation” definition.

But there are other aspects too. If a graph is a symmetric irreflexive binary relation, then the natural notion of substructure gives induced subgraphs, while if it is a set of vertices and a set of edges (either incidence structure or having edges as sets of vertices) the natural notion gives arbitrary subgraphs. In each case, to get the other kind of subgraph we have to put on some extra proviso.

This issue arises as well with the definition of a group. If we define a group to have just the operation of multiplication, and define the identity and inverses from this, then the most natural notion of substructure gives us a sub-semigroup (a subset closed under the operation, and clearly associative); to get a subgroup we have to specify that the substructure is a group, and prove that it has the same identity and inverses as in the whole group. An alternative procedure is to define a group to have a binary operation, a constant (the identity), and a unary operation (inverse); then substructures are subgroups, but we have “defined away” the interesting fact that identity and inverses in a group are unique.

The trend in mathematics in the twentieth century was to define mathematical objects axiomatically. Some people felt that a minimal, or irredundant, set of axioms should be used; sometimes this makes the job harder. The following is not quite an axiomatic definition of a group, but clearly inspired by the group axioms: it is the last exercise in Chapter 2 of Ian D. Macdonald’s book The Theory of Groups, and is given three stars (and there are very few three-star exercises in the book). A semigroup is a structure satisfying the first two group axioms (closure and associativity). Macdonald takes as the definition of a group that it should satisfy these two axioms and have a right identity (an element e such that ae = a for all a) and right inverses (for some such e, for every a there exists x such that ax = e). You see already how he is trying to pare the assumptions down to a minimum.

(Incidentally, right identity and left inverses do not suffice to define a group: if we define the operation by ab = a for all a, then every element is a right identity, and if we choose a particular right identity e, then e is the left inverse of every element.)

The exercise is as follows:

Show that in any semigroup S the following conditions are equivalent.

  • There is an element e such that xe = x for all x in S, and for each such element e and for each x in S there is an element x−1 such that x−1x = e.
  • For each x in S there is an element x−1 such that yxx−1 = y for all y in S.

I think that, in general, this minimality makes our job harder. We want identities and inverses in a group to be two-sided; that is how we think about them.

As in many other situations, there is a tension between the creative and the logical aspect of mathematics.

David Eppstein raises an interesting point, which I will return to later; I have already said more than enough here.

In closing I should say that, even though my definition of a group action as a rank 1 independence algebra throws an interesting light on how mathematics works, I do not think it is very practical; I cannot think of a situation where this would make the job of studying group actions easier.

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About Peter Cameron

I count all the things that need to be counted.
This entry was posted in exposition, mathematics and tagged , , , . Bookmark the permalink.

3 Responses to Definitions

  1. I prefer to define a group as a semigroup where the equations ax=b and ya=b can be solved for all a,b. This somehow seems closer to the intuition of why one wants inverses in the first place.

    • So a group is something that is simultaneously a semigroup and a quasigroup. (Two halves make a whole).
      I doubt, though, whether hard-core group theorists will like this definition.

  2. Pingback: Definition and Determination : 10 | Inquiry Into Inquiry

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