Notes

Chapter 5: Two Dimensions and Beyond

Section 6: Multiway Systems


Semigroups and groups [and multiway systems]

The multiway systems that I discuss can be viewed as representations for generalized versions of familiar mathematical structures. Semigroups are obtained by requiring that rules come in pairs: with each rule such as "ABB" -> "BA" there must also be the reversed rule "BA" -> "ABB". Such pairs of rules correspond to relations in the semigroup, specifying for example that "ABB" is equivalent to "BA". (The operation in the semigroup is concatenation of strings; "" acts as an identity element, so in fact a monoid is always obtained.) Groups require that not only rules but also symbols come in pairs. Thus, for example, in addition to a symbol A, there must be an inverse symbol a, with the rules "Aa" -> "", "aA" -> "" and their reversals.

In the usual mathematical approach, the objects of greatest interest for many purposes are those collections of sequences that cannot be transformed into each other by any of the rules given. Such collections correspond to distinct elements of the group or semigroup, and in general many different choices of underlying rules may yield the same elements with the same properties. In terms of multiway systems, each of the elements corresponds to a disconnected part of the network formed from all possible sequences.

Given a particular representation of a group or semigroup in terms of rules for a multiway system, an object that is often useful is the so-called Cayley graph—a network where each node is an element of the group, and the connections show what elements are reached by appending each possible symbol to the sequences that represent a given element. The so-called free semigroup has no relations and thus no rules, so that all strings of generators correspond to distinct elements, and the Cayley graph is a tree like the ones shown on page 196. The simplest non-trivial commutative semigroup has rules "AB" -> "BA" and "BA" -> "AB", so that strings of generators with A's and B's in different orders are equivalent and the Cayley graph is a 2D grid.

For some sets of underlying rules, the total number of distinct elements in a group or semigroup is finite. (Compare page 945.) A major mathematical achievement in the 1980s was the complete classification of all possible so-called simple finite groups that in effect have no factors. (For semigroups no such classification has yet been made.) In each case, there are many different choices of rules that yield the same group (and similar Cayley graphs). And it is known that even fairly simple sets of rules can yield large and complicated groups. The icosahedral group A5 defined by the rules x^2 ==y^3 ==(x y)^5 == 1 has 60 elements. But in the most complicated case a dozen rules yield the Monster Group, where the number of elements is

808017424794512875886459904961710757005754368000000000

(See also pages 945 and 1032.)

Following work in the 1980s and 1990s by Mikhael Gromov and others, it is also known that for groups with randomly chosen underlying rules, the Cayley graph is usually either finite, or has a rapidly branching tree-like structure. But there are presumably also marginal cases that exhibit complex behavior analogous to what we saw in the main text. And indeed for example, despite conjectures to the contrary, it was found in the 1980s by Rostislav Grigorchuk that complicated groups could be constructed in which growth intermediate between polynomial and exponential can occur. (Note that different choices of generators can yield Cayley graphs with different local subgraphs; but the overall structure of a sufficiently large graph for a particular group is always the same.)


From Stephen Wolfram: A New Kind of Science [citation]