1.6 Structures

Editorial note

A semigroup isomorphism must be bijective. The remark about Planck-scale discreteness is the author’s physical motivation, not an established mathematical consequence.

Physical theories have two aspects, the static and the dynamic. The former refers to the general background in which the theory is set. For example, special relativity takes place in Minkowski space while quantum mechanics is set in Hilbert space. These mathematical structures are, to use J. A. Wheeler’s term, the ‘arena’ in which a physical system evolves; they are of two basic kinds, algebraic and geometric.

In very broad terms, an algebraic structure is a set of binary relations imposed on a set, and ‘algebra’ consists of those results that can be achieved by formal manipulations using the rules of the given relations. By contrast, a geometric structure is postulated as a set of relations on the power set ofa set. The objects in a geometric structure can in some sense be ‘visualized’ as opposed to being formally manipulated. Although mathematicians frequently divide themselves into ‘algebraists’ and ‘geometers’, these two kinds of structure interrelate in all kinds of interesting ways, and the distinction is generally difficult to maintain.

Algebraic structures

A (binary) law of composition on a set S is a binary map

$$\phi :S\times S\to S.$$

For any pair of elements $a,b\in S$ there thus exists a new element $\phi (a,b)\in S$ called their product. The product is often simply denoted by ab, while at other times symbols such as $a\cdot b,a\circ b,a+b,a\times b,a\wedge b,[a,b].$ , etc. may be used, depending on the context.

Most algebraic structures consist ofa set S together with one or more laws ofcomposition defined on S. Sometimes more than one set is involved and the law of composition may take a form such as $\phi :S\times T\to S$ A typical example is the case of a vector space, where there are two sets involved consisting of vectors and scalars respectively, and the law of composition is scalar multiplication (see Chapter 3). In principle we could allow laws of composition that are n-ary maps $(n>2)$ , but such laws can always be thought ofas families of binary maps. For example, a ternary map $\varphi :S^3\to S$ is equivalent to an indexed family of binary maps $\{{\varphi }_a|a\in S\}$ where ${\varphi }_a:S^2\to S$ is defined by ${\varphi }_a(b,c)=\varphi (a,b,c)$

A law of composition is said to be commutative if $ab=ba$ . This is always assumed to be true for a composition denoted by the symbol $+$; that is, $a+b=b+a$ . The law of composition is associative if $a(bc)=(ab)c$ . This is true, for example, of matrix multiplication or functional composition $f\circ (g\circ h)=(f\circ g)\circ h$ , but is not true of vector product $\mathbf{a}\times \mathbf{b}$ in ordinary three-dimensional vector calculus,

$$\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=(\mathbf{a}.\mathbf{c})\mathbf{b}-(\mathbf{a}.\mathbf{b})\mathbf{c}\ne (\mathbf{a}\times \mathbf{b})\times \mathbf{c}.$$

Example 1.12

A semigroup is a set S with an associative law of composition defined on it. It is said to have an identity element if there exists an element $e\in S$ such that

$$ea=ae=a,\forall a\in S.$$

Semigroups are one of the simplest possible examples of an algebraic structure. The theory of semigroups is not particularly rich, and there is little written on their general theory, bu particular examples have proved interesting.

(1) The positive integers N form a commutative semigroup under the operation of addition. If the number 0 is adjoined to this set it becomes a semigroup with identity $e=0;$ 1 denoted Nˆ .

(2) A map $f:S\to S$ ofa set S into itselfis frequently called a discrete dynamical system. The successive iterates of the function $f$ , namely $F=\{f,f^2,...,f^n=f\circ (f^{n-1}),...\}$ form a commutative semigroup with functional iteration as the law of composition. If we include the identity map and set $f^0={\mathrm{id}}_S$ , the semigroup is called the evolution semigroup generated by the function $f,$ denoted $E_f$

The map $\varphi :\hat{\mathbb{N}}\to E_f$ defined by $\varphi (n)=f^n$ preserves semigroup products,

$$\varphi (n+m)=f^{n+m}=f^n\circ f^m.$$

Such a product-preserving map between two semigroups is called a homomorphism. If the homomorphism is a bijective map it is called a semigroup isomorphism. Two semi groups that have an isomorphism between them are called isomorphic; to all intents and purposes they have the same semigroup structure. The map $\varphi$ defined above need not be an isomorphism. For example on the set $S=\mathbb{R}-\{2\}$ , the real numbers excluding the number 2, define the function $f:S\to S$ by

$$f(x)=\frac{2x-3}{x-2}.$$

Simple algebra reveals that $f(f(x))=x$ , so that $f^2={\mathrm{id}}_S$ . In this case $E_f$ is isomorphic with the residue class of integers modulo 2, defined in Example 1.3.

(3) All of mathematics can be expressed as a semigroup. For example, set theory is made up offinite strings of symbols such as $\{\dots |\dots \}$ , and, not, , , etc. and a countable collection of symbols for variables and constants, which may be denoted $x_1,x_2,\dots$ . Given two strings ${\sigma }_1$ and ${\sigma }_2$ made up of these symbols, it is possible to construct a new string ${\sigma }_1{\sigma }_2$ , formed by concatenating the strings. The set of all possible such strings is a semigroup, where ‘product is defined as string concatenation. Of course only some strings are logically meaningful, and are said to be well-formed. The rules for a well-formed string are straightforward to list, as are the rules for ‘universally valid statements’ and the rules ofinference. Gödel’s famous incompleteness theorem states that if we include statements of ordinary arithmetic in the semigroup then there are propositions P such that neither P nor its negation, not P, can be reached from the axioms by any sequence of logically allowable operations. In a sense, the truth of such statements is unknowable. Whether this remarkable theorem has any bearing on theoretical physics has still to be determined.

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• 1.6 StructuresAs already discussed in Example 1.12, a discrete dynamical system is a set S together with a map … .…
• 1.7 Category theoryShow that the class of all semigroups, Example 1.12, forms a category, where morphisms are defined a…

Geometric structures

In its broadest terms, a geometric structure defines certain classes of subsets of S as in some sense ‘acceptable’, together with rules concerning their intersections and unions. Alternatively, we can think of a geometric structure $\mathcal{G}$ on a set S as consisting of one or more subsets of $2^S$ , satisfying certain properties. In this section we briefly discuss two examples: Euclidean geometry and topology.

Example 1.13

Euclidean geometry concerns points (singletons), straight lines, triangles, circles, etc., all of which are subsets of the plane. There is a ‘visual’ quality of these concepts, even though they are idealizations of the ‘physical’ concepts of points and lines that mus have size or thickness to be visible. The original formulation of plane geometry as set out in Book 1 of Euclid’s Elements would hardly pass muster by today’s criteria as a rigorous axiomatic system. For example, there is considerable confusion between definitions and undefined terms. Historically, however, it is the first systematic approach to an area of mathematics that turns out to be both axiomatic and interesting.

The undefined terms are point, line segment, line, angle, circle and relations such as incidence on, endpoint, length and congruence. Euclid’s five postulates are:

1. Every pair of points are on a unique line segment for which they are end points.

2. Every line segment can be extended to a unique line.

3. For every point A and positive number r there exists a unique circle having A as its centre and radius r, such that the line connecting every other point on the circle to A has length r.

4. All right angles are equal to one another.

5. Playfair’s axiom: given any line $\ell$ and a point A not on $\ell$, there exists a unique line through A that does not intersect $\ell$ – said to be parallel to $\ell$.

The undefined terms can be defined as subsets of some basic set known as the Euclidean plane. Points are singletons, line segments and lines are subsets subject to Axioms 1 and 2, while the relation incidence on is interpreted as the relation of set-membership . An angle would be defined as a set $\{A,{\ell }_1,{\ell }_2\}$ consisting of a point and two lines on which it is incident. Postulates 1–3 and 5 seem fairly straightforward, but what are we to make of Postulate 4? Such inadequacies were tidied up by Hilbert in 1921.

The least ‘obvious’ of Euclid’s axioms is Postulate 5, which is not manifestly independent of the other axioms. The challenge posed by this axiom was met in the nineteenth century by the mathematicians Bolyai (1802–1860), Lobachevsky (1793–1856), Gauss (1777–1855) and Riemann (1826–1866). With their work arose the concept of non-Euclidean geometry, which was eventually to be of crucial importance in Einstein’s theory of gravitation known as general relativity; see Chapter 18. Although often regarded as a product of pure thought, Euclidean geometry was in fact an attempt to classify logically the geometrical relations in the world around us. It can be regarded as one of the earliest exercises in mathematical physics. Einstein’s general theory of relativity carried on this ancient tradition of unifying geometry and physics, a tradition that lives on today in other forms such as gauge theories and string theory.

The discovery ofanalytic geometry by René Descartes (1596–1650) converted Euclidean geometry into algebraic language. The cartesian method is simply to define the Euclidean plane as ${\mathbb{R}}^2=\mathbb{R}\times \mathbb{R}$ with a distance function $d:{\mathbb{R}}^2\times {\mathbb{R}}^2\to \mathbb{R}$ given by the Pythagorean formula

$$d((x,y),(u,v))=\sqrt{(x-u{)}^2+(y-v{)}^2}.\text{\hspace{1em}(1.2)}$$

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• 1.6 StructuresThis may look to be a clean distinction, but it is only intended as a guide, for in reality many str…

This theorem is central to the analytic version of Euclidean geometry – it underpins the whole Euclidean edifice. The generalization of Euclidean geometry to a space of arbitrary dimensions ${\mathbb{R}}^n$ is immediate, by setting

$$d(\mathbf{x},\mathbf{y})=\sqrt{\sum _{i=1}^n(x_i-y_i{)}^2}\text{ where }\mathbf{x}=(x_1,x_2,\dots ,x_n),\text{ etc. }$$

The ramifications of Pythagoras’ theorem have revolutionized twentieth century physics in many ways. For example, Minkowski discovered that Einstein’s special theory of relativity could be represented by a four-dimensional pseudo-Euclidean geometry where time is interpreted as the fourth dimension and a minus sign is introduced into Pythagoras’ law. When gravitation is present, Einstein proposed that Minkowski’s geometry must be ‘curved’, the pseudo-Euclidean structure holding only locally at each point. A complex vector space having a natural generalization of the Pythagorean structure is known as a Hilbert space and forms the basis of quantum mechanics (see Chapters 13 and 14). It is remarkable to think that the two pillars of twentieth century physics, relativity and quantum theory, both have their basis in mathematical structures based on a theorem formulated by an eccentric mathematician over two and a half thousand years ago.

Example 1.14

In Chapter 10 we will meet the concept of a topology on a set S, defined as a subset of $2^S$ whose elements (subsets of S) are called open sets. To qualify as a topology, the open sets must satisfy the following properties:

1. The empty set and the whole space are open sets, $\varnothing \in \mathcal{O}$ and $S\in \mathcal{O}$

2. If $U\in \mathcal{O}$ and $V\in \mathcal{O}$ then $U\cap V\in \mathcal{O}$

3. If $\mathcal{U}\subseteq \mathcal{O}$ then $\bigcup \mathcal{U}\in \mathcal{O}$.

The second axiom says that the intersection of any pair of open sets, and therefore of any finite collection ofopen sets, is open. The third axiom says that an arbitrary, possibly infinite, union of open sets is open. According to our criterion, a topology is clearly a geometrical structure on S.

The basic view presented here is that the key feature distinguishing an algebraic structure from a geometric structure on a set S is

$$\text{ algebraic structure }=\text{ a map }S\times S\to S=\text{ a subset of }{S}^3,$$

while

$$\text{ geometric structure }=\text{ a subset of }{2}^S.$$

This may look to be a clean distinction, but it is only intended as a guide, for in reality many structures exhibit both algebraic and geometric aspects. For example, Euclidean geometry as originally expressed in terms of relations between subsets of the plane such as points, lines and circles is the geometric or ‘visual’ approach. On the other hand, cartesian geometry is the algebraic or analytic approach to plane geometry, in which points are represented as elements of ${\mathbb{R}}^2$ . In the latter approach we have two basic maps: the difference map $-:{\mathbb{R}}^2\times {\mathbb{R}}^2\to {\mathbb{R}}^2$ defined as $(x,y)-(u,v)=(x-u,y-v)$ , and the distance map $d:{\mathbb{R}}^2\times {\mathbb{R}}^2\to \mathbb{R}$ defined by Eq. (1.2). The emphasis on maps places this method much more definitely in the algebraic camp, but the two representations of Euclidean geometry are essentially interchangeable and may indeed be used simultaneously to best understand a problem in plane geometry.

Dynamical systems

The evolution of a system with respect to its algebraic/geometric background invokes what is commonly known as ‘laws of physics’. In most cases, particularly when describing a continuous evolution, these laws are expressed in the form of differential equations. Providing they have a well-posed initial value problem, such equations generally give rise to a unique evolution for the system, wherein lies the predictive power of physics. However, exact solutions of differential equations are only available in some very specific cases, and it is frequently necessary to resort to numerical methods designed for digital computers with the time parameter appearing in discrete packets. Discrete time models can also serve as a useful technique for formulating ‘toy models’ exhibiting features similar to those of a continuum theory, which may be too difficult to prove analytically.

There is an even more fundamental reason for considering discretely evolving systems. We have good reason to believe that on time scales less than the Planck time, given by

$$T_{\text{ Planck }}=\sqrt{\frac{G\hslash }{c^5}},$$

the continuum fabric of space-time is probably invalid and a quantum theory of gravity becomes operative. It is highly likely that differential equations have little or no physical relevance at or below the Planck scale.

As already discussed in Example 1.12, a discrete dynamical system is a set S together with a map $f:S\to S$ . The map $f:S\to S$ is called a discrete dynamical structure on $S.$ The complexities generated by such a simple structure on a single set $s$ can be enormous. A well-known example is the logistic map $f:[0,1]\to [0,1]$ defined by

$$f(x)=Cx(1-x)\text{ where }0<C\le 4,$$

and used to model population growth with limited resources or predator–prey systems in ecology. Successive iterates give rise to the phenomena of chaos and strange attractors – limiting sets having a Cantor-like structure. The details of this and other maps such as the Hénon map [8], $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by

$$f(x,y)=(y+1-a{x}^2,bx)$$

can be found in several books on non-linear phenomena, such as [9].

Discrete dynamical structures are often described on the set of states on a given set $S,$ where a state on S is a function $\varphi :S\to \{0,1\}$ . As each state is the characteristic function of some subset of S (see Example 1.7), the set of states on $s$ can be identified with $2^S$ . A discrete dynamical structure on the set of all states on $S$ is called a cellular automaton on S.

Any discrete dynamical system $(S,f)$ induces a cellular automaton $(2^S,f^{\ast }:2^S\to 2^S)$ , by setting $f^{\ast }:\varphi \mapsto \varphi \circ f$ for any state $\varphi :S\to \{0,1\}$ . This can be pictured in the following way. Every state $\varphi$ on S attaches a 1 or 0 to every point $p$ on $S.$ Assign to $p$ the new value $\varphi (f(p))$ , which is the value 0 or 1 assigned by the original state $\varphi$ to the mapped point $f(p)$ This process is sometimes called apullback – it carries state values ‘backwards’ rather than forwards. We will frequently meet this idea that a mapping operates on functions, states in this case, in the opposite direction to the mapping.

Not all dynamical structures defined on $2^S$ , however, can be obtained in the way just described. For example, if S has n elements, then the number of dynamical systems on $s$ is $n^n$ . However, the number of discrete dynamical structures on $2^S$ is the much larger number $(2^n{)}^{2^n}=2^{n{2}^n}$ . Even for small initial sets this number is huge; for example, for $n=4\mathrm{it}\mathrm{is}{2}^{64}\approx 2\times 10^{19}$ , while for slightly larger n it easily surpasses all numbers normally encountered in physics. One of the most intriguing cellular automata is Conway’s game of life, which exhibits complex behaviour such as the existence of stable structures with the capacity for self-reproducibility, all from three simple rules (see [9, 10]). Graphical versions for personal computers are readily available for experimentation.