Group Homomorphisms and Exchange Rates

Introduction

The main purpose of this article is to translate currency exchange systems into a formal algebraic framework, where monetary quantities are represented as elements of additive groups and exchange operations are modeled as group homomorphisms.

Before discussing the economic implications, we first recall some group-theory basics to clarify the discussion.

Group

Let G be a non-empty set equipped with a binary operation:

o : G x G -> G.

The pair (G, ◦) is called a group if the following axioms are satisfied:

(1) for all a, b, c ∈ G: ( a ◦ b ) ◦ c = a ◦ ( b ◦ c ) (associativity axiom).

(2) there is e ∈ G such that e ◦ a = a for all a ∈ G (identity axiom).

(3) for every a ∈ G there exists a⁻¹∈ G such that a⁻¹ ◦ a = e (inverse axiom)

Elementary examples of groups are ( Z , + ) ; (R, + ) and ( {0} , +)

Moreover, a group G is called abelian if for all a, b ∈ G a ◦ b = b ◦ a

Homomorphism

Let (G, ◦) and (H, ∗) be two groups. A map ϕ : G → H is called a homomorphism from (G, ◦) to (H, ∗), if :

For all a, b ∈ G ϕ (a ◦ b) = ϕ (a) ∗ ϕ (b)

For example, consider the groups (R, +), (R\{0}, *) and the map ϕ (x)= eˣ . It is immediate to conclude that:

ϕ (a + b) = eᵃ⁺ᵇ = eᵃ · eᵇ = ϕ (a) · ϕ (b)

Further mathematical foundations will be introduced as needed.

Currency Values as Additive Groups

First, let C denote the set of possible monetary quantities expressed in a given currency, such as euros or dollars, endowed with the usual operation of addition.

The pair (C, +) satisfies all the axioms of an abelian group:

Closure: for all x , y ∈ C we have x+y∈ C;

Associativity: for each x, y, z ∈ C, (x+y)+z=x+(y+z);

Identity element: for each x∈ C, there exists 0∈C such that x+0=x ;

Inverse element: for each x∈C, there exists −x ∈C such that x+(−x )=0 ¹

Commutativity: for each x,y ∈C, we have x+y=y+x

Exchange "Functions" as Group Homomorphisms

Consider two currencies C1 and C2, each modeled by the additive group (C,+).

A fixed exchange rate k>0 is represented by the mapping:

 𝑓ₖ :(C,+)⟶(C,+) where  𝑓ₖ(x)=kx, and x denotes the amount of currency C1​ converted into currency C2.

The map 𝑓ₖ ​ is a group homomorphism, since for all x , y ∈ C :

𝑓ₖ(x+y)=k(x+y)=kx+ky=𝑓ₖ(x) + 𝑓ₖ(y)

The homomorphism ​ 𝑓ₖ is a linear transformation that preserves the additive structure of monetary values. Hence, when the exchange rate is fixed, the sum of two amounts before conversion equals the sum of their converted values.

Moreover, fₖ is bijective, as can be easily verified by observing its graph, which is a straight line through the origin.

Because it is both bijective and a homomorphism from C to itself, fₖ is an automorphism.

Automorphisms and Aut(G)

An automorphism is a homomorphism where G = H  and the map ϕ is bijective.

The set of all automorphisms is denoted by

Aut(G)={ f:G→G ∣f is a bijective group homomorphism}

Hence, every f ∈ Aut(G) is bijective and satisfies f(x + y) = f(x) + f(y) for all x, y ∈ R

Proof that if f∈ Aut(G) then f has the form f(x)=kx

Let G = (R, +).

We want to prove that if f ∈ Aut(G), then f(x) = kx for some constant k ≠ 0 (at least for x ∈ Q, and, under certain assumptions, also for x ∈ R)

First of all, consider 𝑓(nx) with n ∈ N: we aim to prove that if 𝑓∈ Aut(G) then

𝑓(nx)= n𝑓(x) (this result will be useful soon)

We proceed by mathematical induction:

For n=0 𝑓(0x)= 𝑓(0)= 𝑓(0)+𝑓(0) so 𝑓(0)=0 = 0𝑓(x)

So, assuming 𝑓(kx)=k𝑓(x) for some k≥ 1 then:

𝑓((k+1)(x))=𝑓(kx+x)=𝑓(kx)+𝑓(x)=k𝑓(x)+𝑓(x)=(k+1)𝑓(x)

By induction, 𝑓(nx)= n𝑓(x) holds for every n∈ N

It's possible to extend this reasoning both to integers and rational numbers:

Let's define r=p\q with p,q ∈ Z

Hence p=qr, so 𝑓(qrx)= 𝑓(px)=p 𝑓(x). But 𝑓(qrx)= q 𝑓(rx) so this implies that

q𝑓(rx)=p 𝑓(x) and 𝑓(rx)= p/q 𝑓(x) . Now, we define k:=𝑓(1), then

𝑓(rx)=𝑓(r)=𝑓(r *1)=r*𝑓(1)=r · k  for every r∈Q

We therefore conclude that on the rational numbers, the additive function behaves as a multiplication by the constant k=f(1).

In an economic framework, the exchange function is assumed to be continuous since small variations in value should lead to proportionally small variations in the converted amount, avoiding discontinuities or jumps. Under this assumption, the additive relation 𝑓(r)=kr is valid for rational amounts and can be extended to all real values.

So, since the rationals are dense in the reals, for any real value x there exists a sequence of rationals rₙ such that rₙ → x. By the continuity of 𝑓,

𝑓(x) = lim ₙ→∞ 𝑓(rₙ) = limₙ→∞ k · rₙ = k ·x.

Therefore, the same proportional relation holds for all real numbers, 𝑓(x) = k·x for every x ∈ R.

Economic Interpretation

The set Aut(G) represents all ideal and reversible exchange transformations that preserve value in the absence of frictions, transaction costs or asymmetries.

In reality, exchange systems are affected by several factors that distort this ideal linearity and symmetry : for example, every exchange operation carries a service or logistical cost, so the linear map 𝑓(r) = k·r no longer provides an accurate description of real transactions.

In this situation, the exchange function takes the form g(x) = k·x − b, where b is a fixed transaction fee that breaks the linear structure of the model:

g ( x + y )= k ( x +y )- b= kx + ky -b= ( kx - b )+ ( ky-b )+b = g(x) + g(y) + b

so g(x+y) > g(x)+g(y)

This implies that a single aggregate transaction yields a higher value than executing the same exchanges separately.

Hence, g no longer preserves additivity and therefore does not belong to Aut(G) : two separate operations are more expensive than one combined exchange due to the fixed fee applied to each transaction individually.

Composition of Exchange Rates as a Multiplicative Group

It's also interesting to note that the composition of exchange rates can also be interpreted as a multiplicative structure on conversion factors between currencies.

Consider a set of currencies C={USD, EUR, JPY,...} . For every pair (X,Y)∈ C x C define a direct exchange rate rₓᵧ>0, meaning “how many units of Y are needed for 1 unit of X”

The composition of exchanges can be seen as the multiplication of the exchange factors.

If one can go from X to Y and then from Y to Z, the composite exchange is:

rₓ𝓏 = rₓᵧ * rᵧ𝓏

However, exchange rates do not form a group, since composition is not always defined.

It's impossible to compose any two exchange rates , but only those where the intermediate currency matches.

So, for example, a composition from dollars to euros and from pounds to yen is not defined, since the currencies do not align.

This is where the concept of a groupoid comes into play.

Groupoid

To introduce the notion of a groupoid, we first recall the definition of a partial function:

A partial function f : X -> Y is a function defined on a non-empty set A subset of X.

A groupoid is a set G equipped with a unary operation g :G→ G and a partial function ∗ :G × G→ G that satisfies the following axiomatic properties for arbitrary a,b,c∈G:

(1) If a∗b and b∗c are defined, then (a∗b)∗c and a∗(b∗c) are defined and equal.

Conversely, if either of these last two expressions is defined, then so is the other and again they are equal.

(2) a⁻¹∗a and a∗a⁻¹ are always defined

(3) If a∗b is defined, then a∗b∗b⁻¹=a and a⁻¹∗a∗b=b.

Exchange rates as a groupoid

Let G=[rₓᵧ | x,y are currencies}, where every element of G is an exchange rate and an operation * : G x G → G ( which is not binary because it's not defined for all pairs of element of G and it exists only for compatible pairs)

The unary operation g : G→G is, in this case, the inverse of each exchange:

g(rₓᵧ)=(rₓᵧ)⁻¹=rᵧₓ 

It is straightforward to verify that this structure defines a groupoid:

(1) If an exchange from USD to EUR, from EUR to JPY, and from JPY to GBP is defined, then the final exchange does not depend on how these operations are grouped.

(2) For every a both a∈ G, a * a⁻¹  and a⁻¹ ∗ a are defined.

r₍eur,usd₎ * r₍usd,eur₎ = r₍eur,eur₎=1)

(3) Every currency has its own “neutral exchange": if we compose a conversion and then reverse the last one, nothing changes:

[r₍eur,usd₎ * r₍usd,jpy₎] *  [r₍usd,jpy₎]⁻¹=r₍eur,usd₎