Group's Homomorphism and Exchange Rates

Introduction

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

Before discussing the economic implications and interpretations, it is necessary to introduce and recall some mathematical knowledge about group theory that will make our discussion clearer.

Group

Let G be a non-empty set and fix a binary operation ◦ : G × G → G.

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

(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 is a⁻¹∈ G such that a⁻¹ ◦ a = e (inverse axiom)

Elementary examples of groups are ( Z , + ) ; (R, + ) ; ( {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. The 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, considering the groups (R, +) and (R\{0}, *) and the map ϕ (x)= eˣ , it's immediate to conclude that:

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

Other mathematical foundations will be explained as needed.

Currency Values as Additive Groups

First of all, let's define C as the set representing the possible monetary quantities expressed in a given currency like euro or dollar with the usual operation of addition.

The pair (C, +) satisfies all the properties 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= kx and x is the amount of currency C1 that will be converted into C2 currency ( equal to 𝑓ₖ )

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, so, being the exchange rate fixed, the sum of two amounts before conversion is equal to the sum of their converted values

Moreover, fₖ is bijective (it can be verified easily by observing its graph—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 (in our case, G = H = C) 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)² 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 want to prove that if 𝑓∈ Aut(G) then

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

Using 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 possibile 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

So we conclude 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 continuous, because 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)=kx is valid for rational amounts 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) represent all the ideal and reversible exchange systems transformations that conserve value without friction, transaction costs or asymmetry.

In reality, exchange systems are influenced by a variety of factors that distort their 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 , as can be shown:

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)

It means that the value obtained by a single transaction is higher than the total value obtained from performing the same exchanges separately.

Hence, g no longer preserves additivity, and thus g is not an element of 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 be understood as a group structure built on the multiplication of 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

As we have seen, a group is a set where the operation is always defined and every element has an inverse, but exchange rates don’t quite behave like that.

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

So, for example, acomposition 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 completely understand the definition of grupoid it's necessary to know what is a partial function:

A partial function 𝑓 : X→Y is a function 𝑓 : A→ Y for some ∅ ≠ A ⊆ X.

A groupoid is a set G with a unary operation g :G→ G ³ and a partial function

∗ :G × G→ G that satisfy 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's define a non empty set 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 as we've seen it's not define for all pairs of element of G and it exist only for compatible pairs)

The domain of the operation is Dom(∗)={(rₓᵧ,  rᵧ𝓏) ∈ G×G | Y=Y}

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

g(rₓᵧ)=(rₓᵧ)⁻¹=rₓᵧ

Moreover, it's easy to verify that G=[rₓᵧ | x,y are currencies} is a grupoid:

(1) If you can exchange USD→EUR then EUR→JPY, then JPY→GBP,

the final exchange is independent of how you group them.

[r₍usd,eur₎ * r₍eur,jpy₎] * r₍jpy,gbp₎ = r₍usd,eur₎ * [r₍eur,jpy₎ * r₍jpy,gbp₎]

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

Notes

1. A positive value (+x) can be seen as a credit or a positive cash flow, while a negative value (−x) indicates a debit or a cash outflow.

2. More precisely, it may be better to consider Aut(G) = H = {𝑓 ₖ : k > 0}, because negative exchange rates have no economic meaning.

3. Any operation with only one input is called a unary operation