Araara d - Alocu

Concatenation already gives us addition. With the Fraktur alphabet, we can also describe how to choose and combine two arguments. Let’s start by choosing just one.

Choosing an argument

The operator 𝔞 returns the first argument. The operator 𝔟 returns the second. These are the projection functions, usually written π₀ and π₁:

𝔞 = π₀ : Akurto × Akurto → Akurto
(ae, bp)  ↦  ae

𝔟 = π₁ : Akurto × Akurto → Akurto
(ae, bp)  ↦  bp

Here, we use Apar with Akurto as its ambient type: ae and bp name our two parameters. They stand for whichever two Akurto terms we supply. The first projection keeps the first; the second keeps the second.

Putting them together

Now write 𝔞𝔟. Both operators receive the same two arguments, and we concatenate their answers in the order the operators are written. So we take the first argument, then the second:

𝔞𝔟 ae bp  =  ae bp  =  aebp

We will call sequences of Fraktur operators Alocu, after the Globasa word alocu, meaning ‘get changed’. Each Alocu term is understood as a function on two parameters. For now, we are using Akurto as their ambient type.

𝔞𝔟 : Akurto × Akurto → Akurto
(ae, bp)  ↦  aebp

We can make the joining operation visible by writing 𝔞 ⨾ 𝔟 for 𝔞𝔟. Here, ⨾ means pointwise concatenation: give both functions the same pair, then join their answers.

We have already been using this implicitly! Whenever we separated expressions with spaces, we applied 𝔞𝔟 to concatenate them in order, repeating this until everything was concatenated.

Reversing the order

To take the second argument first, use 𝔟𝔞:

𝔟𝔞 ae bp  =  bp ae  =  bpae

The resulting word has a different order, but the same Akurto value: addition is commutative.

Taking additive inverses

We can also take additive inverses. The operator 𝔢 returns the inverse of the first argument; 𝔭 returns the inverse of the second:

𝔢 : Akurto × Akurto → Akurto
(ae, bp)  ↦  ea

𝔭 : Akurto × Akurto → Akurto
(ae, bp)  ↦  pb

So 𝔞𝔭 keeps the first argument and appends the inverse of the second:

𝔞𝔭 ae bp  =  ae − bp  =  ae pb  =  aepb

Subtraction has become concatenation again! Each Fraktur letter tells us what to return, and their sequence tells us how to put the answers together.

Taking multiplicative inverses

The uppercase Fraktur letters take multiplicative inverses. The operator 𝔄 returns the reciprocal of the first argument; 𝔅 returns the reciprocal of the second. With our Apar parameters over Akurto, their answers are in Asoho:

𝔄 : Akurto × Akurto → Asoho
(ae, bp)  ↦  AE

𝔅 : Akurto × Akurto → Asoho
(ae, bp)  ↦  BP

The operators 𝔈 and 𝔓 take both inverses, returning the negative reciprocal:

𝔈 : Akurto × Akurto → Asoho
(ae, bp)  ↦  EA

𝔓 : Akurto × Akurto → Asoho
(ae, bp)  ↦  PB

The same rules work with Asoho arguments. Using Apar over Asoho, our parameters are AE and BP. Lowercase Fraktur chooses an argument or takes its additive inverse, staying in Asoho; uppercase Fraktur takes its reciprocal, returning to Akurto. For example:

𝔢 AE BP  =  EA
𝔄 AE BP  =  ae

In either ambient type, we concatenate the operators’ answers just as before.

Exercises

1. Apply 𝔞 and 𝔟 to the arguments ca and b. Which argument does each return?
2. Write out 𝔞𝔟 ca b and 𝔟𝔞 ca b. Do they produce the same word? Do they have the same Akurto value?
3. Write out 𝔞𝔭 ca b. Then check your result by evaluating it as an integer.
4. Using Apar over Asoho, write out 𝔈𝔅 AE BP. Which inverses does each operator take?