## My research and investigations

### Finite Fields – many Models

The story starts here.
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G: Let the first one introduce himself.

B: George Boole (1815-1864), from England. Did you read my book?
X: Which book?
G: An Investigation of the Laws of Thought of 1854, a classic. I read it just before I logged off.

G: You have another model for GF(2)?
B: Yes, and it isn’t based on numbers.

G: Spell out the details, please.
B: It is based on simple sentences, those you can tell immediately whether it is true or false.

G: This is straight from your book!
B: I also consider simple ways to combine such sentences, only the words and, or, called connectives.

G: That’s two symbols, two operations. Interesting.
B: The connectives can join the simple sentences in 8 possible ways:

Sentence Computations
Or false true
false false true
true true true

And false true
false false false
true false true

X: Exactly how do you compute with sentences?

G: Think logically. I’m here and you’re here: true or false? I’m here or I’m there: true or false?
X: I see. When connected by and, both parts must be true to have true finally. When connected by or, both parts must be false to be false finally.

G: That’s Boole’s Laws of Thoughts!
B: And it gives a model of GF(2) if we use this dictionary:

Boole’s proposed model of GF(2)
Symbol Meaning
0 false
1 true

Operation Meaning
+ logical or
× logical and

X: That’s brilliant! A symbolic model for a symbolic system!

G: Not quite. This model is wrong!
G: I spot the error: in your model: true or true is true. Translate by the dictionary: 1+1=1, which is not correct in GF(2).

B: Oops! How embarrassing! I made a boolean error!
G: That’s the lesson. We need to check carefully to ensure all dictionary translations are perfect.

B: Wait a second. I can fix my error!
G: Don’t make the same mistake twice.

B: No, I won’t. Instead of or, use exclusive or, shortened to xor. I just redo all the computations:

Sentence Computations
Xor false true
false false true
true true false

And false true
false false false
true false true

X: What is xor, or exclusive or?
G: It means either this or that, but not both. Tomorrow either the sun will shine or rain will fall. If tomorrow is foggy or a rainbow appears, my prediction is wrong.

B: And the Boole model of GF(2) is:

Boole model of GF(2)
Symbol Meaning
0 false
1 true

Operation Meaning
+ logical xor
× logical and

G: That’s an impressive quick fix!
G: There are further check … it’s all OK, I declare the Boole model of GF(2) is verified correct.

X: Did you just invent a logical operation to fit into the model?
B: No. It’s all in my book. Using either-or to join sentences is rather common.

E: Euclid of Alexandria (300 BC) from Greece. Excuse me for my dates as I’ve lost count.

X: Which book?
G,G,B: (angry looks) Ssh!

E: My classic text Elements was the bible of geometry for more than 20 centuries, until the rise of modern math.
G: Your method of doing math is still being used in all math textbooks.

E: I try to catch up with modern math, so I’ve been doing a bit of abstract algebra lately.
G: I suppose you have a model of GF(2) based on geometry?

E: Take a non-symmetric alphabet, say the letter E in my name. There are two ways you can play with it: either leave it, or reflect it to ?.
B: See, Euclid is using my xor!

E: The first task, which is doing nothing, I call it identity. The second task, which is flipping over, I call it reverse.
B: I can see that.

E: Given two tasks, you can either do both of them, or just do the first one and ignoring the second one.
B: Another xor. First is diligent, while the second is lazy.

E: There are 8 ways to combine the tasks with the do-both or do-one operations:

Do Both identity reverse
identity identity reverse
reverse reverse identity

Do One identity reverse
identity identity identity
reverse identity reverse

E: This gives a model of GF(2) using the following dictionary:

Euclid’s proposed model of GF(2)
Symbol Meaning
0 identity
1 reverse

Operation Meaning
+ Do -Both
× Do-one

X: Another impressive feat!

G: Look carefully ¡X this model is wrong!
G: If your Do-one means to do the first one, then the Do-one table has an error: assume the verticals denote the first, we should have Do-one(reverse,identity)=reverse, not identity as shown.

E: Oops, I may not be so good in algebra!
B: We do have to check thoroughly,

E: But I think I can fix this ….. Yes, I can. Instead of Do-one meaning to do the first one, let’s change it to do the easy one, assuming the task reverse is harder than the task identity, which is doing nothing.
B: This gives the revised 8 computations:

Do-them-both identity reverse
identity identity reverse
reverse reverse identity

Do -easy-one identity reverse
identity identity identity
reverse identity reverse

E: And with this dictionary we surely have a model of GF(2):

Euclid model of GF(2)
Symbol Meaning

Operation Meaning Name
x+y do x, do y. Do-them-both
x×y do easy(x,y) Do-easy-one

G: Let me do further check …. they’re OK. I declare the Euclid model of GF(2) verified correct.

G: This Euclid model inspire me to have still another model of GF(2).
X: Another model?!

G: Sure. This one is based on numbers, but not modulo 2.
G: Let hear the details.

G: Take the two numbers +1 and 1. Symbolic + is numerically multiply, and symbolic × is take the maximum.
G: This is very interesting. Let me compute:

Numeric Computations
Multiply +1 -1
+1 +1 -1
-1 -1 +1

Maximum +1 -1
+1 +1 +1
-1 +1 -1

B: And use this dictionary for the model of GF(2):

Gauss model of GF(2)
Symbol Meaning
0 +1
1 -1

Operation Meaning
x+y multiply(x,y)
x×y maximum(x,y)

G: Further checks are OK. This is indeed Gauss model of GF(2).

X: How to get this model from Euclid model?
G: When I see that reversing twice give identity, I recall (1)(1)=+1. Euclid used the easier task to pick the identity result in his multiplication, so I try the lesser of 1 and +1. It doesn’t work, but reverting to the bigger of 1 and +1 works.

G: Very good indeed. This inspires me to yet another model of GF(2)!
X: Still more?

G: Yes. Take the symbols 0 and 1 as numbers, but let + to mean difference, and × to mean minimum.
X: What is difference?

G: That’s subtraction without sign: always taking the smaller from the bigger.
X: Oh, I see. Let me do the computations:

Numeric Computations
Difference 0 1
0 0 1
1 1 0

Minimum 0 1
0 0 0
1 0 1

G: To match GF(2), the dictionaries for symbols and operations are:

Galois model of GF(2)
Symbol Meaning
0 Number 0
1 Number 1

Operation Meaning
x+y difference(x,y)
x×y minimum(x,y)

G: That’s excellent.
G: Further checks are OK in this case. This is my Galois model of GF(2).

B: It is thrilling to see so many models of GF(2).
X: Is there a most suitable one among the many choices?

G: That’s a good question. It’s too late. We’ll call it a day.
X: Yes, next time.

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Y: This is way too late.

X: I learn too many models of GF(2) in one day. I’m exhausted.
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