Substructures

**Hello, Bonjour, Salut**

## The dytran-threetran technology

We want to show that if a binary relation is trasitive (ab&bc-->ac), it is also "3-transitive",
meaning ab&bc&cd-->ad.

### phases

#### 1. define the dytran

Let M be a set and **D** a part of M × M having the property :
D1: for all x,y in M, xy & yz → xz (2-transitivity)
#### 2. define the threetran and the counterthreetran

Let **T** be the dytran **D** above satisfying an extra property :
D1: for all x,y in M, xy & yz → xz (2-transitivity, as in the dytran )
T1: for all x, y, z, t in M, xy & yz & zt → xt (the extra 3-transitivity)
A dytran with the negation of T1, say CT1, is called counter-threetran.

#### 3. remember the target : all dytrans are threetrans

By construction, all threetrans and all counter-threetrans are dytrans. A dytran is either a trheetran, or a counter-threetran. We want to show that all threetrans are dytrans or, equivalenttlly, that there are no counter -threetrans.
#### 4. define sub-counter-threetrans

Given a subset N of M, **CT** restricted to N is said to be a sub-counter-threetran of **CT**.

#### 5. remark that all sub-counter-threetrans are counter-threetrans

Remark : all sub-counter-threetrans are counter-threetrans ; their sub-dytrans are dytrans.
#### 6. describe the counter-threetran

Suppose by r.a.a. one has listed a counter-threetran. In this hyphotetic example, D1 and CT1 hold. This counter-threetran contains **a**, **b**, **c**, **d** such that **ab**, **bc**, **cd** but not **ad**.
#### 7. build the smaller sub-counter-threetran

**CT'** = **CT** / {**a**, **b**, **c**, **d**}, defined by the constants of CT1, is a sub-counter-threetran. By the remark above, it is also a counter-threetran, i.e. a dytran that is not a threetran.
#### 8. the verification of a small number of sub-dytrans

It suffice to verify by listing that all small dytrans not richer than **{ 1, 2, 3, 4 }** are threetrans.
#### 9. conclude : all dytrans are threetrans

If the hypothetic example were true, one should find a small counter-example, built from the original. The explicit verification of all cases showed that this is not the case. In conclusion, all dytrans are threetrans.
#### 10. explain to the reader what has been proved

We have proved that a transitive relation is also 3-transitive : ab, bc, cd → ad. Normally one shoud aplly twice the D1 axiom to deduce T1. On the other hand, teacher Boole showed that thinking has a mechanical component in its core. Why think when is so mechanical ?

## The mathematic of cycles

I have an example of actually using a cyc.

McKay's proof of Cauchy's theorem (1959, AMM) uses a p-tuples of elements of G

( x1, x2,....,xp ) and x1.x2......xp = 1

Well, this is a cyc ! (cyclic order does not matter) as usual, when species are present it turns spectaculary.

#### Huntington's cyclic order (1914)

Take as dytran a subset of K×K×K together with the axioms.
Take as threetran a dytran together with a theorem, e.g.
T1) AXB & BYC & CZA → XYZ.
the counter threetran would present {a, b, c, x, y, z} such that axb, byc, cza but not xyz (CT1).
Take the sub-counter threetran **CT** /{a, b, c, x, y, z} with the induced structure, that is also a counter-threetran.
Check all counter-threetrans with no more than 6 elements.
there are no small counter-threetrans, hence there are no counter-threetrans.
The theorem must be true if it is true on cyclic orders with no more than 6 elements.
### limits

• The method is limited to the axioms like:

For all x in N, there is x' in (the same N) that dosomething.
The existential function must be mapped on different sets.
example Putnam 1965 A-4 ; applying both rule, it occurs :
for each boy there is a "succesor", b≠g-b'
for each girls there is a "succesor" g-b≠g'