Is this a valid example of a poset?

Question

I am trying to understand better the basic concepts of partially ordered sets, and tried my own examples by using modular-arithmetic. For this, I tried to define an algorithm to order integers, here is the idea, my question is after the example:

1- Def. set of positive integers $S=\{s_0,s_1,s_2,s_3,..,s_n\}\ /\ S\in\Bbb N $. Repetition of elements is possible, and $0$ is possible.

2- Move all $0's$ to a new list $S0$. They are not evaluated.

3- For every $s_i\ /\ i\in [0..n]$ element of $S-S0$ do as follows:

3.1- Def. $SL=\{s_0..s_{i}\}$ and $SR=\{s_{i+1}..,s_n\}$

3.1.1- For every $s_k \in SR$ calculate $m_k = s_k\ mod\ s_i$

3.1.2- Reorder $SR$ in increasing order according to $m_k$. If two or more elements have the same value, then keep the original order between them (for that reason, if $s_i$ is $1$ then no changes are applied to $SR$).

4- Move back the $0's$ to $S$: $S = S + S0$

E.g.

$S=\{7,34,12,48,0,7\}$

there is one $0$, so $S0=\{0\}$ and $S-S0=\{7,34,12,48,7\}$

Loop in s_i:

for the first element $7$, after sorting $S=\{7,7,12,34,48\}$

for the second element, again $7$, so no changes happened $S=\{7,7,12,34,48\}$

for the third element $12$, $S=\{7,7,12,48,34\}$

we arrived to the element before the last element, $48$, so no more changes are possible.

finally $S=\{7,7,12,48,34\}+\{0\}=\{7,7,12,48,34,0\}$

The point is: this algorithm would sort the data from left to right providing always for every $s_i$ the element of $S$ at the right of $s_i$ which is the closest number to a factor of $s_i$, so it is possible for instance to calculate the average distance of a generic element of the set to a generic factor of the set: $D_S = avg(s_{i+1} \ mod\ s_i)$. This might be useful to verify the randomness of a set of integers.

According to the Wiki definition: A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other. Such a relation is called a partial order to reflect the fact that not every pair of elements need be related: for some pairs, it may be that neither element precedes the other in the poset.

And applied to the sample $S=\{7,7,12,48,34\}$ above:

1. Partial order of $s_2$: $s_2 \le s_{2+k}$ because

$(s_2\ mod\ s_1) \le (s_3\ mod s_1)$ and

$(s_2\ mod\ s_1) \le (s_4\ mod s_1)$ and

$(s_2\ mod\ s_1) \le (s_5\ mod s_1)$.

2. Partial order of $s_3$: $s_3 \le s_{3+k}$ because

$(s_3\ mod\ s_2) \le (s_4\ mod s_2)$ and

$(s_3\ mod\ s_2) \le (s_5\ mod s_2)$.

3. Partial order of $s_4$: $s_4 \le s_{4+k}$ because

$(s_4\ mod\ s_3) \le (s_5\ mod s_3)$.

Would the generated set be a valid example of a poset?

Answer 1

Correct me if it is not what you intent for.

Let $S = \{7, 7, 14, 7\}$. After your procedure, $S$ remains the same. We have $$ s_2 = 7 \leq s_3 = 14 $$ and $$ s_3 = 14 \leq s_4 = 7 $$ implying $7 \leq 14$ and $14 \leq 7$ but $14 \neq 7$, thus violate the antisymmetry of a poset.