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May 8, 2015 / porton

About topological structures corresponding to partial order

Intuitively (not in the sense of comparing cardinalities, but in some other sense), the set of natural numbers is less than the set of whole numbers, which is less than the set of rational numbers, which is less than the set of real numbers, which is less than the set of complex numbers.

First, we could define: poset A is less than the poset B if there is an order embedding from A to B but not vice versa. It plays good, except that the order is commonly just undefined for complex numbers.

To compare reals and complex numbers, we can use topology embedding: Reals can be topologically embedded into complex numbers, but not vice verse.

The problem here is that we need to use two different structures: partial order and topology. We need to “join” it into just one fundamental and natural structure.

Topology in very abstract terms is a description of which elements are near which.

So on \mathbb{N} and \mathbb{Z} there can be defined a structure similar to topology (which points are near which) as relations such as \{ (x;x) \mid x\in\mathbb{Z} \} \cup \{ (x;x+1) \mid x\in\mathbb{Z} \} \cup \{ (x;x-1) \mid x\in\mathbb{Z} \}.

This structure is just a binary relation not a topology. Fortunately, there are abstractions: funcoids and reloids (see my book and these updates for the book) which generalize both binary relations and topologies.

So, our task, refined, is to construct adequate reloids on \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}.

Topologies on \mathbb{N} and \mathbb{Z} are wrong for our purposes: Them are discrete and just say nothing about set structure.

For \mathbb{Q} and \mathbb{R} the structure we generate from the usual partial order should be the same as the usual topology on these sets.

I propose the structure which I call micronization as a generalization for both order and topology for sets \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R} (but not \mathbb{C}, for which we define the correct reloid only directly from topology, as there is no customary order on \mathbb{C}).

Let F be a binary relation on a set U. Then S(F) = \mathrm{id}_U \cup F \cup F^2 \cup F^3 \cup \dots.

Let E be a partial order on a set U. Micronization \mu(E) = \bigcap^{\mathsf{RLD}} \{ f\in\mathscr{P}(A\times A) \mid S(f)=E \}.

I conjecture that for E being the posets \mathbb{Q}, \mathbb{R} we have that \mu(\le) \cup \mu(\ge) coincides with the customary topology. For \mathbb{N} and \mathbb{Z} the micronization probably is \{ (x;x) \mid x\in\mathbb{Z} \} \cup \{ (x;x+1) \mid x\in\mathbb{Z} \} (that is exactly what our feelings describe as natural/whole numbers being “near” to each other).

So we have defined the same kind of structure for all \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C} (for \mathbb{N}, \mathbb{Z} defined from order, for \mathbb{C} defined from topology, and for \mathbb{Q}, \mathbb{R} defined from both order and topology which as I conjecture are the same). Now we can define: A is less than B if there is an embedding from A to B but not vice verse. How to define the term “embedding”? I am yet not sure but probably we should define “embedding” as a continuous injective function (see my book) from a reloid A to a reloid B.

Finally a conjecture:

Conjecture For every poset E we have S^{\ast}(\mu(E)) = E. (S^{\ast} is defined in my book.)

See also http://math.stackexchange.com/questions/1272198/the-least-relation-which-produces-a-partial-order

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