Weak and strong orders

Question

Did I understand it right, that R={⟨X,Y⟩∣X⊆Y} is a weak order and {⟨X,Y⟩∣X⊂Y} is a strong order (not sure about the correct english term)?

Answer 1

Well, I assume, you mean $(X,Y) ∈ R$ iff $X⊂Y$ or $X=Y$. Let us further assume that $⊆$ is a partial order relation, while $⊂$ is irreflexive ($∀X:X\not⊂X$). Then $⊂$ is called “strict (partial) order“ and $⊆$ is a reflexive (partial) order relation. Wikipedia calls it also weak partial order. Because of the duality between sets of pairs and binary relations you can consider the two sets defined above as binary relations that are equivalent to the corresponding defining relations.

I just saw that you link to strict weak order relations in the comments. If you are sure that this is the right definition then you the answer is probably: „No.“ (depending on the missing definition of $⊆$ and $⊂$).

Answer 2

The subset relation $\subseteq$ is a partial order (Halbordnung), and the proper subset relation $\subset$ is a strict partial order (strenge Halbordnung). If the underlying set has at least three elements, $\subset$ is not a strict weak order: if $a,b$, and $c$ are distinct elements, $\{a\}$ is not comparable with $\{b\}$, and $\{b\}$ is not comparable with $\{a,c\}$, but $\{a\}\subset\{a,c\}$, so incomparability with respect to $\subset$ is not an equivalence relation. For essentially the same reason $\subseteq$ is not in general a non-strict weak order (also known as a total preorder or preference relation; German seems to have at least Präferenzrelation, Präferenzordnung, and totale Quasiordnung as possible terms for this concept).