Here is the question that I posted on the Mathematics Chat Room that I was unable to find an answer to:
Question: Under what conditions/properties is a poset ever a wqo (well-quasi-order)? Can we find a definition for wqo involving posets?
Background Definitions
Definition [Partially-Ordered(-Set)]. Let $A$ be a set. Then $\preceq$ on $A$ is partially-ordered $\iff\preceq$ is reflexive, transitive, and antisymmetric.
Definition [Well-Quasi-Ordered(-Set)]. Let $A$ be a set. Then $\preceq$ on $A$ is wqo $\iff$ every infinite sequence of elements of $A$ is good $\iff\exists \ i,j\in\mathbb{Z}:i<j$ and $a_i\preceq a_j$ of the infinite sequence $(a_i)_{i\geq1}$.
Definition (Well-Founded). $\preceq$ is well-founded $\iff\not\exists$ any infinite decreasing chains w.r.t. $\prec$.
(Note: $x\prec y\iff x\preceq y$ and $y\not\preceq x$.)
My Thoughts
By the definitions, I was thinking that essentially we can say that a wqo is essentially a poset that is well-founded. I say this statement because the fact the a wqo makes for a strict, monotonic order such that $i$ is strictly less than $j$ ($i<j$) with a preorder $a_i\preceq a_j$, we can see that defining a well-foundedness on a poset would restrict infinite decreasing chains and will mostly likely make the poset good and possibly a wqo.
Final Comments
- Please feel free to let me know if I have anything errors in my definitions or explanations/thoughts pertaining to the content solely from this question.
- If you also think this question is better suited on MathOverflow, please let me know.
You’re missing part of the definition of a wqo: it must be a quasi-order, meaning that it must be reflexive and transitive. Equivalently, $\preceq$ is a wqo on $A$ iff it is a well-founded quasi-order with no infinite antichain. This means that a partial order with an infinite antichain is not a wqo even if it’s well-founded. For example, the order $\preceq$ on $\Bbb N\times\Bbb N$ given by $\langle k,\ell\rangle\preceq\langle m,n\rangle$ iff $k=m$ and $\ell\le n$ is a well-founded partial order that is not a wqo.
This Wikipedia article is actually fairly helpful.