I'm confused at the definition of 2 similar partially ordered sets and I would like to ask for your help for a clarification please.
I use the book of Halmos (Naive set theory) in which the author gives the following definition:
Two partially ordered sets (which may in particular be totally ordered and even well ordered) are called similar if there exists an order-preserving one-to-one correspondence between them.
More explicitly: to say of the partially ordered sets $X$ and $Y$ that they are similar means that there exists a one-to-one correspondence, say $f$, from $X$ onto
$Y$, such that if $a$ and $b$ are in $X$, then a necessary and sufficient condition
that $f(a) \leq f(b)$ (in $Y$) is that $a \leq b$ (in $X$). A correspondence such as $f$ is sometimes called a similarity
What confuses me is that if we have $f(a) \leq f(b)$ then the relation $\leq$ (which is in fact a set of ordered pairs) contains the pair $(f(a),f(b))$ but there is no reason that $f(a) \in X$ and $f(b) \in X$. In other words, there is no reason that $\leq$ contains $(f(a),f(b))$ which causes a contradiction.
Am i wrong somewhere ? How should I understand the condition "$f(a) \leq f(b)$ (in $Y$) iff $a \leq b$ (in $X$)" in the definition ? Should I understand it as "if the ordered pair $(a,b)$ belongs to some set of ordered pairs, then the ordered pair $(f(a), f(b))$ also belongs to the same set" ? What intuition should I retain here ?
Thank you very much for your enlightment!
Best Answer
In more formal notation we could write, “Posets $P_1 = (X_1, \leq_1)$ and $P_2 = (X_2, \leq_2)$ are similar precisely when there exists a bijection $f$ from $X_1$ to $X_2$ such that for all $x,y \in X_1$, we have $x \leq_1 y$ if and only if $f(x) \leq_2 f(y)$. This clarifies that we are dealing with distinct order relations $\leq_1$ and $\leq_2$.
As Michael Carey wrote though, it is more common to describe such two posets $P_1$ and $P_2$ as isomorphic.