I was reading a paper about well-orderings and this came up:
Suppose (E, ≤) and (F, ≼) are isomorphic well-orderings. Then there exists a unique isomorphism for (E, ≤) to (F, ≼).
I've been scouring the internet for what this symbol means. Someone said it means "precedes", but that led me to wonder if 1 ≼ 2 would be true but then someone else said that X ≼ Y <=> $$X = X\land Y$$ which made no sense to me. Could someone explain the meaning of this symbol? Thanks.
On
The curly versions of the less than and greater than signs are commonly used to denote some other ordering than the one that we are usually talking about. For instance there is a partial ordering on the symmetric matrices, where $A \preccurlyeq B$ if and only if $B-A$ is a nonnegative definite matrix. We write $\preccurlyeq$ instead of $\leq$ to avoid confusion with the ordering that we use more commonly. This would be especially important if we had two orderings on the same set.
But it's just a symbol. It would be perhaps confusing but certainly not wrong to use it some other way.
In this context, "$\le$" and "$\preccurlyeq$" are just names for binary relations on $E$ and $F$. Since these relations are orderings, we use suggestive symbols; but that's all. For instance, maybe $E=\{1, 3, 172\}$ and $\preccurlyeq$ is the relation $\{(1, 3), (172, 1), (172, 3), (172, 172), (1, 1), (3, 3)\}$ - that is, $172\preccurlyeq 1\preccurlyeq 3$. This would be perfectly fine.