apologies in advance for the noob question. (Really rusty when it comes to order theory notation.)

Suppose I have a set $\Omega := \{\alpha,\beta_1,...,\beta_n\} $ (for some fixed $n\in\mathbb{N}$). I would like to succinctly express the following:

$\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad $ "$C$ is a subset of elements from $\Omega$ ordered in the following way:
$\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad$ 1. If $\alpha$ is in $C$, then it is placed first.
$\quad \quad \quad\quad\quad \quad\quad\quad\quad\quad\quad$ 2. All $\beta$'s in $C$ are placed in increasing order of their indeces."

I hope this is sufficiently precise; to help illustrate, here are a couple of examples of what I have in mind:
$\quad \bullet\ $ $C = (\omega, \beta_1,\beta_3)$
$\quad \bullet\ $ $C = (\beta_2,\beta_3)$

If there is no succinct, mathematical way to state this, then no worries! Thanks in any case.


1 Answer 1


The set $\Omega = \{\alpha, \beta_1, \beta_2, \ldots, \beta_n\}$ is ordered by $\leqslant$ where $\alpha \leqslant \beta_1 \leqslant \beta_2 \leqslant \ldots \leqslant \beta_n$.The sequence $C$ is a subset of $\Omega$ where the elements appear in the order given by $\leqslant$. For example $C = (\alpha, \beta_2)$ or $C = (\beta_1, \beta_3, \beta_6)$.


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