What is an order ideal?

Question

I don't understand the example provided for Theorem 2.1 in this paper: https://arxiv.org/pdf/1406.2583v2.pdf, which is on the enumeration of simultaneous core partitions.

It says that the order ideals of $P_{(3,4)}$, a (3,4)-core partition, include {2}, when $P_{(3,4)}$={1,2,5}. If this is an order ideal, and $P_{(3,4)}$ contains a number less than 2, shouldn't {2} not be an order ideal, but instead be {2,1}?

This is based on the following definition of an order ideal, from the paper cited above:

An order ideal of P is a subset I such that if any y ∈ I and x ≤ y in P, then x ∈ I.

Answer 1

According to the example,

It is easily checked that $P_{(3,4)} = \{1, 2, 5\}$ with the partial order 5 > 2 and 5 > 1.

Given this, we do not have $2 > 1$, so $\{2\}$ is an ideal.