partially ordered set

Question

What is the meaning of:

1. $A(x) := \{y ∈ X : x ≤ y\}$
2. $(x ≤ x$ or $x ∈ A(x)$ for all $x ∈ X)$
3. $(A ◦ A ⊂ A$, i.e., $x ≤ y, y ≤ z ⇒ x ≤ z$ for $x, y, z ∈ X)$

in the sentences:

A preorder or partial order or preference relation on a set $X$ is a relation $A$ between elements of $X$, often denoted by $≤$, with $A(x) := \{y ∈ X : x ≤ y\}$ that is reflexive $(x ≤ x$ or $x ∈ A(x)$ for all $x ∈ X)$ and transitive $(A ◦ A ⊂ A$ i.e., $x ≤ y, y ≤ z ⇒ x ≤ z$ for $x, y, z ∈ X)$.

Answer 1

The notation in this question is a bit non-standard, and some of the terms are wrong. First of all, a preorder and a partial order are not the same.

Let us start with a set $X$. A (binary) relation $R$ on $X$ is a subset of the set $X \times X$, where $\times$ denotes the cross product of two sets. In other words, a relation on $X$ is a set of pairs $(x,y)$ such that $x \in X$ and $y \in X$. If two elements $x$ and $y$ are related by the relation $R$, we write $(x,y) \in R$, or sometimes $xRy$.

Now, a preorder is a relation $R$ that is reflexive and transitive. By this we mean the following:

1. Reflexivity: For every element $x \in X$ it must be the case that $(x,x) \in R$. So every element must be related to itself.
2. Transitivity: If $(x,y) \in R$ and $(y,z) \in R$, then it must also be the case that $(x,z) \in R$.

A preorder is sometimes denoted $\leq$, so we can write $x \leq y$ if $x$ and $y$ are related by the preorder $\leq$. Note however, that this notation can be misleading, since it can be the case that $x \leq y$ and $y \leq x$, but not have that $x = y$, as we would expect for example from the usual $\leq$ on natural numbers.

Hence we can also talk about partial orders which are also reflexive and transitive, but furthermore has the anti-symmetry property:

3. Anti-symmetry: If $(x,y) \in R$ and $(y,x) \in R$, then $x = y$.

Partial orders are also often denoted $\leq$, and the usual less-than-or-equal-to relation on the natural numbers is indeed a partial order.