I was going through the topic of Partial Ordering in the text "Discrete Mathematics and its Applications" by Kenneth Rosen where after reading the excerpt below I could make out the intuitive reason as to why partial order is anti-symmetric and transitive, but I could not quite make out the intuitive reason behind partial order is reflexive.
We often use relations to order some or all of the elements of sets. For instance, we order words using the relation containing pairs of words $(x , y)$, where $x$ comes before $y$ in the dictionary. We schedule projects using the relation consisting of pairs $(x , y)$ , where $x$ and $y$ are tasks in a project such that $x$ must be completed before $y$ begins. We order the set of integers using the relation containing the pairs $(x , y )$, where $x$ is less than $y$ . When we add all of the pairs of the form $(x , x )$ to these relations, we obtain a relation that is reflexive, antisymmetric, and transitive. These are properties that characterize relations used to order the elements of sets.
The above excerpt sort of justifies the intuitive logic behind partial order being anti-symmetric and transitive but it sort of forcefully introduces the reflexive property (the line in bold).
As another example of the forceful introduction of the reflexive property,
Lexicographical ordering: We will show how to construct a partial ordering on the Cartesian product of two posets, $(A_1 , \preceq _1 )$ and $(A_2 , \preceq _2 )$. The lexicographic ordering on $A_1 × A_2$ is definedby specifying that one pair is less than a second pair if the first entry of the first pair is less than (in A1) the first entry of the second pair, or if the first entries are equal, but the second entry of this pair is less than (in A2) the second entry of the second pair. In other words, (a1,a2 ) is less than (b1 ,b2), that is, $(a_1,a_2) ≺ (b_1,b_2)$, either if $a_1≺b_1$ or if both $a_1=b_1$ and $a_2≺b_2$. We obtain a partial ordering by adding equality to the ordering $≺$ on $A1 × A2$.
Here again we that the last statement forcefully introduces the reflexive property.
While in the case of ordering we shall not encounter the situation of ordering duplicate objects( handled by the reflexive property) but from my side I feel that the reflexive property is included to aid in the ordering of distinct objects as in the case of lexicographical ordering of strings such as shown below in the line marked as this line,where the use of equality(reflexive property) comes into picture.
Lexicographic ordering of strings. Consider the strings $a_1 a_2 ...a_m$ and $b_1b_2 ...b_n$ on a partially ordered set $S$ . Suppose these strings are not equal. Let $t$ be the minimum of $m$ and $n$. The definition of lexicographic ordering is that the string $a_1 a_2 ...a_m$ is less than $b_1b_2 ...b_n$ if and only if
$(a_1,a_2,..., a_t ) ≺ (b_1,b_2,..., b_t)$ ,
or $(a_1,a_2,..., a_t ) = (b_1,b_2,..., b_t)$ and $m<n$, <------ this line
where $≺$ in this inequality represents the lexicographic ordering of $S^{t}$ .
Another example comes form Category Theory. As you probably know, a category is the datum of a class of objects $A,B,C,\ldots$ together with morphisms $f,g,h,\ldots$ between pair of objects, satisfying some axioms. Every poset may be viewed as a particular category with many objects and few morphisms. Given a poset $(\mathcal{P},\leq )$, you can consider the category whose objects are the elements of $\mathcal{P}$. For every objects $A,B\in\mathcal{P}$, the set of morphisms $\{f:A\to B\}$ is either a singleton or the empty set. It is a singleton precisely when $A\leq B$, it is the empty set otherwise. In this context, reflexivity of $\leq $ is required to guarantee the existence of identity morphisms. Since every object $A$ of a reflexive poset $\mathcal{P}$ satisfies $A\leq A$ then for every object $A$ of the corresponding category the set of morphisms $f:A\to A$ consists of a single element, which is forced to be the identity on $A$ by the category axioms. While transitivity serves to guarantee that morphisms can be composed.
An intuition behind reflexivity might be the following: if you consider orbits of one element in a poset, that is, given $a\in\mathcal{P}$, the set of elements $y\in\mathcal{P}$ such that $x\leq y$, it is somehow convenient to include $x$ between those elements. So that the collection of all orbits results in a covering of the whole of $\mathcal{P}$.