Comparisons
Why
We want language and notation involving order.1
Comparisons
A comparison is a statement (see Statements) involving a partial (which may or may not be total) order.
Notation
Let $A$ be a set. We tend to denote an arbitrary partial order on $A$ by $\preceq$. So $(A, \preceq)$ is a partially ordered set.
As usual (see Relations), we write $a \preceq b$ to mean $(a, b) \in A$. Alternatively, we write $b \succeq a$ to mean $a \preceq b$. In other words, $\succeq$ is the inverse relation (see Converse Relations) of $\preceq$.
Predecessors and successors
If $a \preceq b$ and $a \neq b$, we write $a \prec b$ and say that $a$ precedes $b$. In this case we call $a$ the predecessor of $b$. Alternatively, under the same conditions, we write $b \succ a$ and we say that $b$ succeeds $a$. In this case we call $b$ the successor of $a$.
Induced partial orders
Of course, the object we have defined and denoted by $\prec$ is a relation on $A$. It satisfies (i) for no elements $x$ and $y$ do $x \prec y$ and $y \prec x$ hold simultaneously and (ii) if $x \prec y$ and $y \prec z$, then $x \prec z$ (i.e., $\prec$ is transitive). It is worthwhile to observe that if $S$ is a relation satisfying (i) and (ii), then the relation $R$ defined to mean $(a, b) \in S$ or $a = b$ is a partial order on $A$.
Strict and weak relations
This connection between $\preceq$ and $\prec$ can be generalized. The strict relation corresponding to a relation $R$ on a set $A$ is the relation $S$ on $A$ defined by $(a, b) \in S$ if $(a, b) \in R$ and $a \neq b$. The weak relation corresponding to a relation $S'$ on a set $A$ is the relation $R'$ defined by $(a, b) \in R'$ if $(a, b) \in S'$ or $a = b$. For this reason, a relation is said to partially order a set if it is a partial order or if its corresponding weak relation is one.
- In the present edition, this sheet can be thought of as an extended notation section for Orders. ↩︎