Properties and Special Relations

Given a relation $ R $ we have that

$ R $ is reflexive if for all $ a $ we have that $ a\ R\ a $. That is $ a $ is related to itself. This is typical for the equal relation: we have that $ a = a $. However for the relation "is less then" this does not hold.
$ R $ is symmetric if we have that $$ a\ R\ b \Rightarrow b\ R\ a $$ That is if $ a $ is related to $ b $, then $ b $ is related to $ a $. This is again typical for the equal relation. However for the relation "is less than" this does not hold.
$ R $ is irreflexive, or anti-reflexive, if for no $ a $ we have that $ a\ R\ a $. A typical example is the relation "less than". No $ a $ is less than itself.
$ R $ is anti-symmetrical if $$ a\ R\ b \land b\ R\ a \Rightarrow a = b $$ A typical example is the "less than or equal to". If we have that $ a \leq b $ and $ b \leq a $, then we have that $ a = b $.
$ R $ is asymmetrical if and only if $ R $ is both irreflexive and anti-symmetrical.
$ R $ is transitive if we have that $$ a\ R\ b \land b\ R\ c \Rightarrow a\ R\ c $$ This holds true for both "less than", "less then or equal to" and "equal to". If $ a \lt b $ and $ b \lt c $, then sure $ a \lt c $.
$ R $ is a connex relation (or has the property of connexity) if for all $ a,b $ we have $$ a\ R\ b \lor b\ R\ a $$
$ R $ is a semiconnex relation if for all $ a,b $ we have $$ a \neq b \Rightarrow a\ R\ b \lor b\ R\ a $$

Now we can define special kind of relations.

An equivalence relation is a relation that is:

A partial order is a relation that is

A set with a partial order is called a partially ordered set or a poset.

A preorder relation is a relation that is

A total preorder is a relation that is

A set that is equipped with a preorder is called a preordered set.

A partial order is a relation that is