Definition:Strict Ordering

Also known as
Some sources call this an antireflexive (partial) ordering.

Also see

 * Equivalence of Definitions of Strict Ordering


 * Definition:Strictly Precede
 * Definition:Strictly Succeed

Note that this definition of strict ordering does not demand that every pair of elements of $S$ is related by $\prec$. The way we have defined a strict ordering, they may be, or they may not be, depending on the context.

If it is the case that $\prec$ is a connected relation, i.e. that every pair of elements is related by $\prec$, then $\prec$ is called a strict total ordering.

If it is not the case that $\prec$ is connected, then $\prec$ is called a strict partial ordering.

Beware that some sources use the word partial for an ordering which may or may not be connected, while others insist on reserving the word partial for one which is specifically not connected. It is wise to be certain of what is meant.