The third important type of order relation is the so-called strict total order, a formal definition of which you can find below. It is an order relation, in which each element is not comparable to itself. This means that the order relation is irreflexive. An example of a strictly ordered set is $(\mathbb N,<)$. The irreflexive order relation "$<$" puts all natural numbers into the same order as it was the case in the totally ordered chain $(\mathbb N,\le)$, in which "$\le$" was reflexive. Moreover, a strict order is antisymmetric, i.e. $a < b$ implies $b\not < a$.
A total order "$\preceq$", in which the property of being reflexive is replaced by the property of being irreflexive and the property of being antisymmetric is replaced by the property of being asymmetric is called a strict total order.
A chain $(V,\preceq )$ with a strict total order "$\preceq$$ is called a strictly-ordered set. We will change the notation to $(V,\prec)$, whenever we want to indicate that we deal with a strictly-ordered set and not with a chain.
Definitions: 2 3 4 5 6 7
Examples: 8 9
Explanations: 10 11
Proofs: 13 14 15 16 17
Propositions: 18 19 20