Name of the order with irreflexivity, antisymmetry and transitivity?

Question

I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?

Answer 1

If $R$ is irreflexive, antisymmetric, and transitive, then it is also known as a strict partial order. Notice however, that there is very little difference between strict partial orders and partial orders. The reflexive closure gives a bijection between strict partial orders and partial orders.

Answer 2

As Ittay answers, such a relation is known as a strict partial order.

To read more about the relationship and distinction between a partial order and a strict partial order, see the Wikipedia entry on partial order vs. strict partial order.

Notice that in the linked entry: there exists a one-to-one correspondence between a partial order and the corresponding strict partial order, and so "the number of strict partial orders is the same as that of partial orders" on a set with n elements.

An example often used is relation $\leq$, which is a partial order, and $\lt$ which imposes a strict partial order.