Partially ordered sets with specific property

Question

I'm interested in partially ordered sets with the following property:

For every two elements:

1. There exist an element which is smaller than both of them and an element which is bigger than both of them.

2. The subset of smaller elements includes maximum (an element which is bigger than all the rest) and the subset of bigger elements includes minimum.

I find this structure in some research I make, and I would like to know what can be said about it.

Answer 1

A partially ordered set, in which each pair of elements has a least upper bound and a greatest lower bound, is called a lattice. (Unfortunately, the term lattice is also used in other senses, even in mathematics.)