Looking for an algebraic structure

Question

I'm looking for the name of algebraic structures (in which the elements are partially ordered) with the following properties:

1. Top element defined, bottom optional;
2. Join defined for all elements, meet defined for some;

In other words: I need some of the elements to be atomic (they are minimal, or "bottom-most", elements of the semi-lattice), while others are not (necessarily) minimal in the sense that for such an element $y$ there is an $x$ s.t. $x \leq y$.

Answer 1

I would call this a bounded semi-lattice, although you might have to clarify which bound exists.