An alternate definition which is equivalent to noaseboar's which seems less arbitrary would be to say that a lattice is any poset where sup H and inf H exist for any finite non-empty subset H of L. This is readily seen to be the same. Let the lattice L be defined as noaseboar has defined it above. We can prove the definitions are identical by induction. Say our subset is H={a}, with only one element. Obviously then, sup H and inf H exist and are both a, because of the reflexive property of our partial ordering relation. Say sup Hn=x and inf Hn=y exist for a subset Hn of L of cardinality n. Let us add one element z to Hn, making Hn+1. Obviously sup Hn+1 exists and is simply sup {x, y}, and inf Hn+1 also exists and is inf {y, z}. So by induction both definitions are equivalent. The infimum or supremum of an empty subset of a lattice need not exist.

Yet another way of looking at lattices is as an algebra <L; ∧ ; ∨> with L a non-void set and ∧ and ∨ are binary operations (known as the meet and join operations respectively) on L which are idempotent, commutative, and associative, and obey the absorption identities:

a ∧ (a ∨ b) = a, and

a ∨ (a ∧ b) = a.

This can easily be shown to be equivalent to noaseboar's definition above by setting sup {a, b} = a ∧ b and inf {a, b} = a ∨ b.

It will be noted that the meet and join operators described above have the same properties as the 'or' and 'and' operators in Boolean algebra.