<Dd> 0 ≤ x ≤ 1 for every x in L . </Dd> <P> The greatest and least element is also called the maximum and minimum, or the top and bottom element, and denoted by ⊤ and ⊥, respectively . Every lattice can be converted into a bounded lattice by adding an artificial greatest and least element, and every non-empty finite lattice is bounded, by taking the join (resp., meet) of all elements, denoted by ⋁ L = a 1 ∨ ⋯ ∨ a n (\ displaystyle \ bigvee L = a_ (1) \ lor \ cdots \ lor a_ (n)) (resp . ⋀ L = a 1 ∧ ⋯ ∧ a n (\ displaystyle \ bigwedge L = a_ (1) \ land \ cdots \ land a_ (n))) where L = (a 1,..., a n) (\ displaystyle L = \ (a_ (1), \ ldots, a_ (n) \)). </P> <P> A partially ordered set is a bounded lattice if and only if every finite set of elements (including the empty set) has a join and a meet . For every element x of a poset it is trivially true (it is a vacuous truth) that ∀ a ∈ ∅: x ≤ a (\ displaystyle \ forall a \ in \ varnothing: x \ leq a) and ∀ a ∈ ∅: a ≤ x (\ displaystyle \ forall a \ in \ varnothing: a \ leq x), and therefore every element of a poset is both an upper bound and a lower bound of the empty set . This implies that the join of an empty set is the least element ⋁ ∅ = 0 (\ displaystyle \ bigvee \ varnothing = 0), and the meet of the empty set is the greatest element ⋀ ∅ = 1 (\ displaystyle \ bigwedge \ varnothing = 1). This is consistent with the associativity and commutativity of meet and join: the join of a union of finite sets is equal to the join of the joins of the sets, and dually, the meet of a union of finite sets is equal to the meet of the meets of the sets, i.e., for finite subsets A and B of a poset L, </P> <Dl> <Dd> ⋁ (A ∪ B) = (⋁ A) ∨ (⋁ B) (\ displaystyle \ bigvee \ left (A \ cup B \ right) = \ left (\ bigvee A \ right) \ vee \ left (\ bigvee B \ right)) </Dd> </Dl>

Does the system of all subsets of a finite set under the operation subset of form a lattice