##### Lattice | Least Upper Bound (Join) | Greatest Lower Bound(Meet) | Join semi lattice (∧) | Meet semi lattice(∨) | Bounded Lattice | Complement of an lattice | Complemented lattice | Sub lattice

Least Upper Bound (Join): Let [A;R] be a poset. For a,b∈A, if there exists an element c∈A such that
I. aRc and bRc
II. if there exists any other element 'd' such that aRd and bRd then cRd, then c is called LUB of 'a' and 'b'.
Greatest Lower Bound(Meet): Let [A;R] be a poset. For a,b∈A, if there exists an element c∈A such that
I. cRa and cRb
II. if there exists any other element 'd' such that dRa and dRb then dRc, then c is called GLB of 'a' and 'b'.
Example: LUB (2,3) = 6     GLB (2,3) = 1
LUB (6,9) = 18   GLB (6,9) = 3
Join semi lattice (∧): If LUB exists for every pair of elements in poset.
Meet semi lattice(∨): If GLB exists for every pair of elements in poset.
Lattice: If both LUB and GLB exist for every pair of elements in poset.
Note: Every lattice holds commutative, associative, idempodent, and absorption properties always. If it also holds distributive property then it is called Distributive Lattice.
Bounded Lattice: If upper bound and lower bound exists in a lattice, then it is called a Bounded Lattice.
Complement of an lattice: Let L be an bounded lattice, for any element a∈L, if there exists an element b∈ L, such that a∨b = I(Upper bound) and a∧b =O(Lower bound), then b is called 'complement of a'.
Complemented lattice: Let 'L' be a bounded lattice, if each element of 'L' has a complement in 'L', then L is called a complemented lattice.
Note: In a distributive lattice, complement of an element if exists, is unique.
Sub lattice: Let 'L' be a lattice. A subset 'M' of 'L' is called a sublattice of 'L' if
I. M is a lattice.
II. For any pair of elements a,b∈M, the LUB and GLB are same in M and L.