That said there is no element between those two, which is called immediate predecessor/successor.
The element 1 divides every other element:
1β£2,1β£3,1β£6
Elements 2 and 3 divide 6:
2β£6,3β£6
Thus, the diagram visually represents the partial order defined by divisibility on the set {1, 2, 3, 6}.
Boolean Algebra & Algebra of Sets
A Boolean algebra is a structure (B, β§, β¨, ', 0, 1) where B is a set equipped with two binary operations meet β§ and join β¨, a unary operation complement ', and distinguished elements 0 and 1.
Formally, B is a complemented distributive lattice, i.e.:
1. (B, β§, β¨) is a lattice: for all a, b in B, the meet and join exist.
aβ§b=inf{a,b}βB,aβ¨b=sup{a,b}βB
2. The lattice is distributive, meaning for all a, b, c in B,
aβ§(bβ¨c)=(aβ§b)β¨(aβ§c)
and
aβ¨(bβ§c)=(aβ¨b)β§(aβ¨c)
3. There exist least and greatest elements 0, 1 in B such that for all a in B,
0βͺ―aβͺ―1
where β€ is the partial order induced by the lattice operations:
aβͺ―bβΊaβ§b=a(equivalentlyΒ aβ¨b=b)
4. The lattice is complemented: for every element a in B, there exists a complement a' in B satisfying
aβ§aβ²=0
and
aβ¨aβ²=1
Algebra of Sets
An algebra of sets on a non-empty set X is a non-empty collection A of subsets of X such that:
1. The whole set X is in A.
XβA
2. A is closed under complementation relative to X:
βAβAβΉAc=XβAβA
3. A is closed under finite unions:
βA,BβAβΉAβͺBβA
From these axioms, it follows that A is also closed under finite intersections:
