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A partially ordered set (or ordered set or poset for short) (L, ) is called

a complete lattice if every of its non empty subsets has a least upper

bound (supremum) and a greatest lower bound (infimum) in (L, ).

Moreover, every lattice with a finite set is a complete lattice because

every subset here is finite.

A complete lattice have a least element and a greatest element.

The least and the greatest elements of a lattice are called

bounds(universal bounds) of the lattice and are denoted by 0 and 1respectively.

A lattice which has both elements 0 and 1 is called a bounded

lattice.

Therefore, every finite lattice (L, v, ^) with Ln = {a1,a2, a3, ............. , an} isbounded.

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The power set of a given set, ordered by inclusion. The

supremum is given by the union and the infimum by the intersection

of subsets.

The unit interval [0,1] and the extended real number line, with the

familiar total order and the ordinary suprema and infima. Indeed, a

totally ordered set (with its order topology) is compact as a

topological space if it is complete as a lattice.

The lattice of all transitive relations on a set.

The lattice of all sub-multisets of a multiset.

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1. The Lindenbaum algebra of most logics that support

conjunction and disjunction is a distributive lattice, i.e. "and"

distributes over "or" and vice versa.

2. Every Boolean algebra is a distributive lattice.

3. Every Heyting algebra is a distributive lattice.

4. Every totally ordered set is a distributive lattice with max as

join and min as meet. Note that this is again a specialization of

the previous example.

A lattice (L, ^, v) is distributive if and only if it does notcontain the five element pentagonal or, the diamond

lattice given above as one of its sublattices .

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A lattice (L, :') is said to be modular,

ifa v (b ^ c)= (a v b) ^ cwhen ever a cfor all a, b, c L

Every distributive lattice is modular.

Proof.

Let (L, ) be a distributive lattice and a, b, c L be such thata c.

Thus ifa c ,then a v c = c.

Now

a v (b ^c)= (a v b) ^ (a v c)= (a v b) ^ c

Hence, every distributive lattice is modular.

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All distributive lattices.

The lattice ofnormal subgroups of any

group.

The lattice ofsubmodules of any module.

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In abstract algebra, a Boolean lattice is a boundedcomplemented distributive lattice. This type of algebraic

structure captures essential properties of both set operations

and logic operations. A Boolean algebra can be seen as a

generalization of a power set algebra or a field of sets.

And an algebraic system well defined on Boolean Lattices isknown as Boolean Algebra.

Boolean lattice of subsets

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We say a Boolean lattice system(B , + , . , /) where

+, . And / are the join ,meet and complementoperations respectively is a Boolean Algebra.

For example, (P(S), ) is a complemented

distributive and hence (P(S),U,,/) is a Boolean

algebra where U, and / are the join ,meet andcomplement operations if S is a set within n

elements .

And thus (P(S), ) becomes finite Boolean lattice

so is a finite Boolean Algebra