Partial Orders and Lattices

Relations can be used to order some or all the elements of a set. For instance, the set of Natural numbers is ordered by the relation such that for every ordered pair in the relation, the natural number comes before the natural number unless both are equal. Formally, “A relation on set is called a partial ordering or partial order if it is reflexive, anti-symmetric, and transitive. A set together with a partial ordering is called a partially ordered set or poset. The poset is denoted as .”  

Example: Show that the inclusion relation is a partial ordering on the power set of a set. 

Solution Since every set , is reflexive. If and then , which means is anti-symmetric. It is transitive as and implies . Hence, is a partial ordering on , and is a poset. 

Note: The symbol is used to denote the relation in any poset. The notation is used to denote but . 

Comparability: Let and be the elements of a poset , then and are said to be comparable if either or . Otherwise, and are said to be incomparable. 

• Example – In the poset (where is the set of all positive integers and is the divides relation) are the integers 3 and 9 comparable? Are 7 and 10 comparable? 
• Solution – 3 and 9 are comparable since i.e. 3 divides 9. But 7 and 10 are not comparable since and . 

Total Order: It is possible in a poset that for two elements and neither nor i.e. the elements and are incomparable. But in some cases, such as the poset , every element is comparable to every other element. 

A poset is called totally ordered if every two elements of are comparable. i is called a total order. A totally ordered set is also called a chain. 

Hasse Diagrams: A partial order, being a relation, can be represented by a di-graph. But most of the edges do not need to be shown since it would be redundant. For instance, we know that every partial order is reflexive, so it is redundant to show the self-loops on every element of the set on which the partial order is defined. Every partial order is transitive, so all edges denoting transitivity can be removed. The directions on the edges can be ignored if all edges are presumed to have only one possible direction, conventionally upwards. In general, a partial order on a finite set can be represented using the following procedure is as follows: 

1. Remove all self-loops from all the vertices. This removes all edges showing reflexivity.
2. Remove all edges which are present due to transitivity i.e. if and 
   are in the partial order, then remove the edge . Furthermore if is in partial order, then remove the edge .
3. Arrange all edges such that the initial vertex is below the terminal vertex.
4. Remove all arrows on the directed edges, since all edges point upwards.

For example, the poset would be converted to a Hasse diagram as follows:

The last figure in the above diagram contains sufficient information to find the partial ordering. This diagram is called a Hasse Diagram. 

Extremums in Posets: Elements of posets that have certain extremal properties are important for many applications. 

• Maximal Elements: An element in the poset is said to be maximal if there is no element in the poset such that .
• Minimal Elements:  An element in the poset is said to be minimal if there is no element in the poset such that .

Maximal and Minimal elements are easy to find in Hasse diagrams. They are the topmost and bottommost elements respectively. 
For example, in the hasse diagram described above, “1” is the minimal element and “4” is the maximal element. Since maximal and minimal are unique, they are also the greatest and least elements of the poset. 

Note: If the maximal or minimal element is unique, it is called the greatest or least element of the poset respectively. 

Bounds in Posets: It is sometimes possible to find an element that is greater than or equal to all the elements in a subset of poset . Such an element is called the upper bound of . Similarly, we can also find the lower bound of. These bounds can be further constrained to get the least upper bound and the greatest lower bound. These bounds are elements that are less than or greater than all the other upper bounds or lower bounds respectively. 

Example: Find the least upper bound and greatest lower bound of the following subsets- , , . 

Solution: For the set 
The upper bounds are – . So the least upper bound is . 
The lower bounds are – . So the greatest lower bound is . 
For the set 
The upper bounds are – . So the least upper bound is . 
The lower bounds are – . So the greatest lower bound is . 
For the set 
The upper bounds are – . So the least upper bound is . 
The lower bounds are – . So the greatest lower bound is . 

Lattices: A Poset in which every pair of elements has both, a least upper bound and a greatest 
lower bound is called a lattice. There are two binary operations defined for lattices –  

1. Join: The join of two elements is their least upper bound. It is denoted by , not to be confused with disjunction.
2. Meet: The meet of two elements is their greatest lower bound. It is denoted by , not to be confused with a conjunction.

Sub Lattice: A sublattice of lattice is a subset such that if , , and . 

Identities for join and meet:

Distributive laws may or may not hold true for a lattice: 

1. 
2. 

Note: A lattice is called a distributive lattice if the distributive laws hold for it. 

But Semidistributive laws hold true for all lattices : 

1. 
2. 

Two important properties of Distributive Lattices – 

1. In any distributive lattice and together imply that .
2. If and , where and are the least and greatest elements of lattice, then and are said to be a complementary pair. and are a trivially complementary pair.

complemented lattice: Suppose L is a bounded lattice (with 0 and 1), and a?L(a belongs to L). A complement of a is an element b?L such that :

a?b=0 and a?b=1. 

Remark: Complements may not exist. If L is a non-trivial chain, then no element (other than 0 and 1) has a complement. This also shows that if a is a complement of a non-trivial element b, then a and b form an antichain.

An element in a bounded lattice is complemented if it has a complement.
A complemented lattice is a bounded lattice in which every element is complemented.

GATE CS Corner Questions

Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them. 

1. GATE CS 2007, Question 26
2. GATE CS 2006, Question 4
3. GATE CS 2005, Question 9
4. GATE CS 2004, Question 73
5. GATE CS 2015 Set-1, Question 44

References: 

Partially Ordered Set – Wikipedia
Lattices – Wikipedia