1. Partially Ordered Sets (POSets)

2. Partially Ordered Sets (POSets)
Let R be a relation on a set S. Then R is called a partial order if it is
Reflexive
a R a,  a  S
Antisymmetric
If a R b and b R a  a = b
Transitive
If a R b and b R c  a R c
The set S with partial order is called partially ordered set or poset.

3. Ex. The relation “” on the real numbers, is a partial order.
Sol. Reflexive : a  a for all real numbers
Antisymmetric : If a  b, b  a then a = b
Transitive : If a  b, b  c then a  c
This order relation on N or R is called usual order
Ex. (Z+, | ), the relation “divides” on +ve integers.
Ex. (Z, | ), the relation “divides” on integers.
Ex. (2S,  ), the relation “subset” on set of all
subsets of S.

4. Comparability
Let a and b be the elements in a partially ordered set (S, ). Then a and b are called comparable if a  b or b  a.
They are incomparable or non-comparable, written as a b if neither a  b nor b  a.
Ex. In poset (Z+, |), 3 and 6 are comparable, 6 and 3 are comparable, 3 and 5 are not, 8 and 12 are not.
Dual Order Let  be any partial ordering of set S. If the relation  is also a partial ordering of S, then it is called dual order.

5. Ordered Subsets Let A be any subset of an ordered set S
Suppose a, b  A.
Define a  b as elements of A whenever a  b as elements of S.
This defines a partial ordering of A called the induced order on A.
The subset A with the induced order is called an ordered subset of S.

6. Totally Ordered Set
If (S, ≼) is a poset and every two elements of S are comparable, then S is called totally ordered or linearly ordered.
A totally ordered set is also called a chain.
Ex. The poset (Z, ), is totally ordered, because either a  b or b  a when a and b are integers.
Ex. The poset (Z+, |), is not totally ordered because it contains elements that are incomparable such as 5 and 7.

7. Well-Ordered Set
A poset (S, ≼) is called a well-ordered set if
the order relation ≼ is a total-ordering and
every non-empty subset of S has a least element.
Ex.
The set (Z, ) is not well-ordered
because the set of –ve integers, which is a subset of Z, has no least element.

8. Product Order Suppose S and T are ordered sets
Product Order Suppose S and T are ordered sets. Then  is an order relation on the product set S  T, called the product order such that (a, b)  (a’, b’) if a  a’ and b  b’ Lexicographical Order Suppose S and T are linearly ordered sets. Then the order relation ≼ on the product set S  T, called the lexicographical order such that (a, b) ≺ (a’, b’) if a ≺ a’ or if a = a’ and b ≺ b’ It is also called dictionary order.

9. Ex.
Determine whether (3, 5)  (4, 8),
whether (3, 8)  (4, 5)
whether (4, 9)  (4, 11)
in the poset (ZZ, ≼) is the lexicographic ordering
(1, 2, 3, 5)  (1, 2, 4, 3) is the lexicographic ordering.

10. Lexicographic Ordering on the Set of Strings A string is less than a second string if the letter in the first string in the first position where the strings differ comes before the letter in the second string in this position, or if the first string and the second string agree in all positions, but the second string has more letters. This ordering is the same as that used in dictionaries. Ex. discreet ≺ discrete discreet ≺ discreetness discrete ≺ discretion

11. Hasse Diagram
Let S be a partially ordered set let a, b  S If a  b, then a is called an immediate predecessor of b, or b is known an immediate successor of a, or b is a cover of a, written as a  b but no element in S lies between a and b, i.e., there exists no element c in S such that a  c  b The set of pairs (a, b) such that b covers a is called the covering relation of the poset S.

12. Hasse Diagram Let S be a finite partially ordered set.
The Hasse diagram of S is the directed graph whose vertices are the elements of S and
there is a directed edge from a to b
whenever a  b in S.
(At place of an arrow from a to b, we can place b higher than a and draw a line between them)

13. Ex. Hasse diagram of poset ( {1, 2, 3, 4}, )
Also find the covering relation

14. Ex. Draw the Hasse diagram representing
the partial ordering { (a, b) | a divides b }
on {1, 2, 3, 4, 6, 8, 12} 12 8 4 2 3 6 1
Also find the covering relation

15. Ex. Draw the Hasse diagram for the partial ordering
{ (A,B) | A  B } on the power set P(S)
where S = {a, b, c}
{a,b,c}
{b,c}
{a,b}
{a,c}
{a}
{c}
{b}
{} or 
Also find the covering relation

16. Minimal and Maximal Elements
Let S be a partially ordered set. An element a in S is called minimal if no other element of S strictly precedes a. An element b in S is called maximal if no element of S strictly succeeds b. They are respectively bottom and top elements in the diagram. A Poset can have more than one minimal and more than one maximal elements.

17. Ex. Which elements of the poset
({2, 4, 5, 10, 12, 20, 25}, | ) are maximal, and which are minimal. 20 12 4 2 5 25 10
The maximal elements are 12, 20 and 25, and the minimal elements are 2 and 5.

18. Least and Greatest Elements
An element a in a poset S is called least (or first) element if a precedes every other element of S.
An element b in poset S is called the greatest (or last) element if b succeeds every other element of S.
Ex Find greatest and least elements in Hasse diagrams d c d b d d e b c c e c a b a b a a

19. Supremum Infimum Upper and Lower Bound
Let A be a subset of partially ordered set S.
An element u in S is called an upper bound of A if u succeeds every element of A.
An element l in a poset S is called a lower bound of A if l precedes every element of A.
Supremum
If an upper bound of A precedes every other upper bound of A, then it is called the supremum or least upper bound of A and is denoted by sup(A) or lub(A).
Infimum
If a lower bound of A succeeds every other lower bound of A, then it is called the infimum or greatest lower bound of A and is denoted by inf(A) or glb(A).

20. Upper bounds : e, f, j, h Lower bound : a For {j, h}, No Upper bound.
Ex. Find the lower and upper bounds of the subsets {a, b, c}, {j, h} and {a, c, d, f} in the poset with the given Hasse diagram.
Sol. For {a, b, c},
Upper bounds : e, f, j, h Lower bound : a
For {j, h}, No Upper bound.
Lower bounds : a, b, c, d, e, f
For {a, c, d, f},
Upper bounds : f, h, j
Lower bound : a g d b c j h f e a

21. Ex. Find glb and lub of {b, d, g}, if they exist, in the poset of previous slide.
Sol. The upper bounds of {b, d, g} are g and h.
Since g ≺ h, g is the least upper bound.
The lower bounds of {b, d, g} are a and b.
Since a≺b, b is the greatest lower bound.
Ex. Find glb and lub of the sets {3, 9, 12} and {1, 2, 4, 5, 10}, if they exist, in the poset (Z+, |).
Sol. Greatest lower bound for {3, 9, 12} is 3 and for {1, 2, 4, 5, 10} is 1.
Least upper bound for {3, 9, 12} is 36 and for {1, 2, 4, 5, 10} is 20.

22. Lattice
A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called lattice.
The least upper bound is known as join ().
The greatest lower bound is known as meet ().

23. Ex. Determine whether the posets represented by each of the Hasse diagrams are lattices.

24. Ex. Is the poset (Z+, |) a lattice?
Sol. Let a and b be two positive integers.
The least upper bound of two integers are the L.C.M. and the greatest lower bound of two integers are the greatest common divisor.
 this poset is a lattice.
Ex. Determine whether the poset ({1, 2, 3, 4, 5}, |) and ({1, 2, 4, 8, 16}, |) are lattices.
Ex. Determine whether (P(S), ) is a lattice where S is a set.

25. Lattices
Let L be a nonempty set closed under two binary operations called meet() and join().
Then L is called a lattice if it satisfies:
L1: Commutative Law
a  b = b  a and a  b = b  a
L2: Associative Law
(a  b)  c = a  (b  c) and
(a  b)  c = a  (b  c)
L3: Absorption Law
a  (a  b) = a and a  (a  b) = a
where a, b, c are elements in L

26. Duality
The dual of any statement in a lattice is the statement obtained by interchanging meet and join.
Ex. The dual of a  (b  a) = a  a
is a  (b  a) = a  a
Ex. Show that the operations of meet and join on a lattice is idempotent a  a = a & a  a = a
Proof : a  a
= glb {a, a} = glb {a} = a
a  a
= lub {a, a} = lub {a} = a

27. Ex. Show that the operations of meet and join on a lattice is commutative
a  b = b  a and a  b = b  a
Proof:
a  b
= glb {a, b} = glb {b, a}
= b  a
a  b
= lub {a, b} = lub {b, a}
= b  a

28. Ex. Show that the operations of meet and join on a lattice is associative
(a  b)  c = a  (b  c)
and (a  b)  c = a  (b  c)
Proof :
Since (a  b)  c = lub {a  b, c}
a  b ≤ (a  b)  c ………(1)
and c ≤ (a  b)  c ………(2)
Since a  b = lub {a, b}
 a ≤ a  b ………(3)
and b ≤ a  b ……(4)
From (1) & (3),
a ≤ (a  b)  c [by transitivity] …....(5)
From (1) & (4),
b ≤ (a  b)  c [by transitivity] …...(6)

29. From (2) & (6),
b  c ≤ (a  b)  c
[by definition of join] ………(7) From (5) & (7),
a  (b  c) ≤ (a  b)  c
[by definition of join] ………(8)
Also, a ≤ a  (b  c) ……(9)
b ≤ b  c ≤ a  (b  c) ………(10)
c ≤ b  c ≤ a  (b  c) ……(11)
From (9) & (10),
a  b ≤ a  (b  c) ………(12)
From (11) & (12),
(a  b)  c ≤ a  (b  c) ……(13)
From (8) & (13),
(a  b)  c = a  (b  c) [by anti-symmetry]

30. Ex. Show that the operations of meet and join on a lattice holds Absorption law
a  (a  b) = a and a  (a  b) = a
Proof :
Since a  b = glb {a, b}
a  b ≤ a ………(1)
Also, a ≤ a ………(2)
From (1) & (2),
a  (a  b) ≤ a ………(3)
Also a ≤ a  (a  b) ………(4)
[by definition of lub]
From (3) and (4),
a  (a  b) = a [by anti-symmetry]

31. Sublattice
Let S be nonempty subset of a lattice L. Then S is called a sublattice of L if S itself is a lattice.
Distributive Lattice :
A lattice L is said to be distributive if it satisfies the distributive laws i.e. if a, b, c  L then
a  (b  c) = (a  b)  (a  c)
and a  (b  c) = (a  b)  (a  c)
Otherwise, L is said to be non-distributive.

32. Ex. Show that following lattices are distributive or not?
Sol.
a  (c  d) = a  e = a
(a  c)  (a  d) = b  b = b
 a  (c  d) ≠ (a  c)  (a  d)
d  (a  c) = d  e = d
(d  a)  (d  c) = b  c = e
 d  (a  c) ≠ (d  a)  (d  c)
Ex. Distributive laws hold for the poset (P(A), ) where A = {a, b, c}

33. Theorem : In any Lattice, the semi-distributive laws hold.
x  (y  z)  (x  y)  (x  z)
and x  (y  z)  (x  y)  (x  z)
Theorem : In any distributive lattice,
a  x = a  y and a  x = a  y
together imply x = y.
Proof : x = x  (x  a) = x  (y  a)
= (x  y)  (x  a)
= (y  x)  (y  a)
= y  (x  a) = y  (y  a) = y

34. Ex. Show that every chain is a distributive lattice.
Sol. Let (L, ≼) be a chain and a, b, c  L.
Since L is a chain, either a ≼ b or b ≼ a.
If a ≼ b, then a  b = b and a  b = a.
For any pair of elements in L, lub and glb exist in L
L is a lattice
Case 1. If a ≼ b ≼ c then
a  (b  c) = a  c = a and
(a  b)  (a  c) = a  a = a.
Hence, we have a  (b  c) = (a  b)  (a  c)
Case 2. If c ≼ b ≼ a then
a  (b  c) = a  b = b and
(a  b)  (a  c) = b  c = b.
Hence, a  (b  c) = (a  b)  (a  c).
Similarly, we can prove a  (b  c) = (a  b)  (a  c)

35. Bounded Lattices
A Lattice L is called bounded if L has both a lower bound, denoted by 0 such that 0 ≼ x,  x  L and an upper bound, denoted by 1 such that x ≼ 1,  x  L
The bounds 0 and 1 are called universal bounds.
In bounded lattice,
a  1 = a  1 = a
a  0 = a a  0 = 0
for any element a in L.
Theorem : Every finite lattice L is bounded.

36. Ex. Let L = {1, 2, 3, 6}. Then (L, |) is a bounded lattice
Ex. Let L = {1, 2, 3, 6}. Then (L, |) is a bounded lattice. In this lattice, lower bound is 1 and upper bound is 6.
Ex. The lattice (N, ), where N is the set of natural numbers and  is the ‘less than or equal to’ relation on N, is not bounded because this lattice has no upper bound. Although (N, ) has lower bound namely 1.

37. Complement
Let L be a bounded lattice with lower bound 0 and upper bound 1. Let a be any element of L.
An element x in L is called a complement of a if
a  x = 1 and a  x = 0
Complement of 0 = 1
Complement of 1 = 0
Complements need not exist and need not be unique.
Complemented Lattice
A lattice L is said to be complemented if L is bounded and every element in L has a complement.

38. Ex. The lattice (P(S), ) of the power set of any set S is complemented. In this lattice each element has unique complement.
Ex. Any chain having 3 or more elements is not a complemented lattice.
Ex. The lattice in the diagram is complemented. In this lattice, complements are not unique. Both a and c are complements of d.

39. Theorem : Prove that, in a distributive lattice, if an element has a complement then this complement is unique.
Proof :
Suppose x and y are complements of any element a in L. Then a  x = 1, a  y = 1
a  x = 0, a  y = 0
x = x  0 = x  (a  y)
= (x  a)  (x  y) [Using distributive Law]
= 1  (x  y) = x  y ……..1)
y = y  0 = y  (a  x)
= (y  a)  (y  x) [Using distributive Law]
= 1  (y  x) = y  x ……….2)
From 1) & 2), x = y
Hence, compliment of a is unique

40. Complete Lattice
A lattice L is complete if every subset of L has a glb and lub
A complete lattice is always bounded but a bounded lattice need not be a complete lattice
Every finite lattice is complete.
Ex. Which of the following lattices have universal bounds (Z+, ), ({…, -3, -2, -1, 0}, ), (Z, ) and ({1, 2, 3, …, 100}, )
Ex. Consider the lattice L = {0, 1, 2, 3, 6} under divisibility relation. Then (L, |) is complete lattice.

41. Modular Lattice
A lattice L is modular if
a ≼ c  a  (b  c) = (a  b)  c
Ex. The lattice given by the diagram is modular.

42. Ex. The pentagonal lattice is not modular
Ex. Every chain is a modular lattice.
Sol. Since we can not find any triplet a, b, c in a lattice such that a ≼ c and b is not comparable with a or c
Therefore, every chain is modular.

43. Theorem : Every distributive lattice is modular.
Proof : Let (L, ≼) be a distributive lattice and a, b, c  L be such that a ≼ c.
Then
a  (b  c) = (a  b)  (a  c)
= (a  b)  c
Hence, every distributive lattice is modular.
However, the converse is not true

44. Product of Lattices
Let L and M be two lattices. The set of ordered pairs
{(x, y) : x  L, y  M}
with operations  and  defined by
(x1, y1)  (x2, y2) = (x1  x2, y1  y2)
(x1, y1)  (x2, y2) = (x1  x2, y1  y2)
is the direct product of L and M denoted by L x M

45. Ex. Let L and M be two lattices. The product of two lattices L x M is

46. Homomorphism
Let L and M be lattices. A mapping f : L  M is called homomorphism if it satisfies
f(x  y) = f(x)  f(y)
f(x  y) = f(x)  f(y)
Isomorphism
Let L and M be lattices. A mapping f : L  M is called isomorphism if it satisfies –
f is one-to-one
f is onto
f is homomorphism
Automorphism
Let L be a lattice. A mapping f : L  L is called
automorphism if f is an isomorphism.