1. Chapter 6 Order Relations and Structures
CSCI 115
Chapter 6
Order Relations and Structures

2. §6.1 Partially Ordered Sets
CSCI 115
§6.1
Partially Ordered Sets

3. §6.1 – Partially Ordered Sets
POSET
A relation R on a set A is called a partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set or poset, and is denoted (A,R).

4. §6.1 – Partially Ordered Sets
Dual 
Comparable
Linear order (chain)

5. §6.1 – Partially Ordered Sets
Theorem 6.1.1
If (A, 1) and (B, 2) are posets, then (A x B, ) is a poset where  is defined by: (a, b)  (a’, b’) iff a 1 a’ in A and b 2 b’ in B.
(A x A, ) where 1 = 2 is called the product partial order

6. §6.1 – Partially Ordered Sets
<
a < b if a  b and a  b
Lexicographic (dictionary) order
Let (A, ) and (B, ) be posets. Then defined as (a, b) (a’, b’) iff a < a’ or a = a’ and b  b’ is a partial order called the lexicographic or dictionary order.

7. §6.1 – Partially Ordered Sets
Theorem 6.1.2
The digraph of a partial order has no cycle of length greater than 1

8. §6.1 – Partially Ordered Sets
Hasse Diagram for (A, )
i) Draw digraph of 
ii) Delete all cycles of length 1
iii) Delete all edges implied by transitive property
iv) Draw diagram with all edges pointing up and omit any arrows
v) Replace circles with labeled points
Hasse diagram gives a visual representation with all the implied components removed

9. §6.1 – Partially Ordered Sets
Topological Sorting
Linear order that is an extension of a partial order
Typical notation:
Many topological sortings may exist for a given partial order

10. §6.1 – Partially Ordered Sets
Let (A, ) and (B, ) be posets. Let f:AB. f is called an isomorphism if:
i) f is a 1-1 correspondence
ii) a1, a2  A, a1  a2 iff f(a1)  f(a2)
In this case, we say (A, ) and (B, ) are isomorphic posets.

11. §6.1 – Partially Ordered Sets
Theorem (Principle of correspondence)
Let (A, ) and (B, ) be finite posets and f:AB be a 1-1 correspondence. Let H be the Hasse diagram of (A, ). Then:
i) If f is an isomorphism and each label a of H is replaced by f(a), then H becomes a Hasse diagram for (B, ).
ii) If H becomes a Hasse diagram for (B, ) when each label a of H is replaced by f(a), then f is an isomorphism.

12. §6.2 Extremal Elements of Partially Ordered Sets
CSCI 115
§6.2
Extremal Elements of Partially Ordered Sets

13. §6.2 Extremal elements of posets
Maximal Element
aA is a maximal element of (A,R) if there does not exist cA s.t. a < c
Minimal Element
bA is a minimal element of (A,R) if there does not exist dA s.t. d < b
Theorem 6.2.1
Let (A,) be a poset with A finite and non-empty. Then A has at least one maximal element, and at least one minimal element.

14. §6.2 Extremal elements of posets
Procedure to find a topological sorting of a finite poset (A, ≤)
Declare an array called SORT the size of |A|
Choose a minimal element x of A
Make x the next element in SORT
Repeat steps 2 – 3 until A = {}

15. §6.2 Extremal elements of posets
Greatest Element (Unit Element: 1)
aA is a greatest element of (A,R) if xA x  a.
Least Element (Zero Element: 0)
bA is a least element of (A,R) if xA b  x.
Theorem 6.2.2
A poset has at most one greatest element, and at most one least element.

16. §6.2 Extremal elements of posets
Let (A, ) be a poset, with B A.
Upper Bound (UB)
aA is an upper bound of B if b  a bB.
Least Upper Bound (LUB)
aA is a least upper bound of B if a is an upper bound for B, and a  a’ whenever a’ is an upper bound of B.
Lower Bound (LB)
aA is a lower bound of B if a  b bB.
Greatest Lower Bound (GLB)
aA is a greatest lower bound of B if a is a lower bound for B, and a’  a whenever a’ is a lower bound of B.

17. §6.2 Extremal elements of posets
Theorem 6.2.3
Let (A, ) be a poset. Then a subset B of A has at most one LUB and at most one GLB.

18. §6.2 Extremal elements of posets
Theorem 6.2.4
Suppose (A, ) and (B, ) are isomorphic posets under f:AB. Then:
i) If a is a max (min) element of (A, ), then f(a) is a max (min) element of (B, ).
ii) If a is a greatest (least) element of (A, ), then f(a) is a greatest (least) element of (B, ).
iii) If a is an UB (LB, LUB, GLB) of (A, ), then f(a) is an UB (LB, LUB, GLB) of (B, ).
iv) If every subset of (A, ) has a LUB (GLB), then every subset of (B, ) has a LUB (GLB).

19. CSCI 115
§6.3
Lattices

20. §6.3 – Lattices
Lattice
Poset (L, ) where every subset of 2 elements has a LUB and GLB
Join of 2 elements
a  b = LUB ({a, b})
Meet of 2 elements
a  b = GLB ({a, b})

21. §6.3 – Lattices
Theorem 6.3.1
If (L1, 1) and (L2, 2) are lattices, then (L, ) is a lattice where L = L1 x L2 and  is the product partial order
Let (L, ) be a lattice. A non-empty subset S of L is called a sublattice of L if a  b  S and a  b  S  a, b  S

22. §6.3 – Lattices Isomorphic Lattices
If f:L1  L2 is an isomorphism from the poset (L1, 1) to the poset (L2, 2), and if L1 and L2 are Lattices, then L1 and L2 are isomorphic lattices.

23. §6.3 – Lattices Theorem 6.3.2 Theorem 6.3.3 – 6.3.7 in book
Let L be a lattice.  a, b  L we have: i) a  b = b iff a  b ii) a  b = a iff a  b iii) a  b = a iff a  b = b
Theorem – in book
We will not cover special types of lattices
Bounded, distributive, complemented

24. §6.4 Finite Boolean Algebras
CSCI 115
§6.4
Finite Boolean Algebras

25. §6.4 – Finite Boolean Algebras
Theorem 6.4.1
If S1 = {x1, x2, …, xn} and S2 = {y1, y2, …, yn} are 2 finite sets with n elements, then the lattices (P(S1), ) and (P(S2), ) are isomorphic lattices. Consequently, the Hasse diagram of these lattices may be drawn identically.

26. §6.4 – Finite Boolean Algebras
If the Hasse diagram of a lattice corresponding to a set with n elements is labeled by a sequence of 0s and 1s of length n, then the resulting lattice is called Bn.

27. §6.4 – Finite Boolean Algebras
If x = a1a2…an and y = b1b2…bn are 2 elements of Bn, then the properties of Bn can be described by:
i) x  y iff ak  bk for k = 1, 2, 3, …, n
ii) x  y = c1c2…cn where ck = min{ak, bk}
iii) x  y = d1d2…dn where dk = max{ak, bk}

28. §6.4 – Finite Boolean Algebras
A finite lattice is called a Boolean Algebra if it is isomorphic to Bn for some nZ+
Theorem (modified)
Dn is a boolean algebra iff n = p1p2…pk where the pi are all distinct primes
Theorem and in book

29. §6.5 Functions on Boolean Algebras
CSCI 115
§6.5
Functions on Boolean Algebras

30. §6.5 – Fns on Boolean Algebras
Boolean Polynomials
Let x1, x2, …, xn be a set of n variables. A Boolean Polynomial p(x1, x2, …, xn) in the variables xk is defined by the following:
i) x1, x2, …, xn are all boolean polynomials
ii) 0 and 1 are boolean polynomials
iii) If p(x1, x2, …, xn) and q(x1, x2, …, xn) are both boolean polynomials in the variables xk, then p(x1, x2, …, xn)  q(x1, x2, …, xn) and p(x1, x2, …, xn)  q(x1, x2, …, xn) are also boolean polynomials
iv) If p(x1, x2, …, xn) is a boolean polynomial, then so is If p(x1, x2, …, xn)’
v) Only polynomials generated by rules 1 – 4 are boolean polynomials

31. §6.5 – Fns on Boolean Algebras
Manipulations
Not responsible for manipulations
Boolean Functions
Similar to polynomial functions
Accept arguments, and return values
Evaluates to true or false

32. §6.5 – Fns on Boolean Algebras
Schematic representations of boolean polynomials
Used in circuitry, and other technical areas
AND gates
OR gates
NOT inverters

33. §6.5 – Fns on Boolean Algebras
The AND gate
Accepts 2 arguments, and evaluates to true or false according to the logical rules for AND

34. §6.5 – Fns on Boolean Algebras
The OR gate
Accepts 2 arguments, and evaluates to true or false according to the logical rules for OR

35. §6.5 – Fns on Boolean Algebras
The NOT inverter
Accepts 1 argument, and evaluates to true or false according to the logical rules for NOT