1. CSCI 115 Chapter 6 Order Relations and Structures

2. 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)  a 1, a 2  A, a 1  a 2 iff f(a 1 )  f(a 2 ) In this case, we say (A,  ) and (B,  ) are isomorphic posets.

11. §6.1 – Partially Ordered Sets Theorem 6.1.3 (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. 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, ≤) 1.Declare an array called SORT the size of |A| 2.Choose a minimal element x of A and remove x from A 3.Make x the next element in SORT 4.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 (L 1,  1 ) and (L 2,  2 ) are lattices, then (L,  ) is a lattice where L = L 1 x L 2 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:L 1  L 2 is an isomorphism from the poset (L 1,  1 ) to the poset (L 2,  2 ), and if L 1 and L 2 are Lattices, then L 1 and L 2 are isomorphic lattices.

23. §6.3 – Lattices Theorem 6.3.2 –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 6.3.3 – 6.3.7 in book We will not cover special types of lattices –Bounded, distributive, complemented

24. CSCI 115 §6.4 Finite Boolean Algebras

25. §6.4 – Finite Boolean Algebras Theorem 6.4.1 –If S 1 = {x 1, x 2, …, x n } and S 2 = {y 1, y 2, …, y n } are 2 finite sets with n elements, then the lattices (P(S 1 ),  ) and (P(S 2 ),  ) 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 B n.

27. §6.4 – Finite Boolean Algebras If x = a 1 a 2 … a n and y = b 1 b 2 … b n are 2 elements of B n, then the properties of B n can be described by: –i) x  y iff a k  b k for k = 1, 2, 3, …, n –ii) x  y = c 1 c 2 … c n where c k = min{a k, b k } –iii) x  y = d 1 d 2 … d n where d k = max{a k, b k }

28. §6.4 – Finite Boolean Algebras A finite lattice is called a Boolean Algebra if it is isomorphic to B n for some n  Z + Theorem 6.4.2 (modified) –D n is a boolean algebra iff n = p 1 p 2 … p k where the p i are all distinct primes Theorem 6.4.3 and 6.4.4 in book

29. CSCI 115 §6.5 Functions on Boolean Algebras

30. §6.5 – Fns on Boolean Algebras Boolean Polynomials –Let x 1, x 2, …, x n be a set of n variables. A Boolean Polynomial p(x 1, x 2, …, x n ) in the variables x k is defined by the following: i) x 1, x 2, …, x n are all boolean polynomials ii) 0 and 1 are boolean polynomials iii) If p(x 1, x 2, …, x n ) and q(x 1, x 2, …, x n ) are both boolean polynomials in the variables x k, then p(x 1, x 2, …, x n )  q(x 1, x 2, …, x n ) and p(x 1, x 2, …, x n )  q(x 1, x 2, …, x n ) are also boolean polynomials iv) If p(x 1, x 2, …, x n ) is a boolean polynomial, then so is If p(x 1, x 2, …, x n )’ 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