CSCI 115

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) 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 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, ≤) • Declare an array called SORT the size of |A| • Choose a minimal element x of A and remove x from 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 • Let L be a lattice.  a, b  L we have:i) a  b = b iff a  bii) a  b = a iff a  biii) 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 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 6.4.2 (modified) • Dn is a boolean algebra iff n = p1p2…pk where the pi 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 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