CSCI 115

Chapter 6

Order Relations and Structures

CSCI 115

§6.1

Partially Ordered Sets

§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).

§6.1 – Partially Ordered Sets

• Dual• • Comparable• Linear order (chain)

§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

§6.1 – 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.

§6.1 – Partially Ordered Sets

• Theorem 6.1.2– The digraph of a partial order has no cycle of

length greater than 1

§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

§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

§6.1 – Partially Ordered Sets

• Let (A, ) and (B, ) be posets. Let f:AB. 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.

§6.1 – Partially Ordered Sets

• Theorem 6.1.3 (Principle of correspondence)– Let (A, ) and (B, ) be finite posets and f:AB 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.

CSCI 115

§6.2

Extremal Elements of Partially Ordered Sets

§6.2 Extremal elements of posets

• Maximal Element– aA is a maximal element of (A,R) if there does not

exist cA s.t. a < c• Minimal Element

– bA is a minimal element of (A,R) if there does not exist dA 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.

§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 A3. Make x the next element in SORT4. Repeat steps 2 – 3 until A = {}

§6.2 Extremal elements of posets

• Greatest Element (Unit Element: 1)– aA is a greatest element of (A,R) if xA x a.

• Least Element (Zero Element: 0)– bA is a least element of (A,R) if xA b x.

• Theorem 6.2.2– A poset has at most one greatest element, and at

most one least element.

§6.2 Extremal elements of posets

• Let (A, ) be a poset, with B A.– Upper Bound (UB)

• aA is an upper bound of B if b a bB.– Least Upper Bound (LUB)

• aA 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)• aA is a lower bound of B if a b bB.

– Greatest Lower Bound (GLB)• aA 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.

§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.

§6.2 Extremal elements of posets

• Theorem 6.2.4– Suppose (A, ) and (B, ) are isomorphic posets under

f:AB. 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).

CSCI 115

§6.3

Lattices

§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})

§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

§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.

§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

CSCI 115

§6.4

Finite Boolean Algebras

§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.

§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.

§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}

§6.4 – Finite Boolean Algebras

• A finite lattice is called a Boolean Algebra if it is isomorphic to Bn for some nZ+

• 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

CSCI 115

§6.5

Functions on Boolean Algebras

§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

§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

§6.5 – Fns on Boolean Algebras

• Schematic representations of boolean polynomials– Used in circuitry, and other technical areas– AND gates– OR gates– NOT inverters

§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

§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

§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