hass poset diagrams

BITS PilaniPilani Campus


Discrete Structures for Computer Science(CS F222/IS F222)

SK Hafizul Islam, Ph.D

[email protected]


Lecture No. –15Date –07/09/2015

Time –2:00 PM –3:00 PM

BITS Pilani, Pilani Campus

POSET


• Consider the renovation of a hostel. In this process several tasks were undertaken

– Remove Asbestos

– Replace windows

– Paint walls

– Refinish floors

– Assign offices

– Move in office furniture

Introduction

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CS F222/IS F222 8 September 2015


• Clearly, some things had to be done before others could begin– Asbestos had to be removed before anything (except assigning offices)

– Painting walls had to be done before refinishing floors to avoid ruining them, etc.

• On the other hand, several things could be done concurrently:– Painting could be done while replacing the windows

– Assigning offices could be done at anytime before moving in office furniture

• This scenario can be nicely modeled using partial orderings

Introduction

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• Definitions:

– A relation R on a set A is called a partial order if it is

• Reflexive [for any a A, (a, a) R]

• Antisymmetric [For a, bA, if (a, b)R and (b, a)R, then a = b]

• Transitive [For a, b, cA, if (a, b)R and (b, c)R, then (a, c)R]

– The set A together with a partial order relation R is called a partially ordered set (POSET).

Partial Order

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• Partial orderings are used to give an order to setsthat may not have a natural one

• In our renovation example, we could define anordering such that (a, b)R if “a must be donebefore b can be done”.

Partial Order

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• A POSET is denoted by (A p).

• We use the notation:

– apb, when (a, b)R and ab [a strictly precedes b]

• The notation p is not “less than”.

• The notation p is used to denote any partial ordering

POSET

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• The relation “less than or equal to” defined over aset of real numbers is a POSET, i.e., (R, ) is a POSET.

• Which of the following sets with the given relationsfrom POSETs?

– (Z, ≥)

– (Z, >)

– (P, ), where P = {, {a}, {b}, {a, b}}

– (P, )

– (Z, |), where I is the divisibility relation

– (Z+, I)

POSET

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Cover:

• Let (A, p) be a POSET and a, bA. The element b iscalled the cover of a if apb and no cA exists suchthat apcpb.

– b is the cover of a if b is the immediate successor of a.

• Example: A = {2, 3, 6, 12, 18, 36} with the partialorder relation “divides”.– 2p6, 6 is the cover of 2, but 3 is not the cover of 2.

– 36 is not a cover of 6, since 6p12p36 or 6p18p36.

Hasses Diagram

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Covering relation:

• For a finite poset (A, p), we can find the cover of

every element in A.

• The set of pairs (a, b) such that “b covers a” is calledthe covering relation of (A, p).

• Example: A = {2, 3, 6, 12, 18, 36} with the partialorder relation “divides”.– The covering relation of (A, p) = {(2, 6), (3, 6), (6, 12), (6,

18), (12, 36), (18, 36)}.

Hasses Diagram

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• The Hasses diagram of a finite poset (A, p) is a graph

in which

– The elements are represented as vertices

– If b is a cover of a, then this relation is shown by placing bhigher than a and providing a edge between them.

Hasses Diagram

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• A = {2, 3, 6, 12, 18, 36} with the partial order relation“divides”.– The covering relation of (A, p) = {(2, 6), (3, 6), (6, 12), (6, 18),

(12, 36), (18, 36)}.

Hasses Diagram

36

12 18

6

2 3

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Thank You

hass poset diagrams

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