A partially ordered set is a set equipped with a partial order relation.

What are the axioms of a non-strict partial order?

Reflexivity, antisymmetry, and transitivity.

How does a partial order differ from a total order?

A partial order may contain incomparable pairs; in a total order every pair is comparable.

How are strict and non-strict orders converted?

Define $a<b$ by $a\leq b$ and $a\neq b$; define $a\leq b$ by $a<b$ or $a=b$.

What is the dual of an order relation?

The dual reverses comparisons: $aR^{\mathrm{op}}b$ exactly when $bRa$.

Discrete mathematicsLesson 5 of 5 in the course Set theory and relations

Partial Orders: Axioms, Structure, Maps, and Intervals

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

A partial order describes when one element precedes another, while allowing some pairs to remain incomparable. This lesson introduces partially ordered sets, their defining properties, and the relationship between strict and non-strict orders.

A partially ordered set, or poset, is a pair $(P,\leq)$ consisting of a set $P$ and a partial order relation on its elements. The set is the ground set; the relation tells us which elements precede others.

All flashcards

What is a poset?: A partially ordered set is a set equipped with a partial order relation.
What are the axioms of a non-strict partial order?: Reflexivity, antisymmetry, and transitivity.
How does a partial order differ from a total order?: A partial order may contain incomparable pairs; in a total order every pair is comparable.
How are strict and non-strict orders converted?: Define $a<b$ by $a\leq b$ and $a\neq b$ ; define $a\leq b$ by $a<b$ or $a=b$ .
What is the dual of an order relation?: The dual reverses comparisons: $aR^{\mathrm{op}}b$ exactly when $bRa$ .

Quiz questions

Which properties must a non-strict partial order satisfy?Answer: Reflexivity, antisymmetry, and transitivity.
What does an upward edge in a Hasse diagram represent?Answer: A cover relation, with no element strictly between its endpoints.
When does $(a,b)\leq(c,d)$ hold in the product order on two posets?Answer: When $a\leq c$ and $b\leq d$ .
What is a map that is both order-preserving and order-reflecting called?Answer: An order-embedding.
For a finite poset represented by a directed acyclic graph, what does topological sorting find?Answer: A total order consistent with every existing comparison.

Based on the Wikipedia article Partially ordered set, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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