partially ordered set: meaning, definition, pronunciation and examples
C2Technical/Academic
Quick answer
What does “partially ordered set” mean?
A set equipped with a partial order, which is a binary relation that is reflexive, antisymmetric, and transitive, but does not necessarily compare every pair of elements.
Audio
Pronunciation
Definition
Meaning and Definition
A set equipped with a partial order, which is a binary relation that is reflexive, antisymmetric, and transitive, but does not necessarily compare every pair of elements.
A fundamental structure in order theory and discrete mathematics, representing a collection of objects where some objects are comparable (one precedes another) and others may be incomparable. Also called a poset.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or spelling differences. Both use the identical term.
Connotations
Identical technical connotations in mathematics and computer science.
Frequency
Equally low-frequency outside specialised fields in both regions.
Grammar
How to Use “partially ordered set” in a Sentence
[be] a partially ordered set[form/constitute] a partially ordered set[endow/equip] a set [with] a partial order[refer to] as a partially ordered setVocabulary
Collocations
Examples
Examples of “partially ordered set” in a Sentence
adjective
British English
- The partially-ordered-set structure is crucial for the proof.
- We need a partially ordered set model.
American English
- The partially ordered set structure is crucial for the proof.
- We need a poset model.
Usage
Meaning in Context
Business
Never used.
Academic
Core term in order theory, lattice theory, discrete mathematics, and theoretical computer science.
Everyday
Virtually never used.
Technical
Precisely defined foundational concept in mathematics, logic, and computer science.
Vocabulary
Synonyms of “partially ordered set”
Neutral
Weak
Vocabulary
Antonyms of “partially ordered set”
Watch out
Common Mistakes When Using “partially ordered set”
- Using 'partially ordered set' to mean a set that is 'somewhat' or 'incompletely' ordered in a non-mathematical sense.
- Confusing it with a 'totally ordered set' where every pair is comparable.
- Omitting the hyphen when using it as a compound adjective (e.g., 'partially-ordered-set properties' is less common than 'properties of a partially ordered set').
FAQ
Frequently Asked Questions
It means that not every pair of distinct elements in the set is comparable. Some elements may be unrelated by the order relation.
The standard shorthand is 'poset'.
Yes. A totally ordered set is a special case of a partially ordered set where every pair of elements is comparable.
Primarily in advanced mathematics (especially order theory, lattice theory), theoretical computer science (domain theory, semantics), and occasionally in formal logic or philosophy.
A set equipped with a partial order, which is a binary relation that is reflexive, antisymmetric, and transitive, but does not necessarily compare every pair of elements.
Partially ordered set is usually technical/academic in register.
Partially ordered set: in British English it is pronounced /ˈpɑːʃəli ˈɔːdəd sɛt/, and in American English it is pronounced /ˈpɑːrʃəli ˈɔːrdərd sɛt/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a corporate hierarchy: the CEO is above managers, managers are above staff, but two managers in different departments may be INcomparable – that's a PARTIAL order.
Conceptual Metaphor
A HIERARCHY WITH SIDE BRANCHES (not a simple ladder).
Practice
Quiz
Which of the following is a key property of a partially ordered set?