semigroup: meaning, definition, pronunciation and examples
Very LowHighly Technical / Academic
Quick answer
What does “semigroup” mean?
A set equipped with a binary associative operation but not necessarily possessing an identity element or inverses.
Audio
Pronunciation
Definition
Meaning and Definition
A set equipped with a binary associative operation but not necessarily possessing an identity element or inverses.
In abstract algebra and theoretical computer science, an algebraic structure fundamental to the study of formal languages, automata theory, and semigroup theory. It can model sequential processes where closure and associativity hold.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or usage differences. Minor variations may exist in preferred textbooks or notation.
Connotations
Purely mathematical/technical. No cultural connotations.
Frequency
Exclusively used in advanced mathematics and theoretical computer science contexts in both regions.
Grammar
How to Use “semigroup” in a Sentence
The [adjective] semigroupA semigroup [verb phrase, e.g., 'under concatenation']Semigroup of [noun, e.g., 'transformations']Vocabulary
Collocations
Examples
Examples of “semigroup” in a Sentence
adjective
British English
- The semigroup properties are fundamental to the proof.
- This leads to a semigroup-theoretic approach.
American English
- The semigroup properties are fundamental to the proof.
- This yields a semigroup-theoretic approach.
Usage
Meaning in Context
Business
Not used.
Academic
Primary context. Used in advanced courses in pure mathematics (abstract algebra) and theoretical computer science (automata, formal languages).
Everyday
Never used.
Technical
Core term in specific technical fields as above.
Vocabulary
Synonyms of “semigroup”
Neutral
Vocabulary
Antonyms of “semigroup”
Watch out
Common Mistakes When Using “semigroup”
- Confusing it with a 'monoid' (which has an identity) or a 'group'.
- Assuming the operation is commutative (it need not be).
- Using it in non-mathematical contexts.
FAQ
Frequently Asked Questions
A group requires all semigroup properties (closure, associativity) plus an identity element and inverse elements for every element. A semigroup lacks these last two requirements.
Primarily in theoretical domains: the foundation of finite state automata in computer science, in the study of partial differential equations (via operator semigroups), and in abstract algebra.
No. A monoid is a semigroup *with* an identity element. Therefore, all monoids are semigroups, but not all semigroups are monoids.
The set of positive integers {1, 2, 3, ...} under addition. Adding any two gives another positive integer (closure), addition is associative, but there is no identity element (0 is not in the set), so it's a semigroup, not a monoid or group.
Semigroup is usually highly technical / academic in register.
Semigroup: in British English it is pronounced /ˈsɛmɪɡruːp/, and in American English it is pronounced /ˈsɛmɪˌɡrup/ ˈsɛmaɪˌɡrup/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think 'SEMI-group' – it's only HALFway to being a full 'group' because it lacks an identity and inverses, but it keeps associativity.
Conceptual Metaphor
A machine or process that combines two items into one, always producing a valid result from the same set, and where the grouping of operations doesn't matter, but you can't 'undo' steps or start from a neutral state.
Practice
Quiz
Which of the following is a necessary property of a semigroup?