different between semigroup vs idempotent
semigroup
English
Etymology
From semi- +? group, reflecting the fact that not all the conditions required for a group are required for a semigroup. (Specifically, the requirements for the existence of identity and inverse elements are omitted.)
Noun
semigroup (plural semigroups)
- (mathematics) Any set for which there is a binary operation that is closed and associative.
- 1961, Alfred Hoblitzelle Clifford, G. B. Preston, The Algebraic Theory of Semigroups (page 70)
- If a semigroup S contains a zeroid, then every left zeroid is also a right zeroid, and vice versa, and the set K of all the zeroids of S is the kernel of S.
- 1988, A. Ya A?zenshtat, Boris M. Schein (translator), On Ideals of Semigroups of Endomorphisms, Ben Silver (editor), Nineteen Papers on Algebraic Semigroups, American Mathematical Society Translations, Series 2, Volume 139, page 11,
- It follows naturally that various classes of ordered sets can be characterized by semigroup properties of endomorphism semigroups.
- 2012, Jorge Almeida, Benjamin Steinberg, Syntactic and Global Subgroup Theory: A Synthesis Approach, Jean-Camille Birget, Stuart Margolis, John Meakin, Mark V. Sapir (editors), Algorithmic Problems in Groups and Semigroups, page 5,
- If one considers the variety of semigroups, one has the binary operation of multiplication defined on every semigroup.
- 1961, Alfred Hoblitzelle Clifford, G. B. Preston, The Algebraic Theory of Semigroups (page 70)
Hypernyms
- (set for which a closed associative binary operation is defined): magma
Hyponyms
- (set for which a closed associative binary operation is defined): group, monoid
Derived terms
- semigroup homomorphism
Translations
semigroup From the web:
idempotent
English
Etymology
Latin roots, idem (“same”) +? potent (“having power”) – literally, “having the same power”.
Coined 1870 by American mathematician Benjamin Peirce in context of algebra.
Pronunciation
- (US) IPA(key): /a?.d?m?po?.t?nt/, /?.d?m?po?.t?nt/
Adjective
idempotent (not comparable)
- (mathematics, computing) Said of a function: describing an action which, when performed multiple times on the same subject, has no further effect on its subject after the first time it is performed.
- A projection operator is idempotent.
- (mathematics) Said of an element of an algebraic structure with a binary operation (such as a group or semigroup): when the element operates on itself, the result is equal to itself.
- Every finite semigroup has an idempotent element.
- Every group has a unique idempotent element: namely, its identity element.
- (mathematics) Said of a binary operation: such that all of the distinct elements it can operate on are idempotent (in the sense given just above).
- Since the AND logical operator is commutative, associative, and idempotent, then it distributes with respect to itself.
- (mathematics) Said of an algebraic structure: having an idempotent operation (in the sense above).
Usage notes
See the Usage notes section of nullipotent.
Coordinate terms
- nilpotent
- nullipotent
Related terms
- idempotence
- nilpotent
- nullipotent
- unipotent
Translations
Noun
idempotent (plural idempotents)
- (mathematics) An idempotent element.
- (mathematics) An idempotent structure.
References
- “idempotent” at FOLDOC
German
Pronunciation
Adjective
idempotent
- idempotent
Swedish
Adjective
idempotent
- idempotent
Turkish
Adjective
idempotent
- idempotent
idempotent From the web:
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