different between semigroup vs idempotent

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)

1. (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.

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

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)

1. (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.

2. (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.

3. (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.

4. (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)

1. (mathematics) An idempotent element.
2. (mathematics) An idempotent structure.

References

• “idempotent” at FOLDOC

German

Pronunciation

Adjective

idempotent

1. idempotent

Swedish

Adjective

idempotent

1. idempotent

Turkish

Adjective

idempotent

1. idempotent