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monoid (Definition)

A monoid is a semigroup $ G$ which contains an identity element; that is, there exists an element $ e \in G$ such that $ e \cdot a = a \cdot e = a$ for all $ a \in G$.

If $ e$ and $ f$ are identity elements of a monoid $ G$, then $ e=e\cdot f=f\cdot e=f$, so we may speak of “the” identity element of $ G$.

A monoid homomorphism from monoids $ G$ to $ H$ is a semigroup homomorphism $ f:G\to H$ such that $ f(e_G)=e_H$, where $ e_G,e_H$ are identity elements of $ G$ and $ H$ respectively.



"monoid" is owned by djao. [ full author list (2) ]
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See Also: semigroup

Other names:  homomorphism
Also defines:  monoid homomorphism

Attachments:
identity element is unique (Theorem) by pahio
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Cross-references: semigroup homomorphism, identity element, contains
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This is version 3 of monoid, born on 2001-10-19, modified 2008-05-13.
Object id is 389, canonical name is Monoid.
Accessed 11080 times total.

Classification:
AMS MSC: 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
Pending Errata and Addenda
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