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semigroup (Definition)

A semigroup $ G$ is a set together with a binary operation $ \cdot: G \times G \longrightarrow G$ which satisfies the associative property: $ (a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $ a,b,c \in G$.

The set $ G$ is not required to be nonempty.

Let $ G,H$ be two semigroups. A semigroup homomorphism from $ G$ to $ H$ is a function $ f:G\to H$ such that $ f(ab)=f(a)f(b)$.



"semigroup" is owned by djao. [ full author list (2) ]
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See Also: groupoid, band, subsemigroup, submonoid, and subgroup, null semigroup, zero elements, monoid

Other names:  homomorphism
Also defines:  semigroup homomorphism

Attachments:
regular semigroup (Definition) by yark
cancellative semigroup (Definition) by yark
semigroup with two elements (Example) by rspuzio
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Cross-references: function, property, associative, binary operation
There are 58 references to this entry.

This is version 6 of semigroup, born on 2001-10-19, modified 2008-05-13.
Object id is 388, canonical name is Semigroup.
Accessed 11266 times total.

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