Semigroup - Wikipedia

Abstraction: Algebraic structure with associative binary operation only

Key points:

• A semigroup is a set with a single associative binary operation; unlike a group it need not have an identity or inverses.
• A monoid is a semigroup with an identity element; a group is a monoid with inverses — the hierarchy is semigroup ⊂ monoid ⊂ group.
• Finite semigroup theory links to finite automata via the syntactic monoid; Krohn–Rhodes theory decomposes finite semigroups analogously to Jordan–Hölder for groups.
• In functional analysis, one-parameter C₀-semigroups model time-evolution of PDEs (e.g., exp(tA) for the heat equation on a Hilbert space).
• Green's relations — five equivalence relations on ideals — are fundamental tools not present in group theory.
• Anatoly Maltsev (1937) gave necessary and sufficient conditions for embedding a semigroup into a group.

Connections: Anton Sushkevich · Anatoly Maltsev · Semigroup · Abstract Algebra · Operator Semigroup

Source: http://en.wikipedia.org/wiki/Semigroup