Definition:Semigroup

Definition

Let $\left({S, \circ}\right)$ be a magma.

Then $\left({S, \circ}\right)$ is a semigroup iff $\circ$ is associative on $S$.

That is, a semigroup is an algebraic structure which is closed and associative.

Also known as

Some older texts have this as semi-group.

Warning

Some sources call this a monoid, but this term usually has a more precise meaning.

Make sure you understand which is being used.

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Chapter $\text{I}$: $\S 1$: Definition $1$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 4.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 7$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$: Definitions $1.1 \ \text{(a)}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 26 \alpha$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 29$
• John F. Humphreys: A Course in Group Theory (1996): $\S 3$: Remark