Semigroup

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tags: - colorclass/functional analysis ---A semigroup is a basic algebraic structure that captures the essence of associativity in binary operations. It’s a foundational concept in abstract algebra and has important applications across mathematics, including in analysis, topology, and the theory of computation. Understanding semigroups helps in studying more complex structures like Groups, rings, and fields.

Definition

Formally, a semigroup $(S,\cdot )$ consists of a set $S$ equipped with a binary operation $\cdot :S\times S\to S$ that satisfies the following property:

- Associativity: For all $a,b,c\in S$, the equation $(a\cdot b)\cdot c=a\cdot (b\cdot c)$ holds.

Note that unlike a group, a semigroup does not require the existence of an identity element or inverses for its elements.

Examples

1. Natural numbers under addition: The set $\mathbb{N}$ (including 0) with the operation of addition is a semigroup because addition is associative. 2. Matrix multiplication: The set of all $n\times n$ matrices over a field (or ring) with the operation of matrix multiplication forms a semigroup. Matrix multiplication is associative but not all matrices have inverses. 3. Strings under concatenation: Given a set of characters, the set of all finite strings formed from these characters, including the empty string, under the operation of concatenation, is a semigroup. This example is particularly notable in computer science, especially in the theory of formal languages.

Properties

- Identity and Subsemigroups: If a semigroup has an element that acts as an identity element (meaning $e\cdot a=a\cdot e=a$ for all $a\in S$), it is called a monoid. A subset of a semigroup that is itself a semigroup under the same operation is known as a subsemigroup. - Commutativity: A semigroup is called commutative or abelian if its operation is commutative ($a\cdot b=b\cdot a$ for all $a,b\in S$).

Importance and Applications

- Algebraic Structures: Semigroups provide insight into the more general behavior of algebraic operations and serve as the stepping stones towards understanding more complex structures like groups and monoids. - Functional Analysis: In functional analysis, semigroups of operators are used to solve differential equations and to model dynamical systems. - Computer Science: Semigroups appear in the study of automata and formal languages, where the concatenation of strings or paths in a graph follows semigroup properties. - Topology: Semigroups are used in topological dynamics and the study of continuous transformation semigroups, providing tools for understanding the behavior of dynamical systems over time.

Semigroup theory thus serves as a foundational area of mathematics with deep theoretical significance and wide-ranging applications across disciplines.