# Category:Semigroups

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This category contains results about **Semigroups**.

Definitions specific to this category can be found in Definitions/Semigroups.

- A
**semigroup**is an algebraic structure which is closed and whose operation is associative.

## Subcategories

This category has the following 28 subcategories, out of 28 total.

### C

- Constant Operation (6 P)

### E

### G

- Graded Rings (empty)

### I

- Inverse Semigroups (empty)

### L

### M

### O

### P

### S

- Semigroup Automorphisms (4 P)

## Pages in category "Semigroups"

The following 70 pages are in this category, out of 70 total.

### C

- Cancellable Finite Semigroup is Group
- Cancellable Infinite Semigroup is not necessarily Group
- Commutative Semigroup is Entropic Structure
- Commutativity of Powers in Semigroup
- Composition of Left Regular Representation with Right
- Composition of Left Regular Representations
- Composition of Regular Representations
- Composition of Right Regular Representations
- Condition for Group given Semigroup with Idempotent Element
- Condition for Semigroup to be Internal Direct Product of Subgroup and Subsemigroup with Right Operation
- Construction of Inverse Completion

### E

- Element Commutes with Product of Commuting Elements
- Element Commutes with Square in Semigroup
- Element has Idempotent Power in Finite Semigroup
- Epimorphism Preserves Semigroups
- Extension Theorem for Distributive Operations
- Extension Theorem for Homomorphisms
- Extension Theorem for Isomorphisms
- Extension Theorem for Total Orderings
- External Direct Product of Semigroups

### I

- Identity Property in Semigroup
- Index Laws for Semigroup
- Index Laws for Semigroup/Product of Indices
- Index Laws for Semigroup/Sum of Indices
- Index Laws/Product of Indices/Semigroup
- Index Laws/Sum of Indices/Semigroup
- Inverse Completion is Unique
- Inverse Completion Theorem
- Isomorphism Preserves Semigroups

### L

### P

- Power of Element of Semigroup
- Power of Element/Semigroup
- Power of Product of Commutative Elements in Semigroup
- Power of Product of Commuting Elements in Semigroup equals Product of Powers
- Power Structure of Group is Semigroup
- Power Structure of Semigroup is Semigroup
- Powers of Commutative Elements in Semigroups
- Powers of Commuting Elements of Semigroup Commute
- Powers of Semigroup Element Commute
- Product is Left Identity Therefore Left Cancellable
- Product is Right Identity Therefore Right Cancellable
- Product of Commuting Idempotent Elements is Idempotent
- Product of Semigroup Element with Left Inverse is Idempotent
- Product of Semigroup Element with Right Inverse is Idempotent

### R

- Real Numbers under Subtraction do not form Semigroup
- Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection
- Regular Representations in Semigroup are Permutations then Structure is Group
- Regular Representations wrt Element are Permutations then Element is Invertible
- Restriction of Associative Operation is Associative
- Right Identity in Semigroup may not be Unique
- Right Identity while exists Right Inverse for All is Identity
- Right Inverse for All is Left Inverse
- Right Regular Representation wrt Right Cancellable Element on Finite Semigroup is Bijection
- Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors

### S

- Semigroup is Group Iff Latin Square Property Holds
- Semigroup is Subsemigroup of Itself
- Set of Normal Subgroups of Group is Subsemigroup of Power Set Semigroup
- Set of Normal Subgroups of Group is Subsemigroup of Power Set under Intersection
- Structure Induced by Permutation on Semigroup is not necessarily Semigroup
- Structure Induced by Semigroup Operation is Semigroup
- Structure is Group iff Semigroup and Quasigroup
- Subset Product within Semigroup is Associative