### Table of Contents

## Monoids

Abbreviation: **Mon**

### Definition

A \emph{monoid} is a structure $\mathbf{M}=\langle M,\cdot ,e\rangle $, where $\cdot $ is an infix binary operation, called the \emph{monoid product}, and $e$ is a constant (nullary operation), called the \emph{identity element} , such that

$\cdot $ is associative: $(x\cdot y)\cdot z=x\cdot (y\cdot z)$

$e$ is an identity for $\cdot $: $e\cdot x=x$, $x\cdot e=x$.

##### Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be monoids. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:Marrow N$ that is a homomorphism:

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$

### Examples

Example 1: $\langle X^{X},\circ ,id_{X}\rangle $, the collection of functions on a sets $X$, with composition, and identity map.

Example 1: $\langle M(V)_{n},\cdot ,I_{n}\rangle $, the collection of $n\times n$ matrices over a vector space $V$, with matrix multiplication and identity matrix.

Example 1: $\langle \Sigma ^{\ast },\cdot ,\lambda \rangle $, the collection of strings over a set $\Sigma $, with concatenation and the empty string. This is the free monoid generated by $\Sigma $.

### Basic results

### Properties

Classtype | Variety |
---|---|

Equational theory | decidable in polynomial time |

Quasiequational theory | undecidable |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | no |

Congruence modular | no |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | |

Definable principal congruences | |

Equationally def. pr. cong. | no |

Amalgamation property | no |

Strong amalgamation property | no |

Epimorphisms are surjective | no |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &2

f(3)= &7

f(4)= &35

f(5)= &228

f(6)= &2237

f(7)= &31559

\end{array}$