# Differences

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+ | =====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}$ | ||

+ | |||

+ | ====Subclasses==== | ||

+ | [[Cancellative monoids]] | ||

+ | |||

+ | [[Commutative monoids]] | ||

+ | |||

+ | ====Superclasses==== | ||

+ | [[Semigroups]] | ||

+ | |||

+ | [[Partial monoids]] | ||

+ | |||

+ | |||

+ | ====References==== | ||

+ | |||

+ | [(Ln19xx> | ||

+ | )] |

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