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monoids [2010/07/29 18:30] (current)
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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>
 +)]