g-sets - MathStructures

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=====G-sets=====

Abbreviation: **Gset**

====Definition====
A \emph{G-set} is a structure $\mathbf{A}=\langle A,f_g (g\in G)\rangle$, where $\langle G, \cdot, ^{-1}, 1\rangle$ is a [[group]], such that

$f_1$ is the identity map:  $1x=x$ and

the group action associates:  $(g\cdot h)x=g(hx)$

Remark: 
$f_g(x)=gx$ is a unary operation called \emph{the group action by $g$}.

If follows from the associativity that
$f_{g^{-1}}$ is the inverse function of $f_g$.

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It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: 
$h(x ... y)=h(x) ... h(y)$

====Definition====
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle
...\rangle$ such that

$...$ is ...:  $axiom$
  
$...$ is ...:  $axiom$

====Examples====
Example 1: 

====Basic results====


====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

^[[Classtype]]                        |(value, see description) [(Ln19xx)]  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

$\begin{array}{lr}
  f(1)= &1\\
  f(2)= &\\
  f(3)= &\\
  f(4)= &\\
  f(5)= &\\
\end{array}$     
$\begin{array}{lr}
  f(6)= &\\
  f(7)= &\\
  f(8)= &\\
  f(9)= &\\
  f(10)= &\\
\end{array}$


====Subclasses====
  [[...]] subvariety

  [[...]] expansion


====Superclasses====
  [[...]] supervariety

  [[...]] subreduct


====References====

[(Ln19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]] 
)]

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