G-sets

Abbreviation: Gset

Definition

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

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

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


1) F. Lastname, Title, Journal, 1, 23–45 MRreview