Abbreviation: MSet

An \emph{$\mathbf M$-set} is a structure $\mathbf{A}=\langle A,f_m (m\in M)\rangle$, where $\mathbf M=\langle M,\cdot,1\rangle$ is a monoid, such that

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

the monoid action associates: $(m\cdot n)x=m(nx)$

Remark: $f_m(x)=mx$ is a unary operation called \emph{the monoid action by $m$}.

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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.

Let $\mathbf{A}$ and $\mathbf{B}$ be $\mathbf M$-sets. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(f_m^{\mathbf A}(x))=f_m^{mathbf B}(h(x))$.

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

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

Example 1:

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.

$\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}$

[[G-sets]]

[[R-modules]]

[[Unary algebras]]