M-sets

Abbreviation: MSet

Definition

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 $m$}.

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

Definition

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:

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 variety

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

[[G-sets]]
[[R-modules]]

Superclasses

[[Unary algebras]]