M-sets

Abbreviation: MSet

Definition

An $\mathbf M$-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 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.

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 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

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

Superclasses

[[Unary algebras]]

References