# Differences

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m-sets [2010/07/29 15:46] (current)
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+=====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 $m$}.
+
+This is a template.
+
+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 $\mathbf{A}=\langle A,...\rangle$ 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]]                        |variety  |
+^[[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====
+  [[G-sets]]
+
+  [[R-modules]]
+
+
+====Superclasses====
+  [[Unary algebras]]
+
+
+====References====
+
+[(Ln19xx>
+F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
+)]
+
+