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abelian_partially_ordered_groups [2010/07/29 14:43]
jipsen created
abelian_partially_ordered_groups [2010/07/29 14:44] (current)
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-h+=====Abelian partially ordered groups===== 
 + 
 +Abbreviation: **APoGrp** 
 + 
 +====Definition==== 
 +An \emph{abelian partially ordered group} is a [[partially ordered group]] $\mathbf{A}=\langle A,+,-,0,\le\rangle$ such that 
 + 
 +$\cdot$ is \emph{commutative}:  $xy=yx$ 
 + 
 +Remark: This is a template. 
 +If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page. 
 + 
 +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==== 
 +A \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]]                        |quasivariety [(Lastname19xx)]  | 
 +^[[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==== 
 +  [[Abelian lattice-ordered groups]] expanded type 
 + 
 + 
 +====Superclasses==== 
 +  [[Partially ordered groups]] 
 + 
 +  [[Abelian groups]] reduced type 
 + 
 + 
 +====References==== 
 + 
 +[(Lastname19xx> 
 +F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]  
 +)] 
 + 
 +