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ordered_abelian_groups [2014/09/13 10:04] (current)
jipsen created
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 +=====Ordered abelian groups=====
 +
 +Abbreviation: **OGrp**
 +
 +====Definition====
 +An \emph{ordered abelian group} is an [[ordered group]] $\mathbf{G}=\langle G,+,-,0,\le\rangle$ such that
 +
 +$+$ is \emph{commutative}:  $x+y=y+x$
 +
 +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 ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a orderpreserving homomorphism:
 +$h(x + y)=h(x) + h(y)$,
 +$x\le y\Longrightarrow h(x)\le h(y)$.
 +
 +====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]]                        |universal  |
 +^[[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====
 +one-element group
 +
 +====Subclasses====
 +  [[Abelian ordered groups]]
 +
 +
 +====Superclasses====
 +  [[Partially ordered groups]]
 +
 +  [[Ordered monoids]] reduced type
 +
 +
 +====References====
 +
 +[(Lastname19xx>
 +F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]
 +)]
 +
 +