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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]]
		+	)]
		+
		+