# Differences

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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]] | ||

+ | )] | ||

+ | |||

+ | |||

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