abelian_ordered_groups - MathStructures

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=====Abelian ordered groups=====

Abbreviation: **AoGrp**


====Definition====
An \emph{abelian ordered group} is an [[ordered group]] $\mathbf{A}=\langle A,+,-,0,\le\rangle$ such that

$+$ is commutative: $x+y=y+x$

==Morphisms==
Let $\mathbf{A}$ and $\mathbf{B}$ be abelian ordered groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is an orderpreserving homomorphism: 
$h(x + y)=h(x) + h(y)$ and $x\le y\Longrightarrow h(x)\le h(y)$.


====Examples====
Example 1: $\langle\mathbb Z,+,-,0,\le\rangle$, the integers with the usual ordering.


====Basic results====
Every ordered group with more than one element is infinite.


====Properties====
^[[Classtype]]                        |universal |
^[[Equational theory]]                |decidable |
^[[Quasiequational theory]]           |decidable |
^[[First-order theory]]               | |
^[[Locally finite]]                   |no |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |


====Finite members====
None

====Subclasses====


====Superclasses====
[[Ordered groups]]

[[Abelian lattice-ordered groups]]


====References====
[(Lastname19xx>
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45
)]

Trace: • topological_groups • commutative_regular_rings • lattice-ordered_semigroups • abelian_groups • properties • weakly_representable_relation_algebras • transitive_relations • complemented_modular_lattices • finite_commutative_semigroups • commutative_lattice-ordered_semigroups