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Abelian ordered groups
Abbreviation: AoGrp
Definition
An \emph{abelian ordered group} is an ordered group A=⟨A,+,−,0,≤⟩ such that
+ is commutative: x+y=y+x
Morphisms
Let A and B be abelian ordered groups. A morphism from A to B is a function h:A→B that is an orderpreserving homomorphism: h(x+y)=h(x)+h(y) and x≤y⟹h(x)≤h(y).
Examples
Example 1: ⟨Z,+,−,0,≤⟩, 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