nLab theory of abelian groups

Idea

The theory of abelian groups $T$ is the logical theory whose models in a cartesian category are the abelian group objects.

Definition

The theory of abelian groups $T$ is the theory over the signature with one sort $X$, one constant $0$, two function symbols $+$ (of sort $X\times X\to X$) and $-$ (of sort $X\to X$) and equality with axioms:

• $\top\vdash_x x+0=x \quad$
• $\top\vdash_{x,y,z} x+(y+z)=(x+y)+z\quad$,
• $\top\vdash_x x+(-x)=0\quad$.
• $\top\vdash_{x,y} x+y=y+x\quad$.

The theory of abelian groups is algebraic. In the following we list some extensions that use increasingly stronger fragments of geometric logic.

Extensions of the theory of abelian groups

The theory of torsion-free abelian groups $T^\infty$ results from $T$ by addition of the following axioms:

• For all $n\geq 2$: $((nx=0))\vdash_x (x=0))\quad$.

The resulting theory is a Horn theory.

The theory of divisible abelian groups $T^\backslash$ results from $T$ by addition of the following regular axioms:

• For all $n\geq 2$: $\top\vdash_x (\exists y) (ny=x)\quad$.

The theory of divisible torsion-free abelian groups $T^{\backslash\infty}$ results from $T^\backslash$ by adding the above axioms for torsion-freeness. The resulting theory is cartesian.

The theory of torsion abelian groups $T^~$ is an infinitary geometric theory resulting from $T$ by addition of:

• $\top\vdash_x \bigvee_{n\geq 2} (nx=0)\quad$.

Related concepts

• abelian group

• torsion

• geometric theory

• theory of objects

• theory of decidable objects

References

• Peter Johnstone, Sketches of an Elephant II, Oxford UP 2002. pp.812-815.