Is division axiomatizable?

Question

Consider a set $G$ with a group operation. We can define a division operation $a*(b^{-1})$ and call it $\operatorname{div}$. Is the class of division operations first order axiomatizable? And if so, is it finitely axiomatizable?

Answer 1

Let $\star$ be your operator. On a group, this can be axiomatized as:

$$\forall x(1\star(1\star x)=x)\text{ (A)}\\ \forall x(x\star x = 1)\text{ (B)}\\\forall x,y,z\left((x\star y)\star z = x\star(z\star(1\star y))\right)\text{ (C)}$$

We can quickly show:

$$\begin{align} x\star 1 &= (1\star(1\star x))\star 1 \text{ (A)}\\ &=1\star(1\star(1\star(1\star x)))\text{ (C)}\\ &=1\star(1\star x)\text{ (A)}\\ &=x\text{ (A)} \end{align} $$ Then if you define $x^{-1}=1\star x$ and $x\cdot y = x\star(1\star y)$, we can show:

$$x\cdot x^{-1} = x\star(1\star(1\star x)) = x\star x = 1\\ x^{-1}\cdot x = (1\star x)\star (1\star x)=1\\ 1\cdot x = 1\star(1\star x)=x\\ x\cdot 1 = x\star (1\star 1) = x\star 1 = 1\\ \begin{align}(x\cdot y)\cdot z&=(x\star(1\star y))\star(1\star z)\\ &=x\star\left((1\star z)\star(1\star(1\star y))\right)\\ &=x\star\left((1\star z)\star y\right)\\ &=x\star(1\star (y\star(1\star z)))\\ &= x\cdot(y\cdot z) \end{align} $$ and finally:

$$a\cdot b^{-1} = a\star(1\star(1\star b)) = a\star b$$

So you've got all your group axioms.