## Every diassociative $$A$$-loop is Moufang.(English)Zbl 0990.20044

Given a loop $$(L,\cdot)$$, for any $$x\in L$$ let $$L(x)$$ and $$R(x)$$ be the left and the right translation by $$x$$, let $$\text{Mlt}(L):=\langle L(x),R(x)\mid x\in L\rangle$$ be the subgroup of $$\text{Sym }L$$ generated by all left and right translations and let $$\text{Mlt}_1(L):=\{\phi\in\text{Mlt}(L)\mid\phi(1)=1\}$$. $$\text{Mlt}_1(L)$$ is called the inner mapping group of the loop $$(L,\cdot)$$ and, in the associative case, it is the group of inner automorphisms of $$(L,\cdot)$$. A loop $$(L,\cdot)$$ is called an $$A$$-loop if $$\text{Mlt}_1(L)\leq\operatorname{Aut}(L,\cdot)$$. Every $$A$$-looop is always power associative (i.e. every $$\langle x\rangle$$ is a group) but not necessarily diassociative (i.e. every $$\langle x,y\rangle$$ is a group). On the other hand, if $$(L,\cdot)$$ is a Moufang loop (i.e. $$\forall x,y,z\in L$$, $$x(y(xz))=((xy)x)z$$) then it is diassociative. In this note the authors shed light on the relationship between Moufang loops and $$A$$-loops, showing that every diassociative $$A$$-loop is a Moufang loop. They resort to computer-aided proofs via McCune’s OTTER program, also commenting on some drawbacks thereof, and the need of “humanization” of counterintuitive procedures.

### MSC:

 20N05 Loops, quasigroups 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010)

OTTER
Full Text:

### References:

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