Generalisation of a loop concept

Question

Suppose that $(M, \circ)$ is a set $M$ over which there is defined a binary operation $\circ$ so that we have:

1) For every $(a,b) \in M \times M$ we have $a \circ b \in M$

2) For every $a \in M$ there exist exactly $k$ different elements $a_1^{-1},...,a_k^{-1} \in M$ so that we have $a \circ a_i^{-1}=a_i^{-1} \circ a = 1$, for every $i=1,...,k$

3) We have $1 \circ a = a \circ 1 =a$ for every $a \in M$

This would be a generalization of a loop concept because, as is easily seen, loops are obtained when $k=1$

Do these generalisations exist for every $k \in \mathbb N$? How to construct them if they do?

Answer 1

No such $M$ exists for any $k>1$. Here is why: Obviously, by your axioms, the element $1\in M$ must be unique. Then $1\circ 1^{-1}=1$, hence $1=1^{-1}$, and $1^{-1}$ is unique as well.

But if you want, say, exactly two inverses to $a\in M$ for all $a\neq 1$ only, then it is possible by defining a three-elemet set $M=\{a_1,a_2, 1\}$ with $a_i\circ a_j=1$ and $a_i\circ 1=1\circ a_i=a_i$. Similarly, you can construct such a $(k+1)$-element set of loops $\{M,\circ \}$ for every $k$.