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Last night I started to read some book that has to do with applications of groups in physics and the question came in my mind about the existence of some structure, which I define in this way:

Suppose that we have set $S$ which has at least countably infinite number of elements which is equipped with the binary operation $*$ and that we have:

1) $\forall x,y \in S$ we have $x*y \in S$

2) There exist one and only one $e \in S$ such that we have $x*e=e*x=x$, for every $x \in S$.

3) For every $x \in S$ there exist $l \in S$ and $r \in S$ such that we have $l*x=e$ and $x*r=e$ and $l \neq r$.

So this structure is similar to the group in that that it satisfies closure axiom and it has unique identity element but it is different from group in that that every element has left and right inverse which do not coincide and we do not assume associativity.

Since I know that there are a lot of structures in mathematics if this structure exists it is probably not something new, but I do not know.

And now the question:

Does structure defined in this way exist?

Najib Idrissi
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Farewell
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1 Answers1

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As posted in the comments, if you insist that rule 3 is valid for $x=e$, the answer is "No." (See details in the comment by Travis.)

However, if you are willing to exclude $x=e$ from rule 3, here's an example of such a structure on $S=\{0, 1, 2, 3, ....\}$:

Define $0*n = n*0 = n$, $1*2 = 2*3 = 3*1 = 0$, $4*5 = 5*6 = 6*4 = 0$, and so on, and define the operation $*$ on the other pairs as you wish.

Catalin Zara
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