Structure associated with the cocycle condition

Let $$I$$ be a set and suppose we have a partially defined multiplication on $$I\times I$$ satisfying "the cocycle condition" $$(i,j)(j,k)=(i,k)$$ for all $$i,j,k\in I.$$ Now I wonder what kind of algebraic structure (group, semigroup, ... etc) does $$I\times I$$ carry with it.

So far only thing I could extract from the cocycle condition is, for a given $$(i,j)$$ we have a left identity $$(i,i)$$ and a right identity $$(j,j).$$ Also if we define $$(i,j)^{-1}=(j,i),$$ then we can derive

• $$(i,j)(i,j)^{-1}=(i,i)$$
• $$(i,j)^{-1}(i,j)=(j,j)$$
• $$((i,j)^{-1})^{-1}=(i,j).$$

But these properties are not enough to determine the structure completely. Also since the multiplication isn't defined for all ordered pairs, I can't see how we can understand the algebraic structure completely. Thank you for your help, in advance.

The part of such a structure only involving multiplications that appear in your axiom is just a trivial groupoid, where $$(i,j)$$ represents the unique morphism $$i\to j$$. The rest of the structure is totally unaffected by the axiom, so there's nothing to say.