Consider the following group-like algebraic structure (don't know its name).
Let $G$ be a non-empty set with a partial binary operation $*:G \times G \rightarrow G$ (i.e. $*$ is not necessarily defined for all possible pairs of elements from $G \times G$) such that the following conditions hold:
Identity: there exists an identity element $e \in G$ such that for every $a \in G$ $e*a$ and $a*e$ are defined and $e*a=a*e=a$.
Inverse: for each $a \in G$ there exists an element $a^{-1} \in G$ such that $a*a^{-1}$ and $a^{-1}*a$ are defined and $a*a^{-1}=a^{-1}*a=e$.
Restricted associativity: for each $a,c \in G$ there exists $b \in G$ such that $(a*b)*c$ and $a*(b*c)$ are defined and $(a*b)*c=a*(b*c)$.
I am interested in: (A) conditions under which $(G,*)$ is actually a group, (B) conditions under which this structure can be extended to a group on $G$.
Any references will be very much appreciated.
Update. Let me introduce the context under which the structure is appeared.
I am trying to solve the transitivity functional equation of the form $u(s,s')=H(u(s,s'),u(s',s''))$, $s,s',s'' \in S$ under the condition $u(s,s)=u(s',s')$ for any $s,s' \in S$.
Here $S$ is a set, $u:S \times S \rightarrow R$ is an unknown function, and $H:u(S,S) \times u(S,S) \rightarrow u(S,S)$ is an unknown partial function. This functional equation is related to the above structure if we put $G:=u(S,S)$, $*:=H$, $e:=u(s,s)$, $a^{-1}=u(s',s)$, where $s,s' \in S$ are such that $u(s,s')=a$, and $a*b$ is defined if and only if there exists a triple $s,s',s'' \in S$ such that $a=u(s,s')$, $b=u(s',s'')$.
I am interested in the case when $(G,*)$ actually constitutes a group, since in this case the equation considered reduces to the Sincov functional equation whose general solution is known.
Thank you.