I saw this image on Wikipedia (Template:Group-like structures, current revision):

Since there are five "properties" that we can have (in this context), namely: totality, associativity, identity, invertibility and commutativity, there are $\binom {5}{1}+ \binom{5}{2}+ \binom {5} {3} + \binom {5}{4} + \binom {5}{5}=5+10+10+5+1=31$ possible structures that we can have and there are only $10$ of them listed here, which are well known.

It could be that among other $21$ structures not listed here some are also studied and have a name but as the number of operations on a structure increases or/and as a number of properties of operations increases, then it would be very hard to remember the names of all structures, so I would like to propose the following:

If we denote totality by $T$, associativity by $A$, identity by $I$, invertibility by $In$ and commutativity by $C$, then we could denote a group as a $(T,A,I,In)$ structure. where it is immediately seen that group has properties of totality, associativity, has an identity, and every element has an inverse.

Another example, loop would be $(T,I,In)$ structure.

All of this can be made more precise but it is just some start, so maybe more rigour is needed.

This could be more generalized to structures over which there are more than one operation defined, but I will not do that here, I just want to ask:

Is this a well-known approach to designation of algebraic structures? Has anything similar been done? Would this lead to simplifications of some sort? What are your thoughts about this approach?