The composition law in a monoid is usually represented using a binary operation (multiplication) and a zeroary operation (unit), but I view it more naturally as an operation (say, a bracket) taking any finite list of arguments and being associative in the sense that brackets can be eliminated in any expression, for example we have the identities

[a,[],b,[[c,d],e],[f]]=[a,b,c,d,e,f]

and

[a]=a.

Then we can define a zeroary operation 1:=[] and a binary operation a*:b=[a,b], and recover the bracket from them, using identities such as [a,b,c,d]=[a,[b,[c,d]]]. These two operations satisfy the usual axioms of a monoid, and any two operations satisfying them can be extended to an associative finitary bracket.

I view the usual representation by a binary and a zeroary operation as an artifact for being able to produce simpler-looking proofs that the structures that we encounter are monoids.

My point is that naturally binary operations are not that common either! Perhaps an example is the Lie bracket.

The composition law in a monoid is usually represented using a binary operation (multiplication) and a zeroary operation (unit), but I view it more naturally as an operation (say, a bracket) taking any finite list of arguments and being associative in the sense that brackets can be eliminated in any expression, for example we have the identities

[a,[],b,[[c,d],e],[f]]=[a,b,c,d,e,f]

and

[a]=a.

Then we can define a zeroary operation 1:=[] and a binary operation a*b:=[a,b], and recover the bracket from them, using identities such as [a,b,c,d]=[a,[b,[c,d]]]. These two operations satisfy the usual axioms of a monoid, and any two operations satisfying them can be extended to an associative finitary bracket.

I view the usual representation by a binary and a zeroary operation as an artifact for being able to produce simpler-looking proofs that the structures that we encounter are monoids.

My point is that naturally binary operations are not that common either! Perhaps an example is the Lie bracket.