Q:

A very low "algebraic" structure

i m thinking about a specific typ of set eq. with a map but i am not shure if it is any kind of alg. structure.
i found the definition of a so called magma, which is a set $M$ with an arbitrary map $\circ: M \times M \longrightarrow M$, which has no other restrictions, only that it has to be closed.
The Set i am thinking of is defined as follows:
$\mathcal{C} := \mathbb{N}^2$, and the map is 
$\circ: \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}$ which exists if and only if one integer is equal to one on the other tuple, like
$(a,b)(a,c) := (a,b+c)$ or $(a,b)(c,b) := (a+c,b)$ and otherwise there exists no morphism!!
Please can someone help me to find out if this is some kind of "structure" in the common sense

A:

I don't know if this has a name besides a set with a partial binary operation. This is not really an exotic concept. The simplest example I can think of is division in a field; you can divide by everything except $0$. Ostensibly this is a very poorly behaved operation since it is neither commutative nor associative. However, we know the truth: it is the composition of two very nice operations, multiplication and inversion in the field. Another relatively tame example is multiplication of matrices of varying sizes, which is associative but not commutative.
There's not much to say about set with an arbitrary operation of which you don't know any specific properties, so you have to find some to latch on to. The fact that magmas have a name doesn't mean that they are useful, and arbitrary partial binary operations are even less useful. You don't necessarily need the usual properties of commutativity or associativity. Lie algebras are not associative but they satisfy the Jacobi identity, which makes them very nice, and my blog has the example of root systems, which are extremely noncommutative and nonassociative but are easy to completely classify.
If you have some reason to investigate the operation you defined, then go ahead. But if you just pulled it out of thin air, there are likely better ways to spend your time.

