If we want to quantify how much an operation $*$ associates under repeated application, we can consider the associator $(a,b,c)_* := (a*b)*c-a*(b*c)$. I understand the distributive law as a bit like associativity, but looking at how two different binary operations associate with each other, instead of how one operations associates with itself.
So, consider a set $X$ with a binary operation, $*$, and addition of elements, $+$, so that $X$ is a ring but without the distributive law, and that that $*$ may or may not be associative. Is there a name for this structure?
Is there some similar definition for a 'distributator' for my structure $X$ which quantifies how much the binary operation distributes over addition in the same way that associator quantifies how much a binary operator associates with itself?
It seems natural to define $(a,b,c)_{*+} := (a*b)+(a*c)-a*(b+c)$; then we can see that $(a,b,c)_{*+} =0$ when the binary operation distributes over addition.
Is this definition used to analyse non-distributive algebras?
Secondly, consider adding in an additional binary operation $\cdot$, and allowing both binary operations to distribute over addition. Is it also possible to define a distributator which quantifies how much one of the binary operations distributes over the other? Again, something along the lines of $(a,b,c)_{*\cdot} := (a*b)\cdot(a* c)-a*(b\cdot c)$. Although in this case it also seems natural to look at $(a,b,c)_{*\cdot} := (a*b)\cdot c-a*(b\cdot c)$, just taking the associator and replacing the second $*$ operation with $\cdot$
I'm also interested in any reading associated with this topic, and any examples of algebras which have the forms that I've described.
Best Answer
(Note: For simplicity, when I say "is missing" I really mean "we do not require".)
Also I misinterpreted your question slightly. The examples I give below all assume that we have multiplicative inverses.
If it has distributivity but is missing associativity it is a semifield.
If it is missing associativity and missing one side of distributivity (so distributes from right but not from left), it is a quasifield.
If it has associativity but is missing one side of distributivity, it is a near-field.
I believe if it is missing both associativity and distributivity, it is called a "Cartesian group".
Dembowski's book Finite Geometries gives a good reference for this (he discusses these as examples of "planar ternary rings" which are used to coordinative projective planes).
I don't know that you will find a lot of material on structures that are missing both associativity and distributivity. But there is a lot of active research on finite semifields (and I believe the finite near-fields are classified). In response to the second part of your question, for example with semifields, we can consider the left, right, and middle nucleus, which are the elements that associate with everything when they are in the left, right, and middle position, respectively; ie the left nucleus of a semifield $\mathbb{S}$ would be $$N_{L}(\mathbb{S}) = \{a : a \in \mathbb{S} \mid (ab)c = a(bc) \mbox{ for all } b,c \in \mathbb{S} \}$$