I was looking at a collection of related closed binary operations on sets (magmas):
Subtraction on the integers, reals, etc.
Set difference
Set symmetric difference
Saturating subtraction on the nonnegative integers
Ceilinged division on the positive integers
Each of these operations has the following properties, where the operation is represented by $\sim$:
There exists an identity element $e$ such that for all elements $a$: $$a \sim a = e \text{ and } a \sim e = a$$
For all elements $a, b, c$: $$(a \sim b) \sim c = (a \sim c) \sim b$$
I call these operations subtraction magmas because they often take the form of some kind of subtraction.
I have two questions:
Do these operations share any other nontrivial properties? That is, is there a tighter characterization of these operations?
Have such operations been studied in the past?