Given a set $S$ with
- An associative binary "addition" operation $+$ with corresponding neutral element $0$ such that $(S, +)$ is a monoid
- A non-associative binary "truncated subtraction" operation $-$ such that $(S, -)$ is a magma (truncating at the element that is neutral with respect to $+$, i.e.: $\forall x,y \in S: y - y = 0$)
Is there a name for such a structure?
- (This is the motivating example) I am trying to define an algebra of (Unix) globs, considered not as limited by the syntax but as an abstraction allowing the the ability to combine the results of two globs, or to remove the results of one glob from another, with a glob definable in the syntax acting as a sort of urelement, I suppose. It appears to me that from an algebraic perspective, a glob is an encoding or a specifier of a set of files. The set is hypothetical in the sense that its actual contents are not known until the glob is evaluated. However, it should be possible to define the algebra at play without dealing with the evaluation.
- (Seemingly analogous example) Sets, where only union and set difference (relative complement) are under consideration.
- (Seemingly analogous example) The naturals, where only addition and truncated subtraction are under consideration.