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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 $\sim$:

  1. There exists an identity element $e$ such that for all elements $a$: $$a \sim a = e \text{ and } a \sim e = a$$

  2. 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:

  1. Do these operations share any other nontrivial properties? That is, is there a tighter characterization of these operations?

  2. Have such operations been studied in the past?

izzyg
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    Your examples are close to quasigroups and loops, but not quite (set symmetric difference is a loop, i think, and so are integers and reals with subtraction, while set difference and saturated subtraction on the naturals are not). – Arthur Jul 03 '18 at 08:46
  • @Arthur Integers and real numbers with subtraction are not loops because they have no identity. $0$ is a right identity for $-$ over $\mathbb{Z}$ or $\mathbb{R}$ but not a left identity: $a=a-0\ne 0-a=-a$ for all $a\in\mathbb{R}$ :) – MattAllegro May 05 '19 at 08:53

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