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Is there a name which is more specific than `unital magma' for a magma whose only requirements are that it should have both an identity and (L/R symmetric) inverses for all elements?

The following Cayley Table is the simplest non-trivial object of the type I am interested in, though of course you could make much more complicated examples:

  | 0 1 2
--+------
0 | 0 1 2
1 | 1 0 1
2 | 2 1 0

This object it is not a quasigroup (or a loop) as the table is not a Latin Square.

This object is not a semigroup as it is not associative. For example: (1*1)*2 = 0*2 = 2 while 1*(1*2) = 1*1 = 0.

It does, however, have an identity, 0. Furthermore, every element has a unique inverse (which is itself).

For what it's worth the multiplication rule is a*b = |a-b| on the set {0,1,2}.

A part of an older question ( Magma with inverse and identity yet not a quasigroup ) could be interpreted as requesting an answer to the same question I am asking today. However, the one answer which that question received did not answer the bit which I would like to see answered. I have therefore judged that it's helpful to ask the missing part here more succinctly to maximise the chance that it gets an answer.

KesterKester
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2 Answers2

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"Cancellative unital magma" is more-specific than "unital magma". This phrase would technically be a valid answer to the question, as invertibility implies cancellation, but I would prefer something more specific. I'd suggest "Invertible unital magma" except that I cannot find any evidence that this is a term that is in use outside of this post, whereas at least "Cancellative magma" has appears from time to time, such as here.

KesterKester
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This paper, for example, uses the terms

  • inverse property magma
  • magma with inverses

for properties defined in two different ways. The second one means a magma where each element has a unique inverse element.

Also, beware because the finite groupoid as in the Cayley table in your question is definitey not cancellative: $$1*0=1*2=2*1=0*1,$$ but $2\ne 0$.

MattAllegro
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    In the opening sentence of the above paper (not including the abstract) it makes clear that it defines the word magma differently than I intended in my question. However, that paper is clear that what it calls a magma is what my question calls a `unital magma', and then it goes on to add the inverses to its magmas, and so that paper is certainly defining things of the sort of thing I am interested in. – KesterKester Mar 15 '21 at 12:15