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Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus {0}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "Schemes over $\mathbb{F}_1$ and Zeta functions" by Connes and Consani. However they don't give these monoids a name.

A very silly idea might be to call them "monoid fields".

Question. How are these monoids called in the literature? If there is no existing terminology yet, which one would you propose?

The answer by BS tells us that in the non-commutative case these are called groups with zero. My question deals with the commutative case. I would like to have a proper name, not just a combination such as "abelian group with zero" (which is confusing anyway).

edited Feb 15 2012 at 14:39

Martin Brandenburg

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It would help to elaborate on the condition M*. For all I know, it looks like a submonoid with (a different) zero. Gerhard "Ask Me About System Design" Paseman, 2012.02.09 – Gerhard Paseman Feb 9 2012 at 17:44

what about "pointed abelian group"? – Dan Petersen Feb 15 2012 at 14:52

@Dan: This is not compatible with the usage of pointed objects (ncatlab.org/nlab/show/pointed+object). – Martin Brandenburg Feb 15 2012 at 15:11

@Martin: point taken. – Dan Petersen Feb 15 2012 at 15:18

Benjamin Steinberg · Answer 1 · 2012-02-10 01:05:28Z UTC

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The usual term in semigroup theory for a group with adjoined zero is a group with zero. See The Algebraic Theory of Semigroups Volume I by Clifford and Preston.

edited Feb 10 2012 at 1:05

Benjamin Steinberg

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Thanks! This seems to be an established terminology, you can also find it in other places. But as you might guess this does not really satisfy me. Since I'm interested in commutative monoids with zero, I would have to call them "commutative groups with zero" or "abelian groups with zero". But this is confusing since abelian groups are usually written additively and the identity element is written as a zero. Also I would like to use a short name (similar to "field", which is very short). – Martin Brandenburg Feb 10 2012 at 15:19

So the upshot is that I am open for neologisms for the commutative case. – Martin Brandenburg Feb 10 2012 at 15:21

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Some people say group with absorbing element. – Benjamin Steinberg Feb 10 2012 at 19:44

James Cranch · Answer 2 · 2012-02-09 22:58:43Z UTC

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One should take seriously the option of simply calling them "abelian groups with an adjoined zero".

edited Feb 9 2012 at 22:58

James Cranch

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Isn't this just a bit clumsy, or rather a characterization? Namely, the functor $A \mapsto A^*$ establishes an isomorphism of categories between "field monoids" and abelian groups. – Martin Brandenburg Feb 9 2012 at 22:44

(Edited to include the word "abelian"). – James Cranch Feb 9 2012 at 23:01

I agree it's clumsy, but I'm not wholly convinced you should disguise the characterisation. "Abelian groups with zero" is not all that long compared to some phrases in mathematics. – James Cranch Feb 9 2012 at 23:02

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Of course, this functor is not quite an equivalence of categories, unless you demand that homomorphisms preserve zero. Otherwise, for example, there's an extra map between any two objects which sends everything (including the zero) to the identity. – James Cranch Feb 9 2012 at 23:02

I don't understand why such comments still arise today (and get upvotes), decades after universal algebra etc. has been established. Of course a homomorphism between monoids with zero is defined to be a map which preserves the whole structure, in particular the zero. There is no reason to apply the forgetful functor to monoids ... – Martin Brandenburg Feb 10 2012 at 15:13

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Terminology for certain monoids which are to monoids like fields are to rings

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