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Let $M = (M,·,1,0)$ be a monoid $(M,·,1)$ together with an distinguished absorbing element $0 ∈ M$, that is such that $∀x ∈ M\colon 0·x = 0 = x·0$.

Does such a structure $M$ have a nice name?

Furthermore, is there a name for such structures $M$, where the units of $M$ are exactly the nonzero elements of $M$, i.e. $M^× = M\setminus \{0\}$?

Example. Every ring is such a structure when only considering multiplication. Fields are then examples where the units are exactly the nonzero elements.

J.-E. Pin
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k.stm
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    On your first question: the absorbing element is unique, right? So I would just call it a "monoid having an absorbing element". – drhab Nov 07 '15 at 11:21
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    @drhab It is necessarily unique as $0'=0'\cdot 0=0$ – Hagen von Eitzen Nov 07 '15 at 11:24
  • I have only seen it the way drhab describes. Most authors would probably not deign to give it a special name. – rschwieb Nov 07 '15 at 21:53
  • Furthermore, is there a name for such structures , where the units of M are exactly the nonzero elements of M? that would be a group with an adjoined absorbing element, right? (Of course, having adjoined the absorbing element, it's no longer a group.) – rschwieb Nov 07 '15 at 21:56
  • @rschwieb Yes, exactly. A little background for this question: Valuations on fields $F$ are morphisms $F → M$ of such structures (with an additional requirement of some subaddivity of “$+$” and “$∨$”), where $M$ is an ordered group with an adjoint absorbing element at the bottom. Then, $M^×$ would be called the valuation group. – k.stm Nov 09 '15 at 21:54
  • You might call it a "monoid with zero" for short. – Zhen Lin Nov 09 '15 at 23:23
  • I think I might just call it “steroid” … – k.stm Nov 10 '15 at 07:15
  • @drhab I think there can be a notion of "an absorbing element" that is different than "the absorbing element". If the absorbing element is the element that absorbs all other elements, then an absorbing element can refer to an element that absorbs some non-identity element. For this reason, I favor ZhenLin's suggestion, "monoid with zero". – mareoraft Nov 17 '15 at 21:47
  • @mareoraft Somewhat shrugging I go along with you and Zhen Lin. The proposed terminology indeed excludes ambiguity. – drhab Nov 18 '15 at 08:53
  • See my identical question http://mathoverflow.net/questions/88014 – Martin Brandenburg Jan 10 '16 at 11:38

2 Answers2

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A monoid with an distinguished absorbing element is called a monoid with zero in the literature. A monoid with zero in which the units are exactly the nonzero elements is called a group with zero [1, p. 5; 2, Def 1.3.1, p. 34; 4] or a $0$-group [3].

[1] A. H. Clifford, G. B. Preston, The algebraic theory of semigroups. Vol. I. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I. 1961 xv+224 pp.

[2] P.M. Higgins, Techniques of semigroup theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+258 pp. ISBN: 0-19-853577-5

[3] J.M. Howie, Fundamentals of semigroup theory. London Mathematical Society Monographs. New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+351 pp. ISBN: 0-19-851194-9

[4] Lallement, Gérard. Semigroups and combinatorial applications. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley & Sons, New York-Chichester-Brisbane, 1979. xi+376 pp. ISBN: 0-471-04379-6

J.-E. Pin
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They are sometimes also called pointed monoids. See arXiv:1909.00297 for instance.

Martin Brandenburg
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