Monus

In mathematics, monus is an operator on certain commutative monoids that are not groups. A commutative monoid on which a monus operator is defined is called a commutative monoid with monus, or CMM. The monus operator may be denoted with the minus sign, "−", because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus sign, "−˙", to distinguish it from the standard subtraction operator.

## Contents

* 1 Notation
* 2 Definition
* 3 Other structures
* 4 Examples
* 4.1 Natural numbers
* 5 Properties
* 6 Notes
* 7 References

## Notation

glyph | Unicode name  | Unicode code point[1] | HTML character entity reference | HTML/XML numeric character references | TeX

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−˙ | DOT MINUS | U+2238  |  | `&#8760;` | `\dot -`
−  | MINUS SIGN | U+2212  | `&minus;` | `&#8722;` | `-`

A use of the monus symbol is seen in Dennis Ritchie's PhD Thesis from 1968.[2]

## Definition

Let (M,+,0) be a commutative monoid. Define a binary relation ≤ on this monoid as follows: for any two elements a and b, define a≤b if there exists an element c such that a+c=b. It is easy to check that ≤ is reflexive[3] and that it is transitive.[4] M is called naturally ordered if the ≤ relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements a and b, a unique smallest element c0 exists such that a≤b+c0, then M is called a commutative monoid with monus[5] and the monus a−˙b of any two elements a and b can be defined as this unique smallest element c0 such that a≤b+c0.

An example of a commutative monoid that is not naturally ordered is (ℤ,+,0), the commutative monoid of the integers with usual addition, as for any a,b∈ℤ there exists c such that a+c=b, so a≤b holds for any a,b∈ℤ, so ≤ is not antisymmetric and therefore not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus.[6]

## Other structures

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[7]) is a semiring where the commutative monoid induced by the addition operator is naturally ordered. When this monoid is a commutative monoid with monus, the semiring is called a semiring with monus, or m-semiring.

## Examples

If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under a+b=a∨b and a−˙b=a∧¬b.[5]

### Natural numbers

The natural numbers including 0 form a commutative monoid with monus, with their ordering being the usual order of natural numbers and the monus operator being a saturating variant of standard subtraction, variously referred to as truncated subtraction,[8] limited subtraction, proper subtraction, doz (difference or zero),[9] and monus.[10] Truncated subtraction is usually defined as[8]

a−˙b={0if a<ba−bif a≥b,

where − denotes standard subtraction. For example, 5−3=2 and 3−5=−2 in regular subtraction, whereas in truncated subtraction 3−˙5=0. Truncated subtraction may also be defined as[10]

a−˙b=max⁡(a−b,0).

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[8]

P(0)=0P(S(a))=aa−˙0=aa−˙S(b)=P(a−˙b).

A definition that does not need the predecessor function is:

a−˙0=a0−˙b=0S(a)−˙S(b)=a−˙b.

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[8] Truncated subtraction is also used in the definition of the multiset difference operator.

## Properties

The class of all commutative monoids with monus form a variety.[5] The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:

a+(b−˙a)=b+(a−˙b),(a−˙b)−˙c=a−˙(b+c),(a−˙a)=0,(0−˙a)=0.

## Notes

1. ↑ Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
2. ↑ Brailsford, Kernighan & Ritchie 2022. sfn error: no target: CITEREFBrailsfordKernighanRitchie2022 (help)
3. ↑ taking c to be the neutral element of the monoid
4. ↑ if a≤b with witness d and b≤c with witness d′ then d+d′ witnesses that a≤c
5. ↑ 5.0 5.1 5.2 Amer 1984, p. 129.
6. ↑ Monet 2016.
7. ↑ Pouly 2010, p. 22, slide 17.
8. ↑ 8.0 8.1 8.2 8.3 Vereschchagin & Shen 2003.
9. ↑ Warren Jr. 2013.
10. ↑ 10.0 10.1 Jacobs 1996.

## References

* Amer, K. (1984). "Equationally complete classes of commutative monoids with monus". Algebra Universalis 18: 129-131. doi:10.1007/BF01182254.

* Brailsford, David F. (2022). "Proceedings of the 22nd ACM Symposium on Document Engineering, DocEng 2022, San Jose, California, USA, September 20–23, 2022". in Wigington, Curtis; Hardy, Matthew; Bagley, Steven R. et al.. Association for Computing Machinery. pp. 2:1–2:10. doi:10.1145/3558100.3563839.

* Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". in Wirsing, Martin. Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. 1101. Springer. pp. 522. ISBN 3-540-61463-X. https://www.cs.ru.nl/B.Jacobs/PAPERS/AMAST96.ps.

* Monet, M. (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". https://math.stackexchange.com/q/1968090. Retrieved 2025-07-30.

* Pouly, Marc (July 2010). "Semirings for breakfast". pp. 27. http://marcpouly.ch/pdf/internal_100712.pdf. Retrieved 2025-07-30.

* Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. American Mathematical Society. pp. 141. ISBN 0-8218-2732-4.

* Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc.. ISBN 978-0-321-84268-8.

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