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, "${\displaystyle -}$", because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus sign, "${\displaystyle \mathbin {\dot {-}} }$", to distinguish it from the standard subtraction operator.

Notation

glyph	Unicode name	Unicode code point[1]	HTML character entity reference	HTML/XML numeric character references	TeX
${\displaystyle \mathbin {\dot {-}} }$	DOT MINUS	U+2238		∸	\dot -
−	MINUS SIGN	U+2212	−	−	-

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

Definition

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

An example of a commutative monoid that is not naturally ordered is ${\displaystyle (\mathbb {Z} ,+,0)}$ , the commutative monoid of the integers with usual addition, as for any ${\displaystyle a,b\in \mathbb {Z} }$  there exists ${\displaystyle c}$  such that ${\displaystyle a+c=b}$ , so ${\displaystyle a\leq b}$  holds for any ${\displaystyle a,b\in \mathbb {Z} }$ , so ${\displaystyle \leq }$  is 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 ${\displaystyle a+b=a\vee b}$  and ${\displaystyle a\mathbin {\dot {-}} b=a\wedge \neg b}$ .^[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]

${\displaystyle a\mathbin {\dot {-}} b={\begin{cases}0&{\mbox{if }}a<b\\a-b&{\mbox{if }}a\geq b,\end{cases}}}$ 

where − denotes standard subtraction. For example, ${\displaystyle 5-3=2}$  and ${\displaystyle 3-5=-2}$  in regular subtraction, whereas in truncated subtraction ${\displaystyle 3\mathbin {\dot {-}} 5=0}$ . Truncated subtraction may also be defined as^[10]

${\displaystyle a\mathbin {\dot {-}} b=\max(a-b,0).}$ 

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

${\displaystyle {\begin{aligned}P(0)&=0\\P(S(a))&=a\\a\mathbin {\dot {-}} 0&=a\\a\mathbin {\dot {-}} S(b)&=P(a\mathbin {\dot {-}} b).\end{aligned}}}$ 

A definition that does not need the predecessor function is:

${\displaystyle {\begin{aligned}a\mathbin {\dot {-}} 0&=a\\0\mathbin {\dot {-}} b&=0\\S(a)\mathbin {\dot {-}} S(b)&=a\mathbin {\dot {-}} b.\end{aligned}}}$ 

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:

${\displaystyle {\begin{aligned}a+(b\mathbin {\dot {-}} a)&=b+(a\mathbin {\dot {-}} b),\\(a\mathbin {\dot {-}} b)\mathbin {\dot {-}} c&=a\mathbin {\dot {-}} (b+c),\\(a\mathbin {\dot {-}} a)&=0,\\(0\mathbin {\dot {-}} a)&=0.\\\end{aligned}}}$ 

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.
3. taking ${\displaystyle c}$  to be the neutral element of the monoid
4. if ${\displaystyle a\leq b}$  with witness ${\displaystyle d}$  and ${\displaystyle b\leq c}$  with witness ${\displaystyle d'}$  then ${\displaystyle d+d'}$  witnesses that ${\displaystyle a\leq c}$ 
5. Amer 1984, p. 129.
6. Monet 2016.
7. Pouly 2010, p. 22, slide 17.
8. Vereschchagin & Shen 2003.
9. Warren Jr. 2013.
10. 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.; Kernighan, Brian W.; Ritchie, William A. (2022), "How did Dennis Ritchie produce his PhD thesis? A typographical mystery" (PDF), in Wigington, Curtis; Hardy, Matthew; Bagley, Steven R.; Simske, Steven J. (eds.), Proceedings of the 22nd ACM Symposium on Document Engineering, DocEng 2022, San Jose, California, USA, September 20–23, 2022, 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" (PS), in Wirsing, Martin; Nivat, Maurice (eds.), Algebraic Methodology and Software Technology, Lecture Notes in Computer Science, vol. 1101, Springer, p. 522, ISBN 3-540-61463-X

• Monet, M. (14 October 2016), "Example of a naturally ordered semiring which is not an m-semiring", Mathematics Stack Exchange, retrieved 30 July 2025

• Pouly, Marc (July 2010), "Semirings for breakfast" (PDF), University of Luxembourg, p. 27, retrieved 30 July 2025

• Vereschchagin, Nikolai K.; Shen, Alexander (2003), Computable Functions, translated by V. N. Dubrovskii, American Mathematical Society, p. 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