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, "${\displaystyle \dot{-}}$", to distinguish it from the standard subtraction operator.

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

Let ${\displaystyle (M,+,0)}$ be a commutative monoid. Define a binary relation ${\displaystyle \le }$ on this monoid as follows: for any two elements ${\displaystyle a}$ and ${\displaystyle b}$, define ${\displaystyle a\le b}$ if there exists an element ${\displaystyle c}$ such that ${\displaystyle a+c=b}$. It is easy to check that ${\displaystyle \le }$ is reflexive^[3] and that it is transitive.^[4] ${\displaystyle M}$ is called naturally ordered if the ${\displaystyle \le }$ 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 ${\displaystyle c_0}$ exists such that ${\displaystyle a\le b+c_0}$, then M is called a commutative monoid with monus^[5] and the monus ${\displaystyle a\dot{-}b}$ of any two elements ${\displaystyle a}$ and ${\displaystyle b}$ can be defined as this unique smallest element ${\displaystyle c_0}$ such that ${\displaystyle a\le b+c_0}$.

An example of a commutative monoid that is not naturally ordered is ${\displaystyle (\mathbb{\mathbb{Z}},+,0)}$, the commutative monoid of the integers with usual addition, as for any ${\displaystyle a,b\in \mathbb{\mathbb{Z}}}$ there exists ${\displaystyle c}$ such that ${\displaystyle a+c=b}$, so ${\displaystyle a\le b}$ holds for any ${\displaystyle a,b\in \mathbb{\mathbb{Z}}}$, so ${\displaystyle \le }$ 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]

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.

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 ${\displaystyle a\dot{-}b=a\wedge \mathrm{\neg }b}$.^[5]

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]

${\displaystyle a\dot{-}b=\{\begin{array}{ll}0& \text{if }a<b\\ a-b& \text{if }a\ge b,\end{array}}$

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

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

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

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

A definition that does not need the predecessor function is:

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

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.

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:

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

• 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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