# Monus

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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 symbol because the natural numbers are a CMM under subtraction; it is also denoted with the symbol to distinguish it from the standard subtraction operator.

## Notation

glyph Unicode name Unicode codepoint[1] HTML character entity reference HTML/XML numeric character references TeX
DOT MINUS U+2238 &#8760; \dot -
MINUS SIGN U+2212 &minus; &#8722; -

## 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 ${\displaystyle b}$, we set ${\displaystyle a\leq b}$ if and only if there exists another element ${\displaystyle c}$ such that ${\displaystyle a+c=b}$. It is easy to check that ${\displaystyle \leq }$ is reflexive (taking ${\displaystyle c}$ to be the neutral element of the monoid) and that it is transitive (if ${\displaystyle a\leq b}$ with witness ${\displaystyle d}$ and ${\displaystyle b\leq c}$ with witness ${\displaystyle d'}$ then ${\displaystyle d+d'}$ witnesses that ${\displaystyle a\leq c}$). We call ${\displaystyle M}$ naturally ordered if the ${\displaystyle \leq }$ relation is additionally antisymmetric, so that ${\displaystyle \leq }$ is a partial order. Further, if for each pair of elements ${\displaystyle a}$ and ${\displaystyle b}$, there exists a unique smallest element ${\displaystyle c}$ such that ${\displaystyle a\leq b+c}$, then we call M a commutative monoid with monus[2]:129 and we can then define the monus a ∸ b of any two elements ${\displaystyle a}$ and ${\displaystyle b}$ as this unique smallest element ${\displaystyle c}$ such that ${\displaystyle a\leq b+c}$.

An example of a commutative monoid which 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 we have ${\displaystyle a\leq b}$ for any ${\displaystyle a,b\in \mathbb {Z} }$, so ${\displaystyle \leq }$ is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.[3]

## Other structures

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[4]) 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.[2]:129

### 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,[5] limited subtraction, proper subtraction, and monus.[6] Truncated subtraction is usually defined as[5]

${\displaystyle a\mathop {\dot {-}} b={\begin{cases}0&{\mbox{if }}a

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[6]

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

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[5]

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

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[5] 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.[2]:129 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{\dot {-}}a)&=b+(a{\dot {-}}b)\\(a{\dot {-}}b){\dot {-}}c&=a{\dot {-}}(b+c)\\(a{\dot {-}}a)&=0\\(0{\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. ^ a b c Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254