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.

glyph	Unicode name	Unicode code point1	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

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 reflexive³ 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 ${\displaystyle c_{0}}$ exists such that ${\displaystyle a\leq b+c_{0}}$, then M is called a commutative monoid with monus⁵ 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 ${\displaystyle c_{0}}$ such that ${\displaystyle a\leq b+c_{0}}$.

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 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 dioid⁷) 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 ${\displaystyle a\mathbin {\dot {-}} b=a\wedge \neg b}$.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 as⁸

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

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 as¹⁰

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

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.

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:

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