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 − symbol because the natural numbers are a CMM under subtraction; it is also denoted with the ${\displaystyle \mathop {\dot {-}} }$ symbol to distinguish it from the standard subtraction operator.

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^[2] and that it is transitive.^[3] ${\displaystyle M}$ 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^[4]: 129 and the monus ${\displaystyle a\mathop {\dot {-}} b}$ 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.^[5]

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid^[6]) 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 ${\displaystyle a\mathop {\dot {-}} b=a\wedge \neg b}$.^[4]: 129

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,^[7] limited subtraction, proper subtraction, doz (difference or zero),^[8] and monus.^[9] Truncated subtraction is usually defined as^[7]

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

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^[9]

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

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):^[7]

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

A definition that does not need the predecessor function is:

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

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.^[7] Truncated subtraction is also used in the definition of the multiset difference operator.

The class of all commutative monoids with monus form a variety.^[4]: 129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:

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