Monus

Template:Short description Template:Use dmy dates Template:CS1 config 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 Template:Nowrap because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus Template:Nowrap to distinguish it from the standard subtraction operator.

Notation

glyph	Unicode name	Unicode code point	HTML character entity reference	HTML/XML numeric character references	TeX
$\dot{-}$	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.Template:Sfn

Definition

Let $(M,+,0)$ be a commutative monoid. Define a binary relation $\le$ on this monoid as follows: for any two elements $a$ and $b$, define $a\le b$ if there exists an element $c$ such that $a+c=b$. It is easy to check that $\le$ is reflexive and that it is transitive. $M$ is called naturally ordered if the $\le$ relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements $a$ and $b$, a unique smallest element $c_0$ exists such that $a\le b+c_0$, then Template:Math is called a commutative monoid with monusTemplate:Sfn and the monus $a\dot{-}b$ of any two elements $a$ and $b$ can be defined as this unique smallest element $c_0$ such that $a\le b+c_0$.

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

Other structures

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioidTemplate:Sfn) 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 Template:Math is an ideal in a Boolean algebra, then Template:Math is a commutative monoid with monus under $a+b=a\vee b$ and $a\dot{-}b=a\wedge \mathrm{\neg }b$.Template:Sfn

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,Template:Sfn limited subtraction, proper subtraction, doz (difference or zero),Template:Sfn and monus.Template:Sfn Truncated subtraction is usually defined asTemplate:Sfn

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

where − denotes standard subtraction. For example, $5-3=2$ and $3-5=-2$ in regular subtraction, whereas in truncated subtraction $3\dot{-}5=0$. Truncated subtraction may also be defined asTemplate:Sfn

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

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function Template:Math (the inverse of the successor function):Template:Sfn

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

A definition that does not need the predecessor function is:

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

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.Template:Sfn 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.Template:Sfn The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:

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

References

• Template:Cite journal

• Template:Cite conference

• Template:Cite book

• Template:Cite web

• Template:Cite web

• Template:Cite book

• Template:Cite book