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 ∸ symbol to distinguish it from the standard subtraction operator.

Notation

glyph	Unicode name	Unicode code point[1]	HTML character entity reference	HTML/XML numeric character references	TeX
∸	DOT MINUS	U+2238		∸	\dot -
−	MINUS SIGN	U+2212	−	−	-

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 ${\displaystyle b}$, 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 a ∸ 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 which are naturally ordered but are not semirings with monus.[5]

Other structures

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.

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.[4]^: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,[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]

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

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.

Properties

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:

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

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. taking ${\displaystyle c}$ to be the neutral element of the monoid
3. if ${\displaystyle a\leq b}$ with witness ${\displaystyle d}$ and ${\displaystyle b\leq c}$ with witness ${\displaystyle d'}$ then ${\displaystyle d+d'}$ witnesses that ${\displaystyle a\leq c}$
4. Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
5. M.Monet (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". Mathematics Stack Exchange. Retrieved 2016-10-14.
6. Semirings for breakfast, slide 17
7. Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
8. Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8.
9. Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice (eds.). Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. 1101. Springer. p. 522. ISBN 3-540-61463-X.