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 codepoint[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 $\leq$ on this monoid as follows: for any two elements $a$. and $b$, we set $a\leq b$ if and only if there exists another element $c$ such that ${\displaystyle a+c=b}$. It is easy to check that $\leq$ is symmetric (taking $c$ to be the neutral element of the monoid) and that it is transitive (if $a\leq b$ with witness $c$ and ${\displaystyle b\leq c}$ with witness $c'$ then ${\displaystyle c+c'}$ witnesses that ${\displaystyle a\leq c}$). We call $M$ naturally ordered if the $\leq$ relation is additionally antisymmetric, so that $\leq$ is a partial order. Further, if for each pair of elements $a$ and $b$, there exists a unique smallest element $c$ 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 $a$ and $b$ as this unique smallest element $c$ 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 $c$ such that ${\displaystyle a+c=b}$, so we have $a\leq b$ for any ${\displaystyle a,b\in \mathbb {Z} }$, so $\leq$ 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 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]

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

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^[6]

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

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

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

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:

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

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. Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
3. math.SE answer
4. Semirings for breakfast, slide 17
5. Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
6. Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice. Algebraic Methodology and Software Technology. Lecture notes in computer science. 1101. Springer. p. 522. ISBN 3-540-61463-X.