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 $(M,+,0)$ 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 $a+c=b$ . It is easy to check that $\leq$ is reflexive (taking $c$ to be the neutral element of the monoid) and that it is transitive (if $a\leq b$ with witness $c$ and $b\leq c$ with witness $c'$ then $c+c'$ witnesses that $a\leq c$ ). 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 $a\leq b+c$ , 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 $a\leq b+c$ .

An example of a commutative monoid which 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 we have $a\leq b$ for any $a,b\in \mathbb {Z}$ , 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 $a + b = a \lor b$ and $a ∸ b = a \land \neg b$ .^[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:

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

Notes

↑ Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
1 2 3 Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
↑ math.SE answer
↑ Semirings for breakfast, slide 17
1 2 3 4 Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
1 2 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.

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