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 [math]\displaystyle{ \mathop {\dot -} }[/math] symbol to distinguish it from the standard subtraction operator.

Notation

Definition

Let [math]\displaystyle{ (M, +, 0) }[/math] be a commutative monoid. Define a binary relation [math]\displaystyle{ \leq }[/math] on this monoid as follows: for any two elements [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], define [math]\displaystyle{ a \leq b }[/math] if there exists an element [math]\displaystyle{ c }[/math] such that [math]\displaystyle{ a + c = b }[/math]. It is easy to check that [math]\displaystyle{ \leq }[/math] is reflexive^[2] and that it is transitive.^[3] [math]\displaystyle{ M }[/math] is called naturally ordered if the [math]\displaystyle{ \leq }[/math] relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], a unique smallest element [math]\displaystyle{ c }[/math] exists such that [math]\displaystyle{ a \leq b + c }[/math], then M is called a commutative monoid with monus^[4]:129 and the monus [math]\displaystyle{ a \mathop {\dot -} b }[/math] of any two elements [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] can be defined as this unique smallest element [math]\displaystyle{ c }[/math] such that [math]\displaystyle{ a \leq b + c }[/math].

An example of a commutative monoid that is not naturally ordered is [math]\displaystyle{ (\mathbb{Z}, +, 0) }[/math], the commutative monoid of the integers with usual addition, as for any [math]\displaystyle{ a, b \in \mathbb{Z} }[/math] there exists [math]\displaystyle{ c }[/math] such that [math]\displaystyle{ a + c = b }[/math], so [math]\displaystyle{ a \leq b }[/math] holds for any [math]\displaystyle{ a, b \in \mathbb{Z} }[/math], so [math]\displaystyle{ \leq }[/math] is not a partial order. There are also examples of monoids that 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 [math]\displaystyle{ a + b = a \vee b }[/math] and [math]\displaystyle{ a \mathop {\dot {-}} b = a \wedge \neg b }[/math].^[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]

[math]\displaystyle{ a \mathop {\dot -} b = \begin{cases} 0 & \mbox{if } a \lt b \\ a - b & \mbox{if } a \ge b, \end{cases} }[/math]

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]

[math]\displaystyle{ a \mathop {\dot -} b = \max(a - b, 0). }[/math]

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

[math]\displaystyle{ \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} }[/math]

A definition that does not need the predecessor function is:

[math]\displaystyle{ \begin{align} a \mathop {\dot -} 0 &= a \\ 0 \mathop {\dot -} b &= 0 \\ S(a) \mathop {\dot -} S(b) &= a \mathop {\dot -} b. \end{align} }[/math]

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:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Notes

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