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	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. Let ≤ be the partial order relation on that monoid, defined so that, for two elements a and b, a ≤ b if and only if there exists another element c such that a + c = b. If for each pair of elements a and b there exists a unique smallest element c such that a ≤ b + c, then M is a commutative monoid with monus. Since the relation ≤ is a preorder in every monoid, the condition that there be a unique smallest element is equivalent to saying that the relation is antisymmetric, i.e. if a ≤ b and b ≤ a then a = b.

In a commutative monoid with monus, the monus a ∸ b of any two elements a and b is the unique smallest element c such that a ≤ b + c.

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.

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, limited subtraction, proper subtraction, and monus. Truncated subtraction is usually defined as

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

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

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

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

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

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

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