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.
Define a binary relation
It is easy to check that
is called naturally ordered if the
relation is additionally antisymmetric and hence a partial order.
Further, if for each pair of elements
, a unique smallest element
, then M is called a commutative monoid with monus[4]: 129 and the monus
can be defined as this unique smallest element
An example of a commutative monoid that is not naturally ordered is
, the commutative monoid of the integers with usual addition, as for any
is not a partial order.
There are also examples of monoids that are naturally ordered but are not semirings with monus.
[5] 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.
If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under
[4]: 129 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] 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] In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[7] A definition that does not need the predecessor function is: 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.
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: