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 commutativemonoid 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
Definition
Let be a commutative monoid. Define a binary relation on this monoid as follows: for any two elements and, define if there exists an element such that. It is easy to check that is reflexive and that it is transitive. is called naturally ordered if the relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements and, a unique smallest element exists such that, then is called a commutative monoid with monus and the monus of any two elements and can be defined as this unique smallest element such that. An example of a commutative monoid that is not naturally ordered is, the commutative monoid of the integers with usual addition, as for any there exists such that, so holds for any, so is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring 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 is an ideal in a Boolean algebra, then is a commutative monoid with monus under and.
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, limited subtraction, proper subtraction, doz, and monus. Truncated subtraction is usually defined as 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 In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function : 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 multisetdifference 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:
