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 formula_1 symbol to distinguish it from the standard subtraction operator.
Definition.
Let formula_2 be a commutative monoid. Define a binary relation formula_3 on this monoid as follows: for any two elements formula_4 and formula_5, define formula_6 if there exists an element formula_7 such that formula_8. It is easy to check that formula_3 is reflexive and that it is transitive. formula_10 is called "naturally ordered" if the formula_3 relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements formula_4 and formula_5, a unique smallest element formula_7 exists such that formula_15, then is called a "commutative monoid with monus" and the "monus" formula_16 of any two elements formula_4 and formula_5 can be defined as this unique smallest element formula_7 such that formula_15.
An example of a commutative monoid that is not naturally ordered is formula_21, the commutative monoid of the integers with usual addition, as for any formula_22 there exists formula_7 such that formula_8, so formula_6 holds for any formula_22, so formula_3 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 (sometimes called a dioid) 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 formula_28 and formula_29.
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 ("difference or zero"), 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 (the inverse of the successor 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 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:
formula_33