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Monus

Truncating subtraction on natural numbers, or a generalization thereof From Wikipedia, the free encyclopedia

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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 minus sign, "", because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus sign, "", to distinguish it from the standard subtraction operator.

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Notation

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A use of the monus symbol is seen in Dennis Ritchie's PhD Thesis from 1968.[2]

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[3] and that it is transitive.[4] 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 M is called a commutative monoid with monus[5]:129 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 that are naturally ordered but are not semirings with monus.[6]

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Other structures

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[7]) 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

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Perspective

If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under and .[5]: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,[8] limited subtraction, proper subtraction, doz (difference or zero),[9] and monus.[10] Truncated subtraction is usually defined as[8]

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[10]

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[8]

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.[8] Truncated subtraction is also used in the definition of the multiset difference operator.

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Properties

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Perspective

The class of all commutative monoids with monus form a variety.[5]:129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:

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Notes

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