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

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glyph Unicode name Unicode code point[1] HTML character entity reference HTML/XML numeric character references TeX
DOT MINUS U+2238 ∸ \dot -
MINUS SIGN U+2212 − − -

Definition

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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[2] and that it is transitive.[3]   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[4]: 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.[5]

Other structures

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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.

Examples

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If M is an ideal in a Boolean algebra, then M is a commutative monoid with monus under   and  .[4]: 129 

Natural numbers

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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.

Properties

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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:

 

Notes

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  1. ^ Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
  2. ^ taking   to be the neutral element of the monoid
  3. ^ if   with witness   and   with witness   then   witnesses that  
  4. ^ a b c Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
  5. ^ M.Monet (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". Mathematics Stack Exchange. Retrieved 2016-10-14.
  6. ^ Semirings for breakfast, slide 17
  7. ^ a b c d Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
  8. ^ Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8.
  9. ^ a b Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice (eds.). Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. Vol. 1101. Springer. p. 522. ISBN 3-540-61463-X.