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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 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.


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


Let be a commutative monoid. Define a binary relation on this monoid as follows: for any two elements . and , we set if and only if there exists another element such that . It is easy to check that is reflexive (taking to be the neutral element of the monoid) and that it is transitive (if with witness and with witness then witnesses that ). We call naturally ordered if the relation is additionally antisymmetric, so that is a partial order. Further, if for each pair of elements and , there exists a unique smallest element such that , then we call M a commutative monoid with monus[2]:129 and we can then define the monus a ∸ b of any two elements and as this unique smallest element such that .

An example of a commutative monoid which is not naturally ordered is , the commutative monoid of the integers with usual addition, as for any there exists such that , so we have for any , so is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.[3]

Other structures[edit]

Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[4]) 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 a + b = a ∨ b and a ∸ b = a ∧ ¬b.[2]:129

Natural numbers[edit]

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 variant of standard subtraction, variously referred to as truncated subtraction,[5] limited subtraction, proper subtraction, and monus.[6] Truncated subtraction is usually defined as[5]

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[6]

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[5]

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[5] Truncated subtraction is also used in the definition of the multiset difference operator.


The class of all commutative monoids with monus form a variety.[2]:129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:


  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. ^ a b c Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254 
  3. ^ math.SE answer
  4. ^ Semirings for breakfast, slide 17
  5. ^ 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. 
  6. ^ a b Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice. Algebraic Methodology and Software Technology. Lecture notes in computer science. 1101. Springer. p. 522. ISBN 3-540-61463-X.