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||HTML character entity reference||HTML/XML numeric character references||TeX|
Let (M, +, 0) be a commutative monoid. Let ≤ be the partial order relation on that monoid, defined so that, for two elements a and b, a ≤ b if and only if there exists another element c such that a + c = b. If for each pair of elements a and b there exists a unique smallest element c such that a ≤ b + c, then M is a commutative monoid with monus. Since the relation ≤ is a preorder in every monoid, the condition that there be a unique smallest element is equivalent to saying that the relation is antisymmetric, i.e. if a ≤ b and b ≤ a then a = b.:129
In a commutative monoid with monus, the monus a ∸ b of any two elements a and b is the unique smallest element c such that a ≤ b + c.
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, limited subtraction, proper subtraction, and monus. Truncated subtraction is usually defined as
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.
The class of all commutative monoids with monus form a variety.:129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:
- Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
- Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis 18: 129–131, doi:10.1007/BF01182254
- Vereschchagin, Nikolai K.; Shen, Alexander (2003). Computable Functions. Translated by V. N. Dubrovskii. American Mathematical Society. p. 141. ISBN 0-8218-2732-4.
- 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.