Monus

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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.

Notation[edit]

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 − − -

Definition[edit]

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.[2]: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.

Examples[edit]

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,[3] limited subtraction, proper subtraction, and monus.[4] Truncated subtraction is usually defined as[3]

a \mathop {\dot -} b =
\begin{cases}
0 & \mbox{if } a < b \\
a - b & \mbox{if } a \ge b,
\end{cases}

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[4]

a \mathop {\dot -} b = \max(a - b, 0).

In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[3]


\begin{align}
P(0)    &= 0 \\
P(S(a)) &= a \\
a \mathop {\dot -} 0 &= a \\
a \mathop {\dot -} S(b) &= P(a \mathop {\dot -} b).
\end{align}

Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[3] Truncated subtraction is also used in the definition of the multiset difference operator.

Properties[edit]

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:


\begin{align}
a + (b \dot - a) &= b + (a \dot - b) \\
(a \dot - b) \dot - c &= a \dot - (b \dot - c) \\
(a \dot - a) &= 0 \\
(0 \dot - a) &= 0. \\
\end{align}

Notes[edit]

  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. ^ 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. 
  4. ^ 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.