Monus

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 because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus {{nowrap|sign, " \mathbin{\dot{-}} ",}} to distinguish it from the standard subtraction operator.

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 because the natural numbers are a CMM under subtraction. It is also denoted with a dotted minus {{nowrap|sign, " \mathbin{\dot{-}} ",}} to distinguish it from the standard subtraction operator. Let (M, +, 0) be a commutative monoid. Define a binary relation \leq on this monoid as follows: for any two elements a and b , define a \leq b if there exists an element c such that a + c = b . It is easy to check that \leq is reflexive and that it is transitive. M is called naturally ordered if the \leq relation is additionally antisymmetric and hence a partial order. Further, if for each pair of elements a and b , a unique smallest element c_0 exists such that a \leq b + c_0 , then is called a commutative monoid with monus and the monus a \mathbin{\dot{-}} b of any two elements a and b can be defined as this unique smallest element c_0 such that a \leq b + c_0 . An example of a commutative monoid that is not naturally ordered is (\mathbb{Z}, +, 0) , the commutative monoid of the integers with usual addition, as for any a, b \in \mathbb{Z} there exists c such that a + c = b , so a \leq b holds for any a, b \in \mathbb{Z} , so \leq is not antisymmetric and therefore not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus. If is an ideal in a Boolean algebra, then is a commutative monoid with monus under a + b = a \vee b and a \mathbin{\dot{-}} b = a \wedge \neg b . a \mathbin{\dot{-}} b = \begin{cases} 0 & \mbox{if } a where − denotes standard subtraction. For example, 5 - 3 = 2 and 3 - 5 = -2 in regular subtraction, whereas in truncated subtraction 3 \mathbin{\dot{-}} 5 = 0 . Truncated subtraction may also be defined as a \mathbin{\dot{-}} b = \max(a - b, 0). In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function (the inverse of the successor function): \begin{align} P(0) &= 0 \\ P(S(a)) &= a \\ a \mathbin{\dot{-}} 0 &= a \\ a \mathbin{\dot{-}} S(b) &= P(a \mathbin{\dot{-}} b). \end{align} A definition that does not need the predecessor function is: \begin{align} a \mathbin{\dot{-}} 0 &= a \\ 0 \mathbin{\dot{-}} b &= 0 \\ S(a) \mathbin{\dot{-}} S(b) &= a \mathbin{\dot{-}} b. \end{align} 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. a + (b \mathbin{\dot{-}} a) &= b + (a \mathbin{\dot{-}} b),\\ (a \mathbin{\dot{-}} b) \mathbin{\dot{-}} c &= a \mathbin{\dot{-}} (b + c),\\ (a \mathbin{\dot{-}} a) &= 0,\\ (0 \mathbin{\dot{-}} a) &= 0.\\ \end{align}

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