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.
Contents
Nhom chea monus la ngong zono
Definition
Let
An example of a commutative monoid which is not naturally ordered is
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid) 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
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.
Natural numbers
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
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
In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):
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.
Properties
The class of all commutative monoids with monus form a variety. The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms: