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
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
; it is also denoted with the
symbol to distinguish it from the standard subtraction operator.
Notation
Definition
Let
be a commutative
monoid. Define a
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
on this monoid as follows: for any two elements
and
, define
if there exists an element
such that
. It is easy to check that
is
reflexive and that it is
transitive.
is called ''naturally ordered'' if the
relation is additionally
antisymmetric and hence a
partial order. Further, if for each pair of elements
and
, a unique smallest element
exists such that
, then is called a ''commutative monoid with monus''
[
] and the ''monus''
of any two elements
and
can be defined as this unique smallest element
such that
.
An example of a commutative monoid that is not naturally ordered is
, the commutative monoid of the
integers with usual
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
, as for any
there exists
such that
, so
holds for any
, so
is not a partial order. There are also examples of monoids which are naturally ordered but are not semirings with monus.
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid
Semirings for breakfast
slide 17) 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 is an ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
in a Boolean algebra, then is a commutative monoid with monus under and .
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 saturating variant of standard subtraction, variously referred to as truncated subtraction,[
] limited subtraction, proper subtraction, doz (''difference or zero''), and monus.[
] Truncated subtraction is usually defined as
:
where − denotes standard subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
. 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
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
, truncated subtraction is defined in terms of the predecessor function (the inverse of the successor function
In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
):
:
A definition that does not need the predecessor function is:
:
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:
Notes
Algebraic structures