Truncating subtraction on natural numbers, or a generalization thereof
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
Definition
Let
be a commutative monoid. Define a binary relation
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[2] and that it is transitive.[3]
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 M is called a commutative monoid with monus[4]: 129 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, as for any
there exists
such that
, so
holds for any
, so
is not a partial order. There are also examples of monoids that are naturally ordered but are not semirings with monus.[5]
Other structures
Beyond monoids, the notion of monus can be applied to other structures. For instance, a naturally ordered semiring (sometimes called a dioid[6]) 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
and
.[4]: 129
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,[7] limited subtraction, proper subtraction, doz (difference or zero),[8] and monus.[9] Truncated subtraction is usually defined as[7]
![{\displaystyle a\mathop {\dot {-}} b={\begin{cases}0&{\mbox{if }}a<b\\a-b&{\mbox{if }}a\geq b,\end{cases}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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[9]
![{\displaystyle a\mathop {\dot {-}} b=\max(a-b,0).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
In Peano arithmetic, truncated subtraction is defined in terms of the predecessor function P (the inverse of the successor function):[7]
![{\displaystyle {\begin{aligned}P(0)&=0\\P(S(a))&=a\\a\mathop {\dot {-}} 0&=a\\a\mathop {\dot {-}} S(b)&=P(a\mathop {\dot {-}} b).\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A definition that does not need the predecessor function is:
![{\displaystyle {\begin{aligned}a\mathop {\dot {-}} 0&=a\\0\mathop {\dot {-}} b&=0\\S(a)\mathop {\dot {-}} S(b)&=a\mathop {\dot {-}} b.\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Truncated subtraction is useful in contexts such as primitive recursive functions, which are not defined over negative numbers.[7] 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.[4]: 129 The equational basis for the variety of all CMMs consists of the axioms for commutative monoids, as well as the following axioms:
![{\displaystyle {\begin{aligned}a+(b\mathop {\dot {-}} a)&=b+(a\mathop {\dot {-}} b),\\(a\mathop {\dot {-}} b)\mathop {\dot {-}} c&=a\mathop {\dot {-}} (b+c),\\(a\mathop {\dot {-}} a)&=0,\\(0\mathop {\dot {-}} a)&=0.\\\end{aligned}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Notes
- ^ Characters in Unicode are referenced in prose via the "U+" notation. The hexadecimal number after the "U+" is the character's Unicode code point.
- ^ taking
to be the neutral element of the monoid - ^ if
with witness
and
with witness
then
witnesses that ![{\displaystyle a\leq c}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- ^ a b c Amer, K. (1984), "Equationally complete classes of commutative monoids with monus", Algebra Universalis, 18: 129–131, doi:10.1007/BF01182254
- ^ M.Monet (2016-10-14). "Example of a naturally ordered semiring which is not an m-semiring". Mathematics Stack Exchange. Retrieved 2016-10-14.
- ^ Semirings for breakfast, slide 17
- ^ 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.
- ^ Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8.
- ^ a b Jacobs, Bart (1996). "Coalgebraic Specifications and Models of Deterministic Hybrid Systems". In Wirsing, Martin; Nivat, Maurice (eds.). Algebraic Methodology and Software Technology. Lecture Notes in Computer Science. Vol. 1101. Springer. p. 522. ISBN 3-540-61463-X.