Monus
Encyclopedia
In mathematics, monus is a variant of subtraction
that can be defined for certain commutative monoid
s that are not groups.
The monus a − b of two elements a and b (also written with a dot above the minus sign) is defined in certain commutative monoids M such that the relation defined by b ≤ a if and only if there is another element c such that b + c = a is a partial order. Since this relation is a preorder
in every monoid, the condition amounts to saying that the relation is antisymmetric
, i.e. if a ≤ b and b ≤ a then a = b. If for any b ≤ a there exists a unique least c such that b + c = a, then M is called a commutative monoid with monus.
In a commutative monoid with monus, the monus a −b of any two elements a and b is defined as follows:
For example the natural number
s including 0 form commutative monoid with monus, the order ≤ being the usual order. In this case the monus of a and b can be defined with reference to the ordinary subtraction of integer
s as max(a − b, 0). The latter definition is commonly encountered in computability theory
.
Subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with...
that can be defined for certain commutative monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
s that are not groups.
The monus a − b of two elements a and b (also written with a dot above the minus sign) is defined in certain commutative monoids M such that the relation defined by b ≤ a if and only if there is another element c such that b + c = a is a partial order. Since this relation is a preorder
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
in every monoid, the condition amounts to saying that the relation is antisymmetric
Antisymmetric relation
In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in Xor, equivalently,In mathematical notation, this is:\forall a, b \in X,\ R \and R \; \Rightarrow \; a = bor, equivalently,...
, i.e. if a ≤ b and b ≤ a then a = b. If for any b ≤ a there exists a unique least c such that b + c = a, then M is called a commutative monoid with monus.
In a commutative monoid with monus, the monus a −b of any two elements a and b is defined as follows:
- a − b = the least c such that b + c = a, if b ≤ a,
- a − b = 0 otherwise.
For example the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s including 0 form commutative monoid with monus, the order ≤ being the usual order. In this case the monus of a and b can be defined with reference to the ordinary subtraction of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s as max(a − b, 0). The latter definition is commonly encountered in computability theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
.