A wheel is a type of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
(in the sense of
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures in general, not specific types of algebraic structures.
For instance, rather than considering groups or rings as the object of stud ...
) where division is always defined. In particular,
division by zero
In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the di ...
is meaningful. The
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s can be extended to a wheel, as can any
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
.
The term ''wheel'' is inspired by the
topological
Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
picture
of the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not int ...
together with an extra point
⊥ (
bottom element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an ele ...
) such that
.
A wheel can be regarded as the equivalent of a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
(and
semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
) where addition and multiplication are not a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
but respectively a
commutative monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
and a
commutative monoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
with
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
.
Definition
A wheel is an
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
, in which
*
is a set,
*
and
are elements of that set,
*
and
are
binary operation
In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation ...
s,
*
is a
unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to ''binary operations'', which use two operands. An example is any function , where is a set; the function is a unary operation ...
,
and satisfying the following properties:
*
and
are each
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
and
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
, and have
and
as their respective
identities.
*
is an
involution
Involution may refer to: Mathematics
* Involution (mathematics), a function that is its own inverse
* Involution algebra, a *-algebra: a type of algebraic structure
* Involute, a construction in the differential geometry of curves
* Exponentiati ...
, for example
*
is
multiplicative, for example
*
*
*
*
*
*
Algebra of wheels
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument
similar (but not identical) to the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
, such that
becomes shorthand for
, but neither
nor
in general, and modifies the rules of
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
such that
*
in the general case
*
in the general case, as
is not the same as the
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
of
.
Other identities that may be derived are
*
*
*
where the negation
is defined by
and
if there is an element
such that
(thus in the general case
).
However, for values of
satisfying
and
, we get the usual
*
*
If negation can be defined as above then the
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
is a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, and every commutative ring is such a subset of a wheel. If
is an
invertible element
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
of the commutative ring then
. Thus, whenever
makes sense, it is equal to
, but the latter is always defined, even when
.
Examples
Wheel of fractions
Let
be a commutative ring, and let
be a multiplicative
submonoid
In abstract algebra, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being .
Monoids are semigroups with identity ...
of
. Define the
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group (mathematics), group, ring (mathematics), ring, or vector space) that is compatible with the structure in the ...
on
via
:
means that there exist
such that
.
Define the ''wheel of fractions'' of
with respect to
as the quotient
(and denoting the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
containing
as