hcju/proofwikips huggingface dataset:
Definition:Imperial/Volume/Pipe
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pipe''' }} {{eqn | r = 2       | c = hogsheads }} {{eqn | r = 0 \cdotp 58189 \, 952       | c = cubic metres }} {{eqn | r = 581 \cdotp 89952       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Hogshead
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hogshead''' }} {{eqn | r = 2       | c = barrels }} {{eqn | r = 0 \cdotp 29094 \, 976       | c = cubic metres }} {{eqn | r = 290 \cdotp 94976       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Barrel
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''barrel''' }} {{eqn | r = 2       | c = kilderkins }} {{eqn | r = 0 \cdotp 14547 \, 488       | c = cubic metres }} {{eqn | r = 145 \cdotp 47488       | c = litres }} {{end-eqn}}
Definition:Litre
The '''litre''' is an SI unit of volume. It is defined as: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''litre''' }} {{eqn | r = 1000       | c = cubic centimetres }} {{eqn | r = 10^{-3}       | c = cubic metres }} {{end-eqn}}
Definition:Imperial/Volume/Kilderkin
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''kilderkin''' }} {{eqn | r = 2       | c = bushels }} {{eqn | r = 0 \cdotp 07273 \, 744       | c = cubic metres }} {{eqn | r = 72 \cdotp 73744       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Bushel
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''bushel''' }} {{eqn | r = 2       | c = demi-bushels }} {{eqn | r = 0 \cdotp 03636 \, 872       | c = cubic metres }} {{eqn | r = 36 \cdotp 36872       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Demi-Bushel
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''demi-bushel''' }} {{eqn | r = 2       | c = pecks }} {{eqn | r = 18 \cdotp 18436       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Peck
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''peck''' }} {{eqn | r = 2       | c = (imperial) gallons }} {{eqn | r = 9 \cdotp 09218       | c = litres }} {{end-eqn}}
Definition:Imperial/Volume/Gallon
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gallon''' }} {{eqn | r = 2       | c = pottles }} {{eqn | r = 4 \cdotp 54609       | c = litres }} {{eqn | r = 4 \, 546 \cdotp 09       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Pottle
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pottle''' }} {{eqn | r = 2       | c = quarts }} {{eqn | r = 2 \cdotp 273045       | c = litres }} {{eqn | r = 2 \, 273 \cdotp 045       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Quart
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quart''' }} {{eqn | r = 2       | c = pints }} {{eqn | r = 1 \cdotp 13652 25       | c = litres }} {{eqn | r = 1 \, 136 \cdotp 5225       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Pint
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pint''' }} {{eqn | r = 2       | c = chopins }} {{eqn | r = 0 \cdotp 5682 \, 6125       | c = litres }} {{eqn | r = 568 \cdotp 26125       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Chopin
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chopin''' }} {{eqn | r = 2       | c = gills }} {{eqn | r = 0 \cdotp 28413 \, 0625       | c = litres }} {{eqn | r = 284 \cdotp 13062 \, 5       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Gill
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gill''' }} {{eqn | r = 5       | c = fluid ounces }} {{eqn | r = 0 \cdotp 14206 \, 53125       | c = litres }} {{eqn | r = 142 \cdotp 06531 \, 25       | c = millilitres }} {{end-eqn}}
Definition:Imperial/Volume/Fluid Ounce
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''fluid ounce''' }} {{eqn | r = \dfrac 1 {20}       | c = pint }} {{eqn | r = 28 \cdotp 41306 \, 25       | c = millilitres }} {{end-eqn}}
Definition:Wallis's Number
'''Wallis's number''' is the real root of the cubic: :$x^3 - 2 x - 5 = 0$ Its approximate value is: :$2 \cdotp 09455 \, 1 \ldots$ {{OEIS|A007493}}
Definition:Stirling's Constant
'''Stirling's constant''' is a name given to the square root of $2$ times $\pi$ (pi). It is approximated by the decimal expansion: :$\sqrt {2 \pi} \approx 2 \cdotp 50662 \, 82746 \, 31000 \, 5 \ldots$ {{OEIS|A019727}} {{NamedforDef|James Stirling|cat = Stirling}}
Definition:Gelfond's Constant
'''Gelfond's constant''' is the number obtained by raising Euler's number $e$ to the power of $\pi$ (pi): :$e^\pi$
Definition:Time/Fourth Dimension
In the Theory of Relativity, it is convenient to consider time as another dimension in addition to, and treated similarly to, the $3$ dimensions of ordinary space.
Definition:Quadratrix of Hippias/Definition 1
The '''quadratrix of Hippias''' is the plane curve defined in Cartesian coordinates as: :$y = x \cot \left({\dfrac {\pi x }{2 a} }\right)$ for some real constant $a \in \R$. :600px The above diagram illustrates the '''quadratrix of Hippias'''.
Definition:Quadratrix of Hippias
=== Definition 1 === {{:Definition:Quadratrix of Hippias/Definition 1}} === Definition 2 === {{:Definition:Quadratrix of Hippias/Definition 2}}
Definition:Orthocenter
Let $\triangle ABC$ be a triangle. The '''orthocenter''' of $\triangle ABC$ is where its three altitudes meet. :500px
Definition:Medial Triangle
Let $\triangle ABC$ be a triangle. The '''medial triangle''' of $\triangle ABC$ is the triangle formed by joining the midpoint of each of the sides of $\triangle ABC$.
Definition:Euler Line
Let $\triangle ABC$ be a triangle which is not equilateral. The '''Euler line''' of $\triangle ABC$ is the straight line through the orthocenter, centroid and circumcenter of $\triangle ABC$. :700px In the above, the line $OGK$ is the '''Euler line''' of $\triangle ABC$.
Definition:Tessellation
A '''tessellation''' is a division of the plane into plane geometric figures, referred to as tiles. The division is made in such a way that the tiles are fitted together with no gaps and no overlaps.
Definition:Tile
A '''tile''' is a plane geometric figure which forms part of a tessellation.
Definition:Regular Tessellation
A '''regular tessellation''' is a tessellation such that: :Its tiles consist of congruent regular polygons :No vertex of one tile may lie against the side of another tile.
Definition:Magic Square
A '''magic square''' is an arrangement of $n^2$ numbers into an $n \times n$ square array such that: :the sum of the elements of each row :the sum of the elements in each column :the sum of the elements along each diagonal are the same. When the elements are not specified, it is conventional for them to be the first $n^2$ (strictly) positive integers.
Definition:Magic Square/Order
An $n \times n$ magic square is called an '''order $n$ magic square'''.
Definition:Coloring/Why Colors?
It is clear that the nature of the actual elements of a coloring $C$ is irrelevant. They are traditionally referred to as '''colors''' because this subfield of graph theory arose from considerations of the coloring of the faces of planar graphs such that adjacent faces have different '''colors'''. This was the origin of the famous Four Color Theorem.
Definition:Coloring/Vertex Coloring
A '''vertex $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as an assignment of one element from a set $C$ of $k$ colors to each vertex in $V$. That is, a '''vertex $k$-coloring''' of the graph $G = \left({V, E}\right)$ is a mapping $c: V \to \left\{{1, 2, \ldots k}\right\}$. A graph with such a coloring is called a '''vertex-colored graph'''.
Definition:Continued Square Root
A '''continued square root''' is an expression of the form: :$a_0 + \sqrt {a_1 + \sqrt {a_2 + \sqrt {a_3 + \sqrt {\cdots} } } }$ where $\left\langle{a_n}\right\rangle$ is an arbitrary sequence of numbers.
Definition:Perfect Power
A '''perfect power''' is an integer which can be expressed in the form: :$a^k$ where both $a$ and $k$ are integers with $a \ge 1$ and $k \ge 2$. === Sequence === {{:Definition:Perfect Power/Sequence}}
Definition:Perfect Power/Sequence
The sequence of perfect powers begins: :$1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, \ldots$
Definition:Zu Chongzhi Fraction
The '''Zu Chongzhi fraction''' is an exceptionally accurate approximation to $\pi$ (pi): :$\pi \approx \dfrac {355} {113}$ whose decimal expansion is: :$\pi \approx 3 \cdotp 14159 \, 292$ {{OEIS|A068079}}
Definition:Squarely Irrational Number
A number is '''squarely irrational''' if both the number itself and its square is irrational. Thus, the set of the '''squarely irrational numbers''' is $\left\{{x \in \R \setminus \Q : x^2 \in \R \setminus \Q}\right\}$.
Definition:Rationally Expressible Number
A number is '''rationally expressible''' if its square is rational. Thus, the set of the '''rationally expressible numbers''' is $\set {x \in \R : x^2 \in \Q}$.
Definition:Square Root/Historical Note
It is suggested by some sources that the symbol $\surd$ (a stylised '''r''' for '''radix''') for the '''square root''' may have originated with {{AuthorRef|René Descartes}}, but there is evidence that it may have been around a lot earlier than that.
Definition:Philosophical Element
The Pythagoreans postulated that the Universe was composed of $4$ '''elements''': :Earth :Water :Air :Fire. Subsequently, the alchemical tradition proposed the existence of a fifth '''element''': quintessence, or aether. === Earth === {{:Definition:Philosophical Element/Earth}} === Water === {{:Definition:Philosophical Element/Water}} === Air === {{:Definition:Philosophical Element/Air}} === Fire === {{:Definition:Philosophical Element/Fire}} === Quintessence === {{:Definition:Philosophical Element/Quintessence}}
Definition:Philosophical Element/Earth
'''Earth''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. All the '''earthy''' matter of the Universe was supposed to collect at its natural place, which was the centre of the Universe, that is, Earth itself.
Definition:Philosophical Element/Air
'''Air''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place around the rim of the sphere of water.
Definition:Philosophical Element/Fire
'''Fire''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place outside the sphere of air.
Definition:Philosophical Element/Water
'''Water''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place about the rim of the surface of Earth.
Definition:Linearly Irrational Number
A number is '''linearly irrational''' if both the number itself is irrational but its square is rational. Thus, the set of the '''linearly irrational numbers''' is $\left\{{x \in \R \setminus \Q : x^2 \in \Q}\right\}$.
Definition:Exponential Order
Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$. {{explain|Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision".}} Then $f$ is said to be of '''exponential order''', denoted $f \in \mathcal E$,  it is of exponential order $a$ for some $a > 0$.
Definition:Philosophical Element/Quintessence
'''Quintessence''' was the fifth element postulated by the Pythagoreans, and was supposed to be the material out of which the heavens were made. It was postuiated in order to explain how the heavenly bodies were in constant motion, and never came to rest like everything else that could be seen in the universe.
Definition:Pythagoreans/Quadrivium
The '''quadrivium''' was the medieval name of the required course of study of the Pythagoreans, which had been adopted by the educational establishments in Europe. The required bodies of knowledge were divided into '''discrete''' and '''continuous''': Thus: :'''Arithmetic''': study of the absolute discrete :'''Music''': study of the relative discrete :'''Geometry''': study of the '''stable''' continuous :'''Astronomy''': study of the '''moving''' continuous.
Definition:Pythagoreans/Trivium
The '''trivium''' was the medieval name of the supplementary course of study of the Pythagoreans, adopted by the educational establishments in Europe. These supplementary bodies of knowledge were: :'''Grammar''' :'''Rhetoric''' :'''Logic'''.
Definition:Pythagoreans/Quadrivium/Linguistic Note
The word '''quadrivium''' is a Latin word meaning '''the four ways''', or '''place where four roads meet''' (that is, literally: '''crossroads''').
Definition:Pythagoreans/Trivium/Linguistic Note
The word '''trivium''' is a Latin word meaning '''the three ways''', or '''place where three roads meet'''. As the '''trivium''' was a preliminary course of study to the quadrivium, and thereby simpler, from this root the word '''trivial''' evolved, which has a meaning in accordance with '''insignificant''', '''self-evident''' and '''commonplace''', and the like.
Definition:Dual Polyhedron
Let $P$ be a polyhedron. The '''dual polyhedron''' $D$ of $P$ is the polyhedron which can be constructed as follows: :$(1): \quad$ The vertices of $D$ are the centroids of the faces of $P$. :$(2): \quad$ For each edge of $P$ which is adjacent to two faces $F_1$ and $F_2$ of $P$, an edge of $D$ is constructed which is adjacent to the vertices of $D$ forming the centroids of $F_1$ and $F_2$.
Definition:Brown Numbers
'''Brown numbers''' are pairs of integers $\left({m, n}\right)$ such that: :$n! + 1 = m^2$ There are only $3$ known such pairs: :$\left({5, 4}\right)$ :$\left({11, 5}\right)$ :$\left({71, 7}\right)$
Definition:Feigenbaum Constants
The '''Feigenbaum constants''' are a pair of real constants which arise in bifurcation theory. === First Feigenbaum Constant === {{:Definition:Feigenbaum Constants/First}} === Second Feigenbaum Constant === {{:Definition:Feigenbaum Constants/Second}} {{NamedforDef|Mitchell Jay Feigenbaum|cat = Feigenbaum}}
Definition:Feigenbaum Constants/First
The '''first Feigenbaum constant''' is the limiting ratio of each bifurcation interval to the next between every period doubling of a one-parameter mapping: :$x_{i + 1} = f \left({x_i}\right)$ where $f \left({x}\right)$ is a function parameterized by the bifurcation parameter $a$. It is given by the limit: :$\displaystyle \delta = \lim_{n \mathop \to \infty} \dfrac{a_ {n - 1} - a_{n - 2} } {a_n - a_{n - 1} } = 4 \cdotp 66920 \, 16091 \, 02990 \, 67185 \, 32038 \, 20466 \, 20161 \, 72 \ldots$ where $a_n$ are discrete values of $a$ at the $n$th period doubling. Its precise value appears to be a topic of research, as various resources quote it differently from the above, including the {{AuthorRef|David Wells}} $1997$ source work {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}}.
Definition:Feigenbaum Constants/Second
The '''secondFeigenbaum constant''' $\alpha$ is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to $\alpha$ when the ratio between the lower subtine and the width of the tine is measured. Its approximate value is given by: :$\alpha \approx 2 \cdotp 50290 \, 78750 \, 95892 \, 82228 \, 39028 \, 73218 \, 21578 \cdots$
Definition:Pythagorean Triangle
A '''Pythagorean triangle''' is a right triangle whose sides all have lengths which are integers.
Definition:Automorphic Number
An '''automorphic number''' is a positive integer all of whose powers end in that number.
Definition:Absolutely Convergent Integral
Let $I = \displaystyle \int_S \map f x \rd x$ be an improper integral, where $S$ is an unbounded closed interval. $I$ is '''absolutely convergent'''  $\displaystyle \int_S \size {\map f x} \rd x$ converges.
Definition:Untouchable Number
An '''untouchable number''' is a (strictly) positive integer that is not the aliquot sum of any integer.
Definition:Polytope
A '''polytope''' is an $n$-dimensional generalization of a polygon and polyhedron, for $n \in \Z_{>0}$.
Definition:Simplex
A '''simplex''' is an $n$-dimensional generalization of a triangle and tetrahedron, for $n \in \Z_{>0}$. A $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices.
Definition:Pentatope
A '''pentatope''' is a $4$-simplex: :400px
Definition:Triangle (Geometry)/Right-Angled/Legs
In a right-angled triangle, the two sides which are not the hypotenuse are referred to as its '''legs'''.
Definition:Fibonacci Number/Negative
The definition of '''Fibonacci numbers''' for negative integers is an extension of the definition for positive integers: :$F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n + 2} - F_{n + 1} & : n < 0 \end{cases}$ for all $n \in \Z$.
Definition:Extension of Sequence
As a sequence is, by definition, also a mapping, the definition of an '''extension of a sequence''' is the same as that for an extension of a mapping: Let: : $\left \langle {a_k} \right \rangle_{k \mathop \in A}$ be a sequence on $A$, where $A \subseteq \N$. : $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ be a sequence on $B$, where $B \subseteq \N$. : $A \subseteq B$ : $\forall k \in A: b_k = a_k$. Then $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ '''extends''' or '''is an extension of''' $\left \langle {a_k} \right \rangle_{k \mathop \in A}$.
Definition:Extension of Sequence/Negative Integers
A sequence on $\N$ can be extended to the negative integers. Let $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ and $\left \langle {b_k} \right \rangle_{k \mathop \in \N}$ be sequences on $\N$. Let $a_0 = b_0$. Let $c_k$ be defined as: :$\forall k \in \Z: c_k = \begin{cases} a_k & : k \ge 0 \\ b_{-k} : & k \le 0 \end{cases}$ Then $\left \langle {c_k} \right \rangle_{k \mathop \in \Z}$ '''extends''' (or '''is an extension of''') $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ to the negative integers.
Definition:Ansatz
An '''Ansatz''' is a guessed solution, or general form of a solution, to a mathematical problem. A mathematician making the '''Ansatz''' will then attempt to prove that the '''Ansatz''' is indeed a solution to the original problem. If he or she succeeds, the '''Ansatz''' is justified retroactively.
Definition:Finite Suprema Set
Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite suprma set''' of $X$, denoted $\operatorname{finsups}\left({X}\right)$, is defined by :$\left\{ {\sup A: A \in \mathit{Fin}\left({X}\right) \land A}\right.$ admits a supremum$\left.{}\right\}$ where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.
Definition:Contour Integral/Complex/Closed
Let $C$ be a closed contour in $\C$. Then the symbol $\displaystyle \oint$ is used for the '''contour integral''' on $C$. The definition remains the same: :$\displaystyle \oint_C f \left({z}\right) \rd z := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t$
Definition:Goldbach Decomposition
Let $n \in \Z_{>0}$ be a positive integer such that $n$ is even. A '''Goldbach decomposition''' for $n$ is a pair of primes $p_1, p_2$ such that: :$p_1 + p_2 = n$
Definition:Cauchy Principal Value
The '''Cauchy principal value''' is an extension of the concept of an improper integral when the latter might not exist. === Real Integral === {{:Definition:Cauchy Principal Value/Real Integral}} === Complex Integral === {{:Definition:Cauchy Principal Value/Complex Integral}} === Contour Integral === {{:Definition:Cauchy Principal Value/Contour Integral}}
Definition:Tesseract
A '''tesseract''' is the $4$-dimensional regular polytope corresponding to the cube: :500px
Definition:Steiner Tree
Let $G$ be a configuration of points in space. A '''Steiner tree''' is a network of lines joining the points of $G$ such that the total length of those lines is a minimum. {{NamedforDef|Jakob Steiner|cat = Steiner}}
Definition:Fermat Quotient
Let $a$ be a positive integer. Let $p$ be an odd prime. The '''Fermat quotient of $a$ with respect to $p$''' is defined as: :$\map {q_p} a = \dfrac {a^{p - 1} - 1} p$
Definition:Rational Diagonal
The number $7$ (seven) was referred to by the ancient Greeks as the '''rational diagonal''' of the $5 \times 5$ square. This was on account of the fact that: :$5^2 + 5^2 = 50 \approx 7^2 = 49$
Definition:Dissection
A '''dissection''' is a partition of a geometric figure into two or more components.
Definition:Frieze
A '''frieze''' is a plane geometrical figure which has translational symmetry in one line only.
Definition:Pseudoprime (Order Theory)
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $p \in S$. Then $p$ is '''pseudoprime'''  :there exists a prime ideal $P$ in $L$: $p = \sup P$
Definition:Cyclotomic Polynomial
Let $n \ge 1$ be a natural number. The '''$n$th cyclotomic polynomial''' is the polynomial :$\displaystyle \Phi_n \paren x = \prod_\zeta \paren {x - \zeta}$ where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$.
Definition:Fano Plane
The '''Fano plane''' is the finite projective plane of order $2$. It can be denoted: :$\operatorname {PG} \left({2, 2}\right)$ where: :$\operatorname {PG}$ stands for '''projective geometry''' :the first parameter $2$ specifies the dimension :the second parameter $2$ specifies the order. $\operatorname {PG} \left({2, 2}\right)$ has the following properties: : It has $7$ points : It has $7$ lines : There are $3$ points on each line : There are $3$ lines through each point. :400px
Definition:Separable Polynomial
=== Definition 1 === {{:Definition:Separable Polynomial/Definition 1}} === Definition 2 === {{:Definition:Separable Polynomial/Definition 2}} === Definition 3 === {{:Definition:Separable Polynomial/Definition 3}}
Definition:Multiplicity (Polynomial)
Let $R$ be a commutative ring with unity. Let $P \in R \left[{X}\right]$ be a nonzero polynomial. Let $a \in R$ be a root of $P$. The '''multiplicity''' of $a$ in $P$ is the largest positive integer $n$ such that $\left({X - a}\right)^n$ divides $P \left({X}\right)$ in $R \left[{X}\right]$. {{refactor|Extract the following into a separate page, transcluded (perhaps an "examples" page)}} A '''double root''' is a root of multiplicity at least $2$.
Definition:Knot Product
The '''product''' of two knots is informally defined as what you get when you tie those two knots together on the same string. {{stub|Yes I know this is sketchy and inadequate -- a lot more background is needed before this can be fleshed out and made rigorous.}}
Definition:Prime Knot
A '''prime knot''' is a knot which is not the product of two or more other knots.
Definition:Monoid Ring
Let $R$ be a ring with unity. Let $\left({G, *}\right)$ be a monoid. Let $R^{\left({G}\right)}$ be the free $R$-module on $G$. Let $\left\{ {e_g: g \in G}\right\}$ be its canonical basis. {{explain|Before the above notation can be properly understood, the precise nature of the canonical basis needs to be expanded so as to make the operations completely explicit.}} By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map: :$\circ: R^{\left({G}\right)} \times R^{\left({G}\right)} \to R^{\left({G}\right)}$ which satisfies: :$e_g \circ e_h = e_{g \mathop * h}$ Then $R \left[{G}\right] = \left({R^{\left({G}\right)}, +, \circ}\right)$ is called the '''monoid ring of $G$ over $R$'''.
Definition:Free Module on Set
Let $R$ be a ring. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\left({R, +, \circ}\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $I$ be an indexing set. The '''free $R$-module on $I$''' is the direct sum of $R$ as a module over itself: :$\displaystyle R^{\left({I}\right)} := \bigoplus_{i \mathop \in I} R$ of the family $I \to \{R\}$ to the singleton $\{R\}$. === Canonical Basis === {{Definition:Free Module on Set/Canonical Basis}} === Canonical Mapping === {{Definition:Free Module on Set/Canonical Mapping}}