Algebra

Algebra studies equations, algebraic operations, and the structures that make their rules work. Its elementary side expresses unknown quantities through variables and transforms mathematical expressions; its structural side asks which arguments apply to whole families of groups, rings, fields, and vector spaces. The useful step is identifying exactly which properties a calculation needs, so that it can be reused in a new setting.

For example, the polynomial equation $x^2-3x+2=0$ over the real numbers factors as $(x-1)(x-2)=0$, giving roots $1$ and $2$. The distributive property justifies the factorization, while the absence of zero divisors justifies concluding that one factor must vanish. Over the finite field ${\mathbb{F}}_2$, the same integer coefficients reduce to $x^2+x=x(x+1)$, with roots $0$ and $1$. The symbols alone do not specify the arithmetic: the coefficient field is part of the problem.

In a ring with zero divisors, a familiar step can fail. Modulo $8$, all four residue classes $1,3,5,7$ satisfy $x^2=1$. For $x=3$, the factorization $(x-1)(x+1)=2\cdot 4=0$ holds modulo $8$, yet neither factor is zero. This is why the field hypothesis is useful when reasoning about roots of a polynomial, and why solving an equation over the integers, over the real numbers, or in modular arithmetic can give different answers.

The structure is also useful beyond scalar arithmetic. Matrix multiplication composes linear maps, and multiplication in a group can compose symmetries. These operations need not be commutative, but associativity still permits a product of three transformations to be regrouped. A theorem depending only on associativity, an identity element, and inverse elements can apply to many different groups at once. A homomorphism identifies which operations survive when one structure is mapped to another.

Historically, al-Khwārizmī's treatise organizes equations involving squares, unknown quantities, and numbers. Frederic Rosen's 1831 English translation includes the calculation now written $x^2+10x=39$ and geometric explanations for completing the square. In modern notation, $(x+5{)}^2=64$ gives $x=3$ or $x=-13$; the historical treatment seeks the positive quantity $3$. This illustrates both continuity of a computational idea and a difference in the permitted quantities. ^[ref]

A substitution replaces a variable or subexpression by another expression. In an equation or differential equation, introducing a new variable can expose a simpler form; any division or inverse transformation must retain its domain restrictions and separately check discarded branches.

The study of algebraic structures such as groups, rings and fields through their operations and laws.

Factorization expresses an object as a product of simpler factors. Integer factorization decomposes an integer; polynomial factorization decomposes a polynomial. The permitted factors depend on the ambient algebraic structure.

Multiplying numerator and denominator of a fraction by the same nonzero expression can remove a square root from the denominator. For instance,when $a\ge 0$ and both denominators are nonzero. This uses the difference-of-squares identity $(\sqrt{a}+b)(\sqrt{a}-b)=a-b^2$. If $a=b^2$, the original expression may still be defined even though the rationalized quotient is not, so that case must be handled separately. The related rationalization of a square-root difference applies the same multiplication to a numerator.

An equation asserts that two mathematical expressions have the same value. Solving it means finding all allowed assignments to its variables that make that assertion true. It converts a relationship into a constraint: a measurement, a geometric intersection, or an equilibrium model can leave an unknown value to determine.

The allowed values matter. The equation $2x=1$ has solution set $\{1/2\}$ over the rational numbers, but the empty set over the integers. Over the real numbers, $x^2=1$ has two solutions, $-1$ and $1$. A mathematical identity, such as $(x+1{)}^2=x^2+2x+1$, holds for every permitted real $x$. The familiar equality sign therefore does not by itself say whether one is checking a particular value, solving for an unknown, or stating an identity.

For a worked linear equation over the real numbers,The first step subtracts three from both sides; the second divides both sides by the nonzero number two. Reversing the steps recovers the original equation, so these are equivalent equations with the same solution set $\{3\}$. Checking the value gives $2\cdot 3+3=9$. The role of reversible operations and the classification by solution sets are described in College Algebra 2e, §2.2: Solving Linear Equations in One Variable.

When the two sides define real-valued functions $f$ and $g$ on a common domain of a function, solutions are exactly the input values at which their function graphs intersect. Equivalently, they are the zeros of a function $f-g$. This gives a useful connection between symbolic algebra and geometry: rearranging an equation can simplify the comparison without changing the inputs where equality holds.

Not every operation is reversible on every domain of a function. Dividing $x(x-1)=0$ by $x$ gives $x-1=0$ only after restricting to $x\ne 0$; it loses the solution $x=0$ if that restriction is forgotten. Squaring both sides can instead introduce an extraneous solution, because distinct real numbers can have the same square. The source equation, its domain of a function, and the direction of each implication should travel together.

An equation which is not a linear equation in the specified unknowns.

A variable whose value is sought subject to an equation or other constraint.

An implicit equation specifies a relation between variables without isolating a designated output as an explicit function of the inputs. For example, $x^2+y^2=1$ specifies a circle. On $-1<x<1$, it determines the two real branches $y=\pm \sqrt{1-x^2}$, each an implicit function when one branch is selected. The implicit function theorem gives local conditions for a unique differentiable branch; implicit differentiation computes its derivative without first solving explicitly.

A system of equations requires one choice of unknowns to satisfy several equations at once. Its solution set is the intersection of the solution sets of the individual equations. For example, $x^2+y^2=1$ and $y=(x-1{)}^2$ ask for intersections of a unit circle and a parabola. A linear system is the special case in which every equation is linear in the unknowns.

Two equations are equivalent on a specified domain of a function if they have the same solution set. Adding the same defined mathematical expression to both sides is reversible. Multiplying or dividing both sides by a mathematical expression is reversible only where that expression is nonzero and invertible in the arithmetic being used. In a field, every nonzero scalar is invertible; a general ring needs an additional check.

For example, $x^2=1$ and $(x-1)(x+1)=0$ are equivalent equations over the real numbers, by the distributive property. Concluding $x=1$ or $x=-1$ additionally uses the absence of zero divisors. The modular example in algebra shows why that extra inference can fail in a ring with zero divisors.

An extraneous solution satisfies a transformed equation but fails the original equation or its restrictions. The term describes a candidate generated during the calculation, rather than a true solution that can be kept in the final solution set.

Consider the real equationThe square root requires $x\ge 0$. Because the square root is nonnegative, any solution must also have $x-2\ge 0$, so $x\ge 2$. Squaring givesThe squared equation has candidates $x=1$ and $x=4$. At $x=4$, the original equality is $2=2$. At $x=1$, it is $1=-1$, which fails: this candidate is an extraneous solution. Thus the original solution set is exactly $\{4\}$.

On the restricted domain of a function $x\ge 2$, both original sides are nonnegative, so squaring is reversible there and the two equations are equivalent. Retaining the sign restriction or checking every candidate in the original equation achieves the same final solution set. The general need to check candidates after raising both sides to a power is explained in College Algebra 2e, §2.6: Solving Radical Equations.

A mathematical identity is an equation that holds for every assignment in a specified domain of a function. For example, the distributive property gives $(x+1{)}^2=x^2+2x+1$ for every real number $x$. An identity can replace one mathematical expression by another without imposing a new restriction on that domain.

The domain remains essential. The equality $(x^2-1)/(x-1)=x+1$ holds for every real $x\ne 1$, but its left side is undefined at $x=1$. Cancelling a factor does not retrospectively define the original mathematical expression at the excluded input. An identity is different from an identity element of an algebraic operation or an identity function, which are separate concepts despite sharing a name.

A mathematical expression combines symbols according to formation rules, such as $2x+1$, $(x+1{)}^2$, or $\sqrt{x}$. With values for its variables and meanings for its algebraic operations, it can be evaluated. An equation relates two mathematical expressions; an expression alone need not assert anything.

For example, $2x+3$ evaluates to seven at $x=2$ and nine at $x=3$. Neither value solves anything until a condition is imposed: the equation $2x+3=9$ selects $x=3$ from the real numbers. This distinction is useful when moving from a formula to a question about which inputs satisfy it. Evaluation by replacing the variables with assigned values is described in College Algebra 2e, §1.1: Evaluating Algebraic Expressions.

Different written mathematical expressions can have the same value for every input in a common domain of a function. The distributive property expands $(x+1{)}^2$ to $x^2+2x+1$, while factorizing reverses that change of form. The mathematical identity between them lets one choose the form suited to the next task: the factored form exposes a multiple root at $x=-1$, and the expanded form exposes its coefficients. Simplifying should preserve values on the stated domain, rather than silently enlarging it.

A mathematical expression given by a finite combination of specified operations and standard functions, rather than an unevaluated recurrence relation or an unbounded procedure. Which functions are admitted is part of the convention. For example, the sum of the first $n$ positive integers has the closed-form expression $n(n+1)/2$.

Algebraic K-theory studies invariants built from projective modules and their automorphisms over rings. Its lowest groups record the additive projective-module classes and stable invertible matrices, respectively. It differs from Topological K-theory, although vector-bundle clutching and module patching have parallel forms.

A cartesian square of unital rings $A=A_1{\times }_{A_0}{A}_2$ is a Milnor square when at least one map $A_i\to A_0$ is surjective. Finitely generated projective modules over $A$ can then be assembled from projective modules over the two upper rings together with an isomorphism of their reductions over $A_0$.

Evaluation of the cyclic generator at $1$ and at a primitive $p$th root gives the displayed pullback. The maps to the common field reduce integers modulo $p$ and set ${\zeta }_p=1$. If a representative polynomial for the second component has value $c$ at one, adding an integer multiple of ${\Phi }_p$ adjusts that value by a multiple of $p$ without changing the second component. This proves surjectivity onto compatible pairs.

The diagonal maps are extension of scalars; the maps to $A_0$ are differences. The connecting map sends $g$ to the difference of its patched free module and the corresponding free $A$-module. Elementary lifting and common free stabilization prove exactness at every internal term. The displayed fragment does not assert injectivity at its first term or surjectivity at its last.

A compatible triple $(P_1,P_2,\varphi )$ patches to $\{(x_1,x_2):\varphi ({\bar{x}}_1)={\bar{x}}_2\}$. It is finitely generated projective and recovers $P_i$ after extension of scalars. For free modules glued by $g$, the block identity $\mathrm{diag}(g,g^{-1})\in E_{\infty }(A_0)$ lifts through the surjective side and makes the direct sum with inverse gluing free. Complements reduce general projective triples to that free case. Conversely an idempotent over $A$ is exactly a compatible pair of idempotents over its two factors.

Stable invertible matrices, with stabilization $g\mapsto \mathrm{diag}(g,1)$, form the stable general linear group. Its elementary subgroup is its commutator subgroup, and the displayed quotient is its abelianization. Elementary matrices lift along any surjective ring map, even when arbitrary invertible matrices do not lift. Over a field, determinant identifies $K_1$ with its multiplicative unit group.

The union of finite invertible matrices under $g\mapsto \mathrm{diag}(g,1)$. Every element and relation occurs in a finite matrix size; adding identity blocks permits comparisons between different sizes without changing the stable class.

The subgroup generated by elementary transvections in finite matrix sizes, with identity stabilization. A third distinct index expresses each transvection as a commutator; stable block identities show that this subgroup is normal and that its quotient in the stable general linear group is abelian. Thus it is precisely the commutator subgroup.

The Grothendieck group of finitely generated projective right $A$-modules under direct sum. An element is a formal difference $[P]-[Q]$. Equality of two positive classes means $P\oplus T\cong Q\oplus T$ for some projective $T$; adding a complement of $T$ converts this into an isomorphism after a common finite free summand. This stabilization criterion is essential in patching arguments.

An algebraic structure is a set equipped with specified algebraic operations satisfying selected axioms. An operation's domain is part of the structure: a binary operation on $S$ takes two elements of $S$ and produces another element of $S$. Groups specify one such operation, rings specify addition and multiplication, and vector spaces specify vector addition and scalar multiplication by a separate field.

Different choices of operations can give the same underlying set different structures. The integers form a group under addition, since every integer has an additive inverse. They do not form a group under multiplication: for example, $2$ has no multiplicative inverse among the integers. Even the nonzero integers fail that inverse requirement. In contrast, nonzero rational numbers form a group under multiplication.

Finite multiplication tables make such distinctions visible. Arithmetic modulo $5$ gives a field; arithmetic modulo $6$ gives a commutative ring with zero divisors. In the former, multiplying by any nonzero element permutes all residue classes. In the latter, multiplying by $2$ reaches only $0,2,4$, and $2\cdot 3=0$ even though both factors are nonzero.

The goal is to extract reusable properties rather than memorize isolated tables. A homomorphism preserves the specified operations, and an isomorphism shows that two structures have identical operation patterns after relabeling.

A vertex operator algebra is an integer-graded vector space equipped with a state-field correspondence $a\mapsto Y(a,z)$, a vacuum, translation operator and conformal vector satisfying vacuum, translation, locality and compatibility axioms. It packages the local operator product expansions of a chiral conformal field theory. A string vertex operator is an individual state insertion, rather than this complete algebraic structure.

An incline has two associative commutative operations $+$ and multiplication, with multiplication distributing over addition, idempotent addition and absorption $x+xy=x$. Neither operation is required here to have an identity. Its natural order is $x\le y⟺x+y=y$, making addition a join. Multiplication is order-preserving, and products lie below both factors. A finitely generated incline is well-quasi-ordered by its reversed natural order, via the reverse-inclusion well-quasi-ordering of monomial ideals applied to exponent supports.

Expand every term in finitely many generators as a finite nonempty sum of monomials, and associate its exponent support with the upward-closed subset of ${\mathbb{N}}^r$ that it generates. The reverse-inclusion well-quasi-ordering of monomial ideals supplies an earlier support upset containing a later one. Every later monomial is then a multiple of an earlier monomial, hence is absorbed by it under $a+ab=a$. The earlier sum absorbs the later sum. Thus every infinite term sequence has an earlier element preceding a later one in the reverse natural order. Reflexivity, antisymmetry and transitivity follow directly from commutative idempotent addition.

A homomorphism is a map between algebraic structures preserving their specified operations. For a single binary operation, this means $f(a\ast b)=f(a)\star f(b)$, where the source and target may use different operation symbols. A group homomorphism preserves the group operation; a ring homomorphism preserves addition and multiplication, with the identity convention specified; a module homomorphism preserves addition and scalar multiplication.

Reduction modulo $5$ is a ring homomorphism $\mathbb{Z}\to \mathbb{Z}/5\mathbb{Z}$: adding or multiplying first and then reducing gives the same residue as reducing first and then operating. For example, $12\cdot 14=168\equiv 3(\mathrm{mod}5)$, while the reduced operands give $2\cdot 4=8\equiv 3(\mathrm{mod}5)$. This lets a complicated integer calculation be projected into simpler finite arithmetic. It also explains why a polynomial equation over the integers imposes necessary modular arithmetic conditions on any integer solution.

A homomorphism can lose information: $0$ and $5$ have the same image under that reduction. An isomorphism loses none of the structure because it has an operation-preserving inverse. Preservation of the operation, rather than preservation of arbitrary labels or sizes, is the essential condition.

A kernel records the information lost by a homomorphism. For a group homomorphism it is the preimage of the identity element, giving the kernel of a group homomorphism; for a ring homomorphism or linear map it is the preimage of zero, giving the kernel of a ring homomorphism or kernel of a linear map. More generally, the relation $a\sim b⟺f(a)=f(b)$ records which inputs the homomorphism identifies. In the group case, its equivalence classes are cosets of the group kernel, as explained by fibres of a group homomorphism are cosets of its kernel.

An algebraic derivation of an algebra over a field is a linear map satisfying the Leibniz rule for its multiplication: $D(xy)=D(x)y+xD(y)$. This definition makes sense even when the multiplication is not associative. A derivation of an algebra uses an associative product; a derivation of a Lie algebra uses the Lie bracket as its product.

A normed algebra is an algebra over a field with a norm satisfying $\Vert ab\Vert \le \Vert a\Vert \Vert b\Vert$. Its completion is a Banach algebra, since multiplication extends continuously. Completeness is additional structure, rather than part of the definition of a normed algebra.

A finite-dimensional commutative real algebra with a Euclidean multiplicative norm has dimension at most two. Automatic real bilinearity from a multiplicative norm supplies bilinearity if it was not part of the initial description. On its unit sphere, squaring has derivative $v\mapsto 2xv$, injective because there are no zero divisors. The only equal squares are antipodal pairs, so the map induces an injective immersion ${\mathbb{RP}}^{n-1}\to S^{n-1}$. Compactness and the inverse function theorem make it a diffeomorphism when $n\ge 2$. For $n\ge 3$, the fundamental groups $\mathbb{Z}/2$ and zero contradict this. The real and complex number algebras attain dimensions one and two.

Suppose a ring has an underlying real normed vector space and its norm is multiplicative, without initially assuming real bilinearity of multiplication. For fixed $x$, distributivity makes $y\mapsto xy$ additive, while $\Vert xy-xz\Vert =\Vert x\Vert \Vert y-z\Vert$ makes it continuous. Additivity gives rational scalar compatibility, and continuity extends it to every real scalar. Apply the same argument in the other variable to obtain a bilinear map. The norm identity also excludes nonzero zero divisors.

A normed division algebra is a nonzero unital normed algebra in which every nonzero element has a two-sided multiplicative inverse. The complex Gelfand-Mazur theorem forces such an algebra to be $\mathbb{C}$, even when completeness is not initially assumed: embed it in its Banach algebra completion and use nonemptiness of the Banach-algebra spectrum.

For an additive semigroup $D$ with identity zero, the semigroup algebra $k[D]$ has a $k$-basis of formal symbols $t^q$, $q\in D$, and multiplication $t^q{t}^{q^{\prime }}=t^{q+q^{\prime }}$. For $D=\mathbb{Z}[1/2{]}_{\ge 0}$ it is the integral domain ${\bigcup }_{h\ge 0}k[t^{1/2^h}]$. Its positive-exponent ideal is a nonzero idempotent ideal, and the strictly ascending chain $(t)⊊(t^{1/2})⊊\cdots$ shows that the algebra is not a Noetherian ring.

A semiring has a commutative monoid operation $(S,+,0)$ and a monoid operation $(S,\cdot ,1)$, with multiplication distributing over addition on both sides and $0$ absorbing multiplication. Additive inverses are not required. The natural numbers with their usual addition and multiplication are a basic example. Regarding multiplication as composition yields a one-object category with commutative-monoid enrichment.

The isomorphism theorems relate homomorphisms, their kernels and images, and quotient constructions in algebra. For groups, the first isomorphism theorem identifies a homomorphism's image with the quotient group by its group kernel. Analogous statements include the first isomorphism theorem for rings and the First isomorphism theorem for vector spaces: identifying precisely the inputs with the same output preserves the remaining algebraic structure.

Bernoulli's inequality states that $(1+x{)}^r\ge 1+rx$ for every integer $r\ge 0$ and real $x\ge -1$. In particular, $(1-x{)}^r\ge 1-rx$ for $0\le x\le 1$.

An algebraic operation combines one or more elements of a set to produce another element.

A binary operation on a set $S$ is a function $\ast :S\times S\to S$. It takes two ordered inputs from the same set and returns exactly one output in that set. The codomain is part of the definition: ordinary subtraction is a binary operation on the integers, but not on the positive integers, since $2-3=-1$ leaves that set. Ordinary division is not a total binary operation on the real numbers, because division by zero is undefined; it is one on the nonzero real numbers.

The same set can carry several different binary operations. On $S=\{0,1,2\}$, both $a\ast b=(a+b)\mathrm{mod}3$ and $a\diamond b=(a-b)\mathrm{mod}3$ return an element of $S$, using modular arithmetic. Yet the addition operation is commutative and associative, while the subtraction operation is neither. Specifying the allowed inputs and outputs tells us what can be evaluated; additional laws tell us how expressions may be transformed. ^[ref].

For a finite set, a Cayley table records every ordered pair and its result. This turns abstract laws into checkable examples. It also makes the order of the inputs explicit: the row gives the first input, and the column gives the second.

Left cancellation means $ax=ay\Rightarrow x=y$; right cancellation means $xa=ya\Rightarrow x=y$. A group has both laws, since multiplication by an inverse element cancels the common factor.

An algebraic magma is a set equipped with a binary operation. No associative property, commutativity, or identity element is required. A semigroup adds the associative property; a monoid also adds an identity element.

A subset $A\subseteq S$ is closed under a binary operation on $S$ when the result of combining any two elements of $A$ is again in $A$. The even integers are closed under addition, but the odd integers are not: $1+3=4$ is even. Closure permits restricting the binary operation to $A\times A\to A$. It is different from being a closed set in topology.

A Cayley table records a binary operation on a finite set: the cell at row $a$, column $b$ contains $a\ast b$. Every cell must contain an element of the specified set. With the same ordering of row and column labels, the table is symmetric across its main diagonal exactly when the binary operation is commutative. Checking the associative property instead requires comparing $(a\ast b)\ast c$ with $a\ast (b\ast c)$ for every triple; symmetry alone cannot establish it.^[ref]

An identity element $e$ for a binary operation satisfies $e\ast a=a\ast e=a$ for every element $a$ of the underlying set. It is a two-sided identity: merely satisfying one of the two equations is insufficient. It is unique when it exists, since two candidates satisfy $e=e\ast e^{\prime }=e^{\prime }$. This uniqueness does not require the associative property.

Zero is the identity element for addition of integers, one for multiplication of integers, and the identity function for function composition of self-maps. The empty word is the identity element for concatenation. An associative operation with an identity element makes a monoid; inverse elements are an additional requirement for a group.

An identity element for addition: $0+a=a+0=a$. In a vector space, it is the zero vector; among matrices of fixed dimensions, it is the zero matrix.

For two binary operations, multiplication distributes over addition on the left or right when the respective displayed identity holds. If multiplication is commutative, either law gives the other. Repeated distribution expands expressions in an incline into finite sums of monomials. This property is distinct from associativity of either operation.

For a binary operation, idempotence means repeating an input does not change it: $x\ast x=x$. An associative commutative idempotent operation defines a join order by $x\le y⟺x\ast y=y$.

A binary operation is commutative if $a\ast b=b\ast a$ for every pair in its domain. A particular pair may commute even when the operation is not commutative. For example, ordinal addition and ordinal multiplication are not generally commutative.

Two elements commute when their two products in opposite orders agree. A particular pair of matrices may commute even when matrix multiplication is not a commutative operation on the whole set of matrices.

A binary operation is noncommutative if some pair of elements fails to commute. Thus reversing their order can change the result. The same operation may still have many commuting elements.

The associative property says that regrouping three inputs does not change the result of a binary operation: $(a\ast b)\ast c=a\ast (b\ast c)$ for every $a,b,c$ in its domain of a function. It changes parentheses while keeping the order of the inputs. Commutativity instead exchanges two inputs; the two properties are different.

For addition modulo three, $(2+1)+1\equiv 2+(1+1)\equiv 1(\mathrm{mod}3)$. For subtraction modulo three,so the second binary operation fails the associative property. One failed triple disproves associativity; checking a few successful triples does not establish it for all inputs.

The useful consequence is that a finite ordered sequence can be combined in chunks. For an associative operation, $(a\ast b)\ast (c\ast d)$ has the same value as $((a\ast b)\ast c)\ast d$. Intermediate results can therefore summarize parts of a computation without retaining their internal parentheses. The order must still be preserved when the binary operation is not commutative. Function composition and concatenation provide examples where grouping is harmless but exchanging entries changes the result.

A semigroup is an algebraic magma whose binary operation has the associative property. Its products can be regrouped while preserving the order of their elements. An identity element and inverse elements are not required.

For example, the positive integers with addition form a semigroup, since $(a+b)+c=a+(b+c)$ and a sum of positive integers is positive. They have no identity element: it would have to be zero, which is outside that set. Nonempty words over an alphabet with concatenation give a noncommutative example without an identity element. Including the empty word instead gives a free monoid.

The usefulness of a semigroup is that an ordered list has one product regardless of its parentheses. This permits replacing a consecutive block by its already computed value. Here the empty set is also allowed as a semigroup; a monoid is necessarily nonempty because it contains an identity element.

A nonempty finite set with an associative binary operation satisfying both cancellation laws is a group. For fixed $a$, cancellation makes $x\mapsto ax$ and $x\mapsto xa$ injective, hence surjective. Choose $e$ with $ae=a$. Then $a(ex)=ax$ gives $ex=x$, and $(xe)x=xx$ gives $xe=x$. Choose $ab=e=ca$; associativity gives $c=c(ab)=(ca)b=b$, so $a$ has a two-sided inverse element.

A numerical semigroup is an additive subset of the nonnegative integers that contains zero and has finite complement. Positive generators with greatest common divisor one generate such a semigroup; the missing nonnegative integers are its gaps.

Suppose an additively closed subset $M$ of the nonnegative integers contains a positive integer $a$ and every integer from $N$ to $N+a-1$. Then it contains every integer at least $N$. The first $a$ values are the base cases. For $n\ge N+a$, the smaller number $n-a\ge N$ belongs to $M$ by strong induction, and closure under addition gives $n=(n-a)+a\in M$. Equivalently, write $n=N+qa+r$ by division with remainder, with $0\le r<a$; start at $N+r\in M$ and add $a$ exactly $q$ times.

A rectangular band is a semigroup satisfying the displayed identities; these also imply associativity. Every nonempty such algebra is a product $B\times C$ with $(b,c)\ast (b^{\prime },c^{\prime })=(b,c^{\prime })$. Fixing $e$, the factor sets are $B=\{x\ast e\}$ and $C=\{e\ast x\}$, and $x\mapsto (x\ast e,e\ast x)$ has inverse $(b,c)\mapsto b\ast c$. Homomorphisms between nonempty products are exactly pairs of maps on the two factors.

The category of rectangular bands is Cartesian closed. For nonempty products $A=B\times C$, $D=E\times F$, the exponential is $E^B\times F^C$. Also $D^0=1$ and $0^A=0$ for nonempty $A$. It has no subobject classifier: the classifier would have two elements, hence would be left-zero or right-zero, while its characteristic maps on a two-by-two product could not single out a one-point subalgebra.

An element of a semigroup is idempotent when multiplying it by itself leaves it unchanged. Every element of a left-zero or right-zero semigroup is idempotent, whereas a group has only its identity as an idempotent. The Ellis–Numakura lemma guarantees such an element in a nonempty compact one-sided topological semigroup. Additive idempotence on ultrafilters gives the idempotent ultrafilter relation $p+p=p$.

A semigroup presentation specifies generators $X$ and equalities between nonempty words. The resulting semigroup is the quotient of the free semigroup $X^{+}$ by the smallest congruence containing these equalities. Two words are equal precisely when one can be transformed into the other by finitely many replacements in contexts, using the defining equalities in either direction.

A semigroup presentation is finite when both its generator set and its list of defining equalities are finite. A finite transition table for a Turing machine can be encoded this way: state-and-tape words simulate transitions, boundary rules supply blank cells, and halting-state rules erase the configuration to one distinguished symbol.

A finite deterministic Turing machine can be represented by a finite semigroup presentation whose generators are tape symbols, state symbols, a boundary marker $h$, and an accept symbol $q$. Defining equalities replace the local scanned symbol and state according to machine instructions, with separate boundary cases for implicit blanks. Only terminal-state equalities erase tape letters and then replace $h{q}_{H}h$ by $q$. A derivation from an initial description to $q$ must encounter a terminal state before its first erasure. Simulation equalities preserve halting membership in either direction: for a deterministic transition $C\to D$, $C$ halts if and only if $D$ halts. Thus reverse use of defining equalities cannot create a false halting certificate. Boundary and erasure rules require explicit checking for every chosen encoding.

A subsemigroup is a nonempty subset of a semigroup closed under its associative operation. In a compact Hausdorff space carrying a left-topological semigroup, descending chains of nonempty compact subsemigroups have a nonempty compact subsemigroup as their intersection. This supplies the minimal object used in the Ellis–Numakura lemma.

A monoid is a semigroup with a two-sided identity element $e$: $e\ast a=a\ast e=a$ for every element $a$. Associativity handles the grouping of a nonempty sequence; the identity element also supplies the result for an empty sequence. This makes the same rules apply to combining zero, one, or many contributions. ^[ref].

The nonnegative integers $\{0,1,2,\dots \}$ with addition form a commutative monoid, with identity element zero. They do not form a group: a positive integer has no additive inverse in that set. All words over an alphabet with concatenation form a free monoid, whose identity element is the empty word. The full transformation monoid consists of all self-maps of a set, with function composition and the identity function.

These examples explain why monoids are useful: accumulation needs an operation and a starting value, but it need not be reversible. Adding counts, joining strings, and composing transformations all share the same grouping and identity laws. Requiring every element to have an inverse element gives a group, a stronger structure suitable for reversible operations.

An inverse element of $a$ in a monoid is an element $b$ satisfying $a\ast b=b\ast a=e$, where $e$ is the identity element. Its notation is $a^{-1}$. Some elements of a monoid can lack an inverse element; a group requires one for every element. Associativity makes a two-sided inverse element unique: if $b,c$ both invert $a$, then $b=b\ast (a\ast c)=(b\ast a)\ast c=c$.

For the full transformation monoid, a self-map has an inverse element exactly when it is a bijection; that element is its inverse function. On $\{0,1\}$, the swap $s(0)=1,s(1)=0$ is its own inverse element, while a constant function has none because it merges the two inputs.

A transformation monoid is a collection of self-maps of a fixed set $X$ that contains the identity function and is closed under function composition. The associative property comes from function composition. A monoid action describes transformations of a set that obey a monoid's multiplication and identity laws.

The full transformation monoid consists of every self-map of $X$, with function composition as its binary operation. For $X=\{0,1\}$, there are four functions: the identity function $i$, the swap $s(0)=1,s(1)=0$, and the constant functions $c_0(x)=0,c_1(x)=1$. Their composition order matters: $c_0\circ s=c_0$, whereas $s\circ c_0=c_1$. Thus associativity does not imply commutativity.

Only $i$ and $s$ are bijections, and their inverse elements are themselves. They form a two-element group within this monoid. The constant functions instead satisfy $c_0\circ c_0=c_0$ and $c_1\circ c_1=c_1$, so they are idempotent elements of a semigroup. Such irreversible transformations explain why the monoid setting is useful even when a group would be too restrictive.

A left monoid action assigns each $m\in M$ a map $a\mapsto m\cdot a$ with $1\cdot a=a$ and $(mn)\cdot a=m\cdot (n\cdot a)$. It need not be invertible or injective. A left M-set is equivalently a covariant functor from the one-object category determined by $M$.

A right monoid action has $x1=x$ and $(xm)n=x(mn)$. It is a contravariant functor from the one-object category of the monoid to sets. The regular right action is multiplication on $M$ itself. Products of right actions use the diagonal action.

Give $M\times X$ the diagonal right monoid action. On its equivariant maps into $Y$, define $(ek)(g,x)=e(kg,x)$. The evaluation map of an exponential object is $e(1,x)$ and currying of $h:Z\times X\to Y$ is $\hat{h}(z)(g,x)=h(zg,x)$. Both maps are equivariant, and they are inverse under evaluation. The left multiplication $kg$ in this right action is essential in a noncommutative monoid.

For a group, evaluation at the identity identifies the monoid exponential with all set functions $X\to Y$. Its inverse is $e_f(g,x)=f(x{g}^{-1})g$, and the resulting function action is the displayed formula. Fixed functions are precisely equivariant maps. The inverse on the input is necessary for equivariance of evaluation and for the right-action law.

An equivariant map $h:A\to B$ between left M-sets obeys $h(m\cdot a)=m\cdot h(a)$. These are the natural transformations of the corresponding functors from the one-object monoid category.

An M-set is a set carrying a monoid action. This terminology here uses left actions. Morphisms are equivariant maps of monoid sets, and the resulting functor category $[M,\mathbf{Set}]$ is an elementary topos.

A monoid homomorphism $h:M\to N$ preserves multiplication and identity: $h(xy)=h(x)h(y)$ and $h(1_M)=1_N$. It therefore preserves every finite ordered product, including the empty product. The monoids with these maps form the category $\mathbf{Mon}$.

A monoid automorphism is a bijective operation- and identity-preserving self-map of a monoid. Its inverse is also a monoid homomorphism. Permuting prime numbers and extending by prime factorization gives all automorphisms of the multiplicative monoid of positive natural numbers; extending by $0\mapsto 0$ gives automorphisms of the multiplicative monoid including zero.

The free monoid ${\Sigma }^{\ast }$ on an alphabet $\Sigma$ consists of all finite words over an alphabet, including the empty word $\epsilon$, with concatenation as its binary operation. Distinct strings represent distinct elements: there are no imposed equalities between different sequences of letters. The empty word is its identity element, and concatenation has the associative property.^[ref]

For $\Sigma =\{a,b\}$, the elements begin $\epsilon ,a,b,aa,ab,ba,bb,\dots$. Both $(ab)a$ and $a(ba)$ give the same word $aba$, but $ab\ne ba$, so this monoid is not commutative. The length of a word over an alphabet obeys $\ell (uv)=\ell (u)+\ell (v)$; it is a monoid homomorphism to the nonnegative integers with addition. A nonempty word has no inverse element, since joining it to any other word cannot decrease its length to zero.

The defining benefit of being free is that letters can stand for arbitrary elements of another monoid. Choose a value $h(a)$ for each letter $a\in \Sigma$; then $h(a_1\cdots a_n)=h(a_1)\ast \cdots \ast h(a_n)$, with $h(\epsilon )=e$, is the unique monoid homomorphism extending those choices. Associativity and the identity element make this evaluation consistent for every finite word. For example, interpreting the letters as self-maps evaluates a word as a composed transformation; interpreting them as nonnegative counts evaluates it by addition.

A commutative monoid is a monoid whose operation is commutative. The natural numbers with addition form a commutative monoid that is not an abelian group, because positive elements have no additive inverses.

The free commutative monoid on $d$ generators consists of the nonnegative integral sums of those generators, with unique coefficients. Its semigroup algebra over a commutative ring is the polynomial ring in $d$ variables. In a smooth toric chart, the dual-cone lattice points form such a monoid, yielding affine-space coordinates.

A commutative monoid is cancellative when $a+c=b+c$ implies $a=b$. Cancellation makes the natural map into its Grothendieck group injective.

For an integer $m\ge 2$, the positive integers congruent to one modulo $m$ form a cancellative commutative monoid under multiplication. The identity is one, and it is the only unit. Proper factors outside this congruence class do not witness reducibility inside the monoid. Hence irreducible elements of a commutative monoid need not coincide with ordinary prime numbers.

The congruence submonoid of the positive integers with modulus ten has four distinct irreducible elements of a commutative monoid $21,221,51,91$ with equal pairwise products. Their ordinary prime factors are respectively $(3,7),(13,17),(3,17),(7,13)$. None of these proper prime factors is congruent to one modulo ten. The Euclid lemma proves that a product of two distinct primes has no other proper positive divisors, so all four are irreducible in this monoid. This gives nonunique factorization without assuming the fundamental theorem of arithmetic.

A nonunit element of a commutative monoid is irreducible if each factorization has a unit factor. In a submonoid of the positive integers under multiplication, the only unit is one. Irreducibility is relative to the allowed factors: a composite integer can be irreducible in a smaller monoid even though it is not a prime number.

A cancellative commutative monoid is torsion-free in this sense when $na=nb$ implies $a=b$ for every positive integer $n$. Its Grothendieck group is then a torsion-free abelian group, because $n(a-b)=0$ forces $na=nb$. This is the condition inherited by a submonoid of a torsion-free abelian group.

This category has commutative monoids and their homomorphisms. Pointwise addition makes each hom-set a commutative monoid, and composition is additive in both variables. Finite cartesian products are also coproducts, using the coordinate injections with zero in the other coordinates and summing images. Therefore it is a semi-additive category, but it is not an additive category. The reflexive pair from the submonoid $\{(m,n):m\le n\le 2m\}$ of ${\mathbb{N}}^2$ to $\mathbb{N}$, using the two projections and section $m\mapsto (m,m)$, has an arrow from $1$ to $2$ and none from $2$ to $1$ in its represented graph. Thus a reflexive pair here need not yield a groupoid.

An additive semigroup is a semigroup whose operation is written as addition.

A face $F$ of an additive monoid $S$ is a submonoid such that $a+b\in F$ implies $a,b\in F$. If $S=C\cap M$ for a rational polyhedral cone, its faces are precisely $S\cap \nu$ for faces $\nu$ of $C$. Finite generation and clearing rational denominators identify these monoid faces with the exposed faces of the cone.

// Target: geometry-and-topology.bigb

A topological semigroup is a semigroup with a topology for which multiplication is jointly continuous.

A left-topological semigroup requires only that every right translation $x\mapsto xa$ be continuous.

Every nonempty compact Hausdorff left-topological semigroup contains an idempotent. Choose a minimal nonempty compact subsemigroup $K$. For $a\in K$, minimality gives $Ka=K$; the nonempty compact subsemigroup $\{x\in K:xa=a\}$ is therefore all of $K$, so $a^2=a$.

A matrix is strictly diagonally dominant when $|a_{ii}|>{\sum }_{j\ne i}|a_{ij}|$ in every row.

Lie theory studies continuous symmetries through Lie groups, Lie algebras, and their representations. A rotation can change by an arbitrarily small angle: its parameter varies continuously, yet composing two rotations obeys an exact group operation. The central question is how such finite transformations are controlled by their velocities at the identity element.

An infinite group need not require infinitely many independent parameters. The planar rotation group has infinitely many elements but only one local angle coordinate. More generally, an $n$-dimensional Lie group has an $n$-dimensional tangent space $\mathfrak{g}=T_{e}G$ at its identity element. A basis of this Lie algebra, together with the Lie brackets between its basis vectors, supplies finite data for infinitesimal composition. These brackets record the first correction to adding small transformation parameters. The Exponential map of a Lie group turns a chosen infinitesimal direction into an actual motion.

Finite tangent data consists of a finite basis and a finite table of Lie brackets. For planar rotation, every generator is $aJ$ with $a\in \mathbb{R}$, using the quarter-turn matrix $J$ computed below: one real coefficient specifies any angular velocity. This Lie algebra has infinitely many vectors, but its dimension is one. Linear combinations of the chosen basis vectors supply all the possible infinitesimal motions.

This passage between geometry and algebra is useful because linear maps and matrix commutators are easier to compute than an entire family of transformations. The local information does not determine every global identification: the additive real line and the circle group have isomorphic one-dimensional Lie algebras, but only the latter wraps around. For the local-versus-global viewpoint and the connection with differential equations, see Meinrenken, Lie groups and Lie algebras, introduction and Section 4.

An affine algebraic group is an affine variety $G$ whose multiplication and inversion are regular maps. Equivalently, its coordinate ring $k[G]$ is a commutative Hopf algebra, with comultiplication induced by multiplication in $G$.

Fix a Chevalley basis and its integral lattice. The adjoint Chevalley group over a field is generated by the root operators obtained from the integral divided-power polynomials $x_{\alpha }(t)={\sum }_j{t}^j(\mathrm{ad}{e}_{\alpha }{)}^j/j!$ on that lattice. The coefficients are first computed over the integers and then reduced to the field; factorials need not be invertible there. This is the adjoint construction; other weight lattices give other isogeny forms of groups of Lie type.

On the sl2 Lie algebra with basis $e=E_{12},h=E_{11}-E_{22},f=E_{21}$, the integral adjoint root polynomials areThese are conjugation by the upper and lower elementary matrices of the special linear group. Those matrices generate $S{L}_2(K)$, and a matrix commuting with both $E_{12}$ and $E_{21}$ is scalar. Thus the kernel is precisely $\{aI:a^2=1\}$ over every field, even in characteristic two, and the image is the displayed quotient.

An algebraic one-parameter subgroup is a regular group homomorphism from the multiplicative algebraic group. In $S{L}_n$, a change of basis makes it $t\mapsto \mathrm{diag}(t^{a_1},\dots ,t^{a_n})$ for integers with sum zero. This is the algebraic multiplicative notion, distinct from an additive real-parameter subgroup of a Lie group.

For every finite-dimensional rational representation $V$ and every nonzero $v\in V^G$, a geometrically reductive algebraic group admits a homogeneous invariant polynomial $F\in k[V{]}^G$ of positive degree with $F(v)\ne 0$. A linearly reductive group can take degree one. In positive characteristic, higher degree is essential: a power $p^r$ can separate a fixed vector even when an invariant linear functional cannot.