Universal algebra

Main article: Universal algebra

Universal algebra is the study of algebraic structures in general. It is a generalization of abstract algebra
that is not limited to binary operations and allows operations with more inputs as well, such as ternary
operations. Universal algebra is not interested in the specific elements that make up the underlying sets
and instead investigates what structural features different algebraic structures have in common.[64]
One of those structural features concerns the identities that are true in different algebraic structures. In
this context, an identity is a universal equation or an equation that is true for all elements of the
underlying set. For example, commutativity is a universal equation that states that

�
∘

{\displaystyle a\circ b} is identical to

�
∘

{\displaystyle b\circ a} for all elements.[65] Two algebraic structures that share all their identities are
said to belong to the same variety.[66] For instance, the ring of integers and the ring of polynomials
form part of the same variety because they have the same identities, such as commutativity and
associativity. The field of rational numbers, by contrast, does not belong to this variety since it has
additional identities, such as the existence of multiplicative inverses.[67]

Besides identities, universal algebra is also interested in structural features associated with quasi-
identities. A quasi-identity is an identity that only needs to be present under certain conditions.[n] It is a
generalization of identity in the sense that every identity is a quasi-identity but not every quasi-identity
is an identity. Algebraic structures that share all their quasi-identities have certain structural
characteristics in common, which is expressed by stating that they belong to the same quasivariety.[68]

Homomorphisms are a tool in universal algebra to examine structural features by comparing two
algebraic structures.[69] A homomorphism is a function from the underlying set of one algebraic
structure to the underlying set of another algebraic structure that preserves certain structural
characteristics. If the two algebraic structures use binary operations and have the form

,
∘

⟩{\displaystyle \langle A,\circ \rangle } and

⟩{\displaystyle \langle B,\star \rangle } then the function

{\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement:

�
∘

⋆
ℎ

)
{\displaystyle h(x\circ y)=h(x)\star h(y)}. The existence of a homomorphism reveals that the operation

⋆{\displaystyle \star } in the second algebraic structure plays the same role as the operation

∘{\displaystyle \circ } does in the first algebraic structure.[70] Isomorphisms are a special type of
homomorphism that indicates a high degree of similarity between two algebraic structures. An
isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship
between the elements of the two algebraic structures. This implies that every element of the first
algebraic structure is mapped to one unique element in the second structure without any unmapped
elements in the second structure.[71]

Venn diagram of a set and its subset

Subalgebras restrict their operations to a subset of the underlying set of the original algebraic structure.

Another tool of comparison is the relation between an algebraic structure and its subalgebra.[72] If

⟩{\displaystyle \langle A,\circ \rangle } is a subalgebra of

⟩{\displaystyle \langle B,\circ \rangle } then the set

{\displaystyle A} is a subset of

{\displaystyle B}.[o] A subalgebra has to use the same operations as the algebraic structure[p] and they
have to follow the same axioms. This includes the requirement that all operations in the subalgebra are
closed in

{\displaystyle A}, meaning that they only produce elements that belong to

�
{\displaystyle A}.[72] For example, the set of even integers together with addition is a subalgebra of the
full set of integers together with addition. This is the case because the sum of two even numbers is again
an even number. But the set of odd integers together with addition is not a subalgebra since adding two
odd numbers produces an even number, which is not part of the chosen subset.[73]

History

Main articles: History of algebra and Timeline of algebra

Rhind Papyrus

The Rhind Papyrus from ancient Egypt, dated around 1650 BCE, is one of the earliest documents
discussing algebraic problems.

The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations
and unknown quantities. These developments happened in the ancient period in diverse regions such as
Babylonia, Egypt, Greece, China, and India. One of the earliest documents is the Rhind Papyrus from
ancient Egypt, which was written around 1650 BCE[q] and discusses how to solve linear equations, as
expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the
quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and
quadratic polynomial equations, such as the method of completing the square.[74]

Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main
interest was geometry rather than algebra, but they employed algebraic methods to solve geometric
problems. For example, they studied geometric figures while taking their lengths and areas as unknown
quantities to be determined, as exemplified in Pythagoras' formulation of the difference of two squares
method and later in Euclid's Elements.[75] In the 3rd century BCE, Diophantus provided a detailed
treatment of how to solve algebraic equations in a series of books called Arithmetica. He was the first to
experiment with symbolic notation to express polynomials.[76] In ancient China, the book The Nine
Chapters on the Mathematical Art explored various techniques for solving algebraic equations, including
the use of matrix-like constructs.[77]

Title page of The Compendious Book on Calculation by Completion and Balancing

Al-Khwarizmi's The Compendious Book on Calculation by Completion and Balancing provided a general
account of how linear and quadratic equations can be solved through the methods of "reducing" and
"balancing".

It is controversial to what extent these early developments should be considered part of algebra proper
rather than precursors. They offered solutions to algebraic problems but did not conceive them in an
abstract and general manner, focusing instead on specific cases and applications.[78] This changed with
the Persian mathematician al-Khwarizmi,[r] who published his The Compendious Book on Calculation by
Completion and Balancing in 825 CE. It presents the first detailed treatment of general methods that can
be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides.[80]
Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra in the 9th
century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries.[81]

In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with
several variables in the 7th century CE. Among his other innovations were the use of zero and negative
numbers in algebraic equations.[82] The Indian mathematicians Mahāvīra in the 9th century and
Bhāskara II in the 12th century further refined Brahmagupta's methods and concepts.[83] In 1247, the
Chinese mathematician Qin Jiushao wrote the Mathematical Treatise in Nine Sections, which includes an
algorithm for the numerical evaluation of polynomials, including polynomials of higher degrees.[84]

Drawing of François Viète

Painting of René Descartes

François Viète and René Descartes invented a symbolic notation to express equations in an abstract and
concise manner.

The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books like
his Liber Abaci.[85] In 1545, the Italian polymath Gerolamo Cardano published his book Ars Magna,
which covered many topics in algebra and was the first to present general methods for solving cubic and
quartic equations.[86] In the 16th and 17th centuries, the French mathematicians François Viète and
René Descartes introduced letters and symbols to denote variables and operations, making it possible to
express equations in an abstract and concise manner. Their predecessors had relied on verbal
descriptions of problems and solutions.[87] Some historians see this development as a key turning point
in the history of algebra and consider what came before it as the prehistory of algebra because it lacked
the abstract nature based on symbolic manipulation.[88]

Photo of Garrett Birkhoff

Garrett Birkhoff developed many of the foundational concepts of universal algebra.

Many attempts in the 17th and 18th centuries to find general solutions[s] to polynomials of degree five
and higher failed.[91] At the end of the 18th century, the German mathematician Carl Friedrich Gauss
proved the fundamental theorem of algebra, which describes the existence of zeros of polynomials of
any degree without providing a general solution.[18] At the beginning of the 19th century, the Italian
mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show
that no general solution exists for polynomials of degree five and higher.[91] In response to and shortly
after their findings, the French mathematician Évariste Galois developed what came later to be known
as Galois theory, which offered a more in-depth analysis of the solutions of polynomials while also laying
the foundation of group theory.[19] Mathematicians soon realized the relevance of group theory to
other fields and applied it to disciplines like geometry and number theory.[92]
Starting in the mid-19th century, interest in algebra shifted from the study of polynomials associated
with elementary algebra towards a more general inquiry into algebraic structures, marking the
emergence of abstract algebra. This approach explored the axiomatic basis of arbitrary algebraic
operations.[93] The invention of new algebraic systems based on different operations and elements
accompanied this development, such as Boolean algebra, vector algebra, and matrix algebra.[94]
Influential early developments in abstract algebra were made by the German mathematicians David
Hilbert, Ernst Steinitz, Emmy Noether, and Emil Artin. They researched different forms of algebraic
structures and categorized them based on their underlying axioms into types, such as groups, rings, and
fields.[95] The idea of the even more general approach associated with universal algebra was conceived
by the English mathematician Alfred North Whitehead in his 1898 book A Treatise on Universal Algebra.
Starting in the 1930s, the American mathematician Garrett Birkhoff expanded these ideas and
developed many of the foundational concepts of this field.[96] Closely related developments were the
formulation of model theory, category theory, topological algebra, homological algebra, Lie algebras,
free algebras, and homology groups.[97]

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