"Algebra not only permeates all mathematics, it invades the domain of formal logic and even metaphysics" (Tobias Dantzig).
"Algebra is the metaphysics of arithmetic" (John Ray).
"Algebra is universal arithmetic" (Newton).
Abstract Algebra
Elementary algebra refers to the mathematical structure formed by the real and complex numbers.
Abstract algebra −also called "modern algebra"− is a generalization of elementary algebra and refers to mathematical structures defined with internal operations on a set, such as: group, semigroup, ring, body (or field), semigroup, monoid, modulus, vector algebra, lattice, Boolean algebra, associative algebra, commutative algebra, Lie algebra, etc.
Abstract algebra differs from elementary algebra in that the latter operates with concrete numbers, and abstract algebra operates with variables that can represent any object: numbers, matrices, permutations, polynomials, geometric transformations, etc.
Abstract algebra is an elevation from the literal (the numerical) to the symbolic, a step towards a higher level of abstraction. Abstract algebra is linked to variables and operations with them, without considering what they represent. Historically it is linked to equations and their resolution. Negative numbers and complex numbers were born from algebra when trying to solve respectively the equations n+x = 0 and x2 = −1. There is a certain relation between these two expressions, since historically, negative numbers were considered imaginary.
Just as arithmetic derived from numbers, the use of variables to represent unknown quantities allowed the birth of abstract algebra, which made it possible to operate with variables (e.g., x+x produced as a result 2*x). The language of algebra has been a unifying element in different fields: geometry (geometric algebra), logic (Boolean algebra or logical algebra), calculus, vector algebra, matrix algebra, etc.
Symbolic algebra was created by François Viète (1540-1603), by using symbols for unknown quantities.
The development of abstract algebra was motivated by the need to make mathematical definitions more precise in a formal way. Today, abstract algebra is used in virtually all areas of mathematics.
In an abstract algebra, one or more inner operations are defined on a set C. The inner operations operate on elements of C and result in an element of C, that is, they are functions of the form Cn → C:
0-ary (null) operations. They are constants.
1-ary (unary) operations. They are functions of C in C.
2-ary (binary) operations. They are 2-element functions of C in C.
Operaciones n-arias. They are functions of n elements of C in C.
Infinitary operations. They are functions of infinite elements of C on C.
Examples of single-operation algebraic structures are groups, quasigroups, semigroups, and monoids. Algebraic systems with two or more inner operations are: ring, body, vector space, lattice (lattice), Boolean algebra and Lie algebra. An example of an infinitary algebraic structure is a complete lattice of infinite elements.
In general, one usually speaks of an algebraic structure of type Ω, where Ω is a sequence of natural numbers representing the "arities" of the inner operations defined on a set. For example, if Ω=(2, 2, 3), two 2-ary operations and one 3-ary operation have been defined.
Examples
Group.
The definition of group in abstract algebra is as follows:
There is a set G.
There is a binary inner operation (symbolized by "*") defined on G, i.e., for every pair of elements, < i>x and y of G, x*y is an element of G: x*y∈G.
There exists an element e (called neutral) such that x*e = < i>e*x = x for every element x of G.
For every element x of G, there exists another element x' (called "inverse of < i>x") such that x*x' = x'*x = e.
The elements of G satisfy the associative property: x*(y*z) = (x*y)*z.
If the operation satisfies the commutative property (x*y) = (y*x), the group is called "abelian".
Examples of groups are the natural numbers under the addition operation, and the real numbers (except zero) under the product operation.
Monoid.
A monoid is an algebraic structure with a binary operation symbolized by "*") that satisfies the properties:
An example of a monoid is matrices under the multiplication operation.
Lattice.
The concept of lattice has its origin in the formalization of propositional logic. A lattice is an algebra with two dual binary operations (∧ and ∨) satisfying the following laws:
Idempotence: x∧x = x∨x = x
Commutative: x∧y = y∧xx∨y = y∨x
Associative: (x∧y)∧z = x∧(y∧z) (x∨y)∨z = x∨(y∨z)
Absorption: x∧(x∨y) = x∨(x∧y) = x
This definition is equivalent to saying that there exists a partially ordered set C (poset) with 2 binary operations (∧ and ∨) such that, for any pair of elements (< i>x and y), x∧y is the minimum or infimum (meet) and x∨y is the maximum or supremum (join), since the 4 previous laws are fulfilled.
A lattice is complete when any subset has an infimum and a supremum.
A partially ordered set (with the relation x≤y) satisfies the laws.
x≤x.
x≤y e y≤x implies x=y.
x≤y e y≤z implica x≤z.
If in addition the property is satisfied that for every pair (x, y) of C, is x ≤y or y≤x, then the set C is totally ordered.
Examples:
Propositional logic, under the operations of logical conjunction and disjunction.
The set of all subsets of a given set, under the operations of union and intersection of sets.
The natural numbers where the minimum of two numbers is the m.c.d. and the maximum is the m.c.m.
Universal Algebra
Universal algebra −also sometimes called "general algebra"− in turn generalizes abstract algebra. It considers all kinds of internal operations on a set and studies the common properties of all algebraic structures.
Mathematicians, always in search of higher level abstractions, discovered that apparently different algebraic structures could be unified by using a small number of axioms. Thus arose universal algebra, which can be considered an algebraic meta-theory.
Before universal algebra, many theorems (mainly the isomorphism theorems) had to be proved separately in each of the abstract algebras. With universal algebra it suffices to prove them only once for all algebraic systems.
Universal algebra is the culmination of a movement in algebra directed toward maximum abstraction and generality, an advance toward what may be called "conceptual mathematics," a mathematics founded on general concepts, free of particular representations, and whose ultimate goal would be the unification of all mathematics.
The axioms of universal algebra take the form of identities (equalities or equivalences) or equational laws. For example, the associative law is
(x*y)*z = x*(y*z)
Existential quantifiers and relations (including inequality) cannot be used in equational laws. Universal quantification is implicit in equational laws. Universal algebra only allows operations (or functions) with symbols, the only relation allowed is equality, and the language that expresses or describes these structures are equations.
Since there can be various interpretations of the laws of universal algebra, universal algebra can be considered as a branch of model theory.
An algebraic structure that can be defined by identities is called a "manifold" (manifold). Some authors consider the manifold to be the only object of universal algebra, while others consider it to be just another algebraic structure. What is true is that most of the algebraic structures (or systems) of mathematics are varieties.
The ideas of universal algebra have had great influence in several areas of computer science, such as: semantics of programming languages, specification of data types, compiler theory, etc.
Formal definition
A universal algebra is formally defined as follows:
A set C, called the "algebra universe".
A set O of functions or operations defined on C.
Functions, in general, are n-ary of the type Cn → C that is, from n arguments (elements of C), another element of C is obtained. A 0-ary (or nullary) operation is a constant.
A set of axioms governing definite operations, in the form of equational laws (e.g., associative law, distributive law, etc.).
Example: Group
In universal algebra one can only define the operations and the axioms that govern these operations. For example, in the definition of group two additional operations are defined and the definition of group in universal algebra is as follows:
There is a set G.
The following operations on G are defined:
An internal operation (*) of binary type on G, i.e., between each pair of elements of G.
A nular operation e (identity operation).
A suffixed unary operation (') that defines the inverse element.
The axioms are as follows:
x*(y*z) = (x*y)*z (associative law)
x*e = e*x = x (law of the neutral element)
x*(x') = (x')*x = e (inverse element law)
Therefore, one passes from the abstract algebra with:
1 binary operation.
1 equational law (associativity), with universal quantization.
2 existentially quantized laws (neutral and inverse element).
to universal algebra, with:
3 operations: a nularia (neutral element), a unary (inverse element) and a binary (inner operation).
3 equational laws (associativity, neutral element and inverse element), with implicit universal quantification.
Thus, the definition of group in universal algebra is more precise because in the traditional definition of group the neutral element is not said to be unique. If there were more than one neutral element there would be ambiguity. However, it is shown that in a group the neutral element is unique. The same is true of the inverse elements, which are also shown to be unique.
Universal algebra vs. Category theory
Two branches of mathematics aim at the unification of all structures: universal algebra and category theory. In both fields the aim is to provide a common language and to produce general results. In universal algebra, emphasis is placed on the internal operations that define the structure. In category theory, the emphasis is on the arrows (morphisms) between structures.
Universal algebra can be formulated in terms of category theory. There are two main formulations. The first is the one described by Lawvere in his 1963 doctoral thesis. The second, which is the one that has finally prevailed, is the universal algebra defined in terms of monads, also called "triples", with which any algebraic structure can be constructed. Monads are defined from adjacent pairs of functors. Monads emerged from algebraic topology, but without initially having any relation to universal algebra.
Given a list of operations and axioms of universal algebra, the corresponding algebras and the homomorphisms between them are the objects and morphisms of a category.
A Brief History of Algebra
The Origin of Algebra
The name "algebra" is derived from "Algebar wal Muquabalah", meaning "restitution and reduction", the 830 book by the Arab mathematician, astronomer and geographer Mohammed Ibn Musa Al-Khwarizmi or Al-Juarismi, which contained the Hindu numbering system and the concept of zero. Finonacci translated this work into Latin with the title "Algoritmi de numero Indorum" (Algoritmi on Hindu numbers), beginning it with the words "Algoritmi dicit", from which the word "algorithm" and "guarismo" come from. Originally, the word "algorithm" referred to the decimal numbering system.
Al-Khwarizmi is considered the father of algebra. He has been called "the Euclid of algebra" because he systematized this discipline and made it an independent branch of mathematics. His book has had a great influence on universal mathematical thought, the main ideas of which are:
An algorithm is a procedure for determining the unknown (the thing, al-shay, "res" in Latin) from the known. In modern notation, x.
Introduces equations (based on equality) and algebraic operations. Equations are formed with units (numbers), unknowns (x) and their squares (x2).
Restitution (al-jabr) is to shift the terms of an equation from one side to the other, changing their sign. Reduction (al-muqabalah) is the cancellation of equal terms on both sides of an equation.
There are 6 types of equations defined a priori (first and second degree) with which all others are constructed.
It does not start from the problems to arrive at the equations, but starts from the primitive equations and their combinations to arrive at the solutions of the problems.
Algebra is a theory (demonstrative discipline) and a practice (algorithmic discipline).
The evolution of algebra towards greater abstraction
Diophantus of Alexandria (3rd century BC), in his famous work "Arithmetic" introduced symbols to represent the unknown of a problem, but this text (as its title indicates) is not an algebra text, but it had a great influence on what today is called "number theory". He was the first to work with negative numbers, establishing the laws of signs (+×+ = +, +×− = −, −×− = +). He used symbols systematically to indicate powers, equalities and negative numbers. He also discovered the rule that is equivalent to today's law of exponents: xn•xm = xn+m. For his original contributions, Diophantus is considered the pioneer of modern algebra.
François Viète (1540-1603) introduced names (letters of the alphabet) for variables, even for the coefficients of an equation. Since then, some authors called "Ars major" to algebra and "ars minor" to arithmetic.
The progress of mathematics is closely related to symbolism and language in general. Algebra advanced by the introduction of better symbolism. The familiar notations of +, −, <, >, =, √ and parentheses were already present in the 16th century, but the most significant change was the introduction of Viète's symbolism.
Descartes improved on Viète's notation. He used the first letters of the alphabet for known quantities and the last letters for unknown ones.
Leibniz can be considered a precursor of universal algebra by conceiving a "Calculus Ratiotinator" (or algebra of logic) and a "Characteristica Geometrica" (or algebra of geometry). His idea was to unify both concepts in a "Characteristica Universalis" encompassing both algebras.
In 1832, Evariste Galois created the theory of groups, thus giving birth to abstract or modern algebra. The groups constituted the germ from which more algebraic structures were created.
In 1833, William Rowan Hamilton introduced the concept of complex number as a formal algebra of pairs of real numbers with operations of addition and product:
(a, b) + (c, d) = (a+c, b+d)
r(a, b) = (ra, rb)
(a, b)(c, d) = (ac−bd, ad+bc)
Hamilton was the first to treat complex numbers as ordered pairs, the embryo of what eventually became the concept of a vector (or multidimensional number), representing a point in a multidimensional geometric space.
In 1843, Hamilton created quaternions, a generalization of complex numbers, with one real and three imaginary parts. Quaternions were the first example of non-commutative algebra. The product of quaternions is associative but not commutative.
Hamilton considered algebra to be closely related to physics. That geometry is the science of space, and algebra the science of time. He was convinced that quaternions were a fundamental tool for the description of physical reality, since time is a scalar and the other three dimensions are coordinates of points in space.
In 1844, Hermann Grassmann published "Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik" (better known simply as "Ausdehnungslehre"), "The Theory of Linear Extension: A New Branch of Mathematics," a theory of extensive geometric quantities, in which he lays the foundations of linear algebra in the modern sense: vector calculus, vector spaces, multivectors, linear transformations, and systems of linear equations.
Grassmann discovered the relationship between geometry and algebra. Grassmann's algebraic language is fundamentally geometric, and can be considered a geometric theory and practice (calculus). Grassmann created a coordinate-free geometric algebra.
Grassmann's abstract approach allowed him to glimpse new ideas, such as n-dimensional space (a generalization of 3D geometric space), multidimensional algebra, the geometric interpretation of imaginary expressions, and noncommutative algebra (the outer product of two vectors). Grassmann defined a new product, called the "outer product", a generalization of the vector product in n dimensions or multivector (an oriented n-dimensional space segment).
Grassmann rewrote his work in 1861 with the definitive exposition of his linear algebra. Both versions were ignored for several reasons: 1) because he departed from the mathematical tradition of his time; 2) because he could not formalize his work since in his time there was no algebraic language with which to express his ideas, and the language he used was philosophical and abstract in style; 3) because he used new and too advanced concepts for his time; 4) because he created some confusion by mixing theory and practice; 5) because of his poor dissemination.
His linear algebra was finally understood and recognized around 1920, when Hermann Weyl and others published a formal definition. According to David Hestenes [1996], "Die Lineale Ausdehnungslehre deserves to stand alongside Euclid's Elements and Descartes' Analytic Geometry in the library of mathematical masterpieces."
The so-called "Grassmann variables" (anticommutative and associative) play a fundamental role in modern supersymmetry and string theory.
In 1854, Boole publishes "Investigation of the Laws of Thought", where he presents the algebra of logic.
In 1878, William Kingdon Clifford generalized the complex numbers, Hamilton's quaternions and Grassmann's linear algebra (including the outer product) by means of his own geometric algebra, which today is called "Clifford's algebra" or simply "geometric algebra". In his work "Elements of Dynamics" (1878) he introduced the notion of geometric product, which merges the scalar (or inner) products and the outer product as the sum of both: ab = a•b + a∧b, so that the inner and outer products can be expressed as a function of the geometric product.
According to David Hestenes, Clifford algebra is a universal language for physics. In fact, it plays an essential role in quantum physics.
In 1884, James Sylvester published the article "Studies on the Principles of Universal Algebra", in which he used the name "universal algebra" for matrix algebra. Sylvester spoke of the realm of algebra as a science of philosophy.
In 1888, Peano drew analogies between Boole's algebras and Grassmann's algebras.
In 1898, Alfred North Whitehead published "A Treatise on Universal Algebra", where he presented an algebra of geometric type constructed from Grassmann's algebra, with points instead of vectors. Whitehead recognized William Rowan Hamilton and August De Morgan as the originators of the concept of universal algebra, and James Sylvester as the coiner of the term.
Whitehead intended to unify all algebraic structures into an algebra of universal type, by means of an equational theory. In particular he wanted to unify the algebraic structures he considered most important: Hamilton's quaternions, Grassman's linear algebra and Boolean algebra. However, he did not achieve any results of a general nature.
Whitehead's central theme was the search for a new geometrical language that could serve physics. Whitehead understood Boolean, Grassmann, and Hamilton algebras as engines for the investigation of the possibilities of thought and reasoning.
Whitehead attempted to formalize the concept of abstract space, whose properties and operations would lead to a general method of interpreting the various algebras. Whitehead's abstract space interpretation was of a qualitative type of an arbitrary number of dimensions, where regions of space correlate with classes of things. For example, the formula a+a = a is interpreted as the mental process of combining the same region of space. And the expression "all a is b" is interpreted as meaning that the region a is included within b.
Its description of space is based on the general principles of action and motion within space. For example, a line (1D region) can be thought of as a variable point. A plane (2D region) is a variable line passing through a point outside a line. And so on for any number of dimensions, to create a variety of points. Each point of a manifold has "intensity" corresponding to a physical quantity (temperature, density, electric potential, etc.). Whitehead wanted to formalize with these concepts Maxwell's laws of electromagnetism and the notion of field in general.
For Whitehead, the conception of abstract space is relational between events. Whitehead's process philosophy is that the key to reality (being, metaphysical reality) is change, dynamism.
Whitehead's algebra did not obtain results of interest because his algebra was really an abstract algebra. Nevertheless, Whitehead is considered the forerunner of universal algebra.
In 1901, Josiah Williard Gibbs presented a 3D vector algebra in his work "Vector Analysis", a simplified version of Grassman's algebra, which was widely accepted for its clarity, mathematical rigor and simplicity.
Emmy Noether was the most influential figure in the movement toward abstraction and generality of algebra and culminating in universal algebra. He developed a number of important ideas, such as the concept of the "group with operators" (a generic concept that includes groups, rings, and linear spaces) and the three fundamental theorems on isomorphism, theorems that groups with operators satisfy.
Noether pioneered what is called "conceptual mathematics" (Legriffliche Mathematik). He attempted to completely remake algebra, giving priority to general algebraic concepts (homomorphisms, groups, rings, ideals, modules, etc.) over calculus. In particular, he freed himself from the matrices and determinants of linear algebra, replacing them with the concepts of homomorphism and modulus.
Noether did not arrive at abstractions by generalization from known concrete examples, but worked directly with universally valid general concepts. This mathematical style was later adopted by other mathematicians, a style that flourished in new developments such as category theory.
Noether developed the most important ideas of universal algebra on the basis of the group-theoretic concept and its associated concepts (subgroup, homomorphism, isomorphism, quotient group, etc.). The modern concepts of ring, ideal and module over a ring have their origin in Noether. With this conceptual approach, he contributed to the discovery of unifying algebraic principles. Noeher pioneered what we now call "commutative algebra". He also conceived the fundamental ideas that led to the development of algebraic topology.
Universal algebra experienced its great development 37 years after Whitehead's publication. In 1935, Garrett Birkhoff published the article "On the Structure of Abstract Algebras", where the formal definition of universal algebra and the main ideas on this subject appear for the first time. Birkhoff admitted that he took the term "universal algebra" from Whitehead because it seemed appropriate. According to Birkhoff, the unification of algebra is due to Noether. Birkhoff's paper is considered to mark the formal birth of universal algebra. Birkhoff is recognized as the father of universal algebra.
Birkhoff emphasized the theme of the analogy existing between families of formal laws and families of algebras satisfying these laws. These families of algebras correspond to what today is called a "variety" of algebras.
The field of universal algebra was later enriched by developments in metamathematics, category theory and model theory.
MENTAL vs. Universal Algebra
The limitations of universal algebra
Universal algebra is not really universal because it is not general enough; it has limitations:
It does not allow ordered sets because it only admits the equality relation.
Algebraic structures are static and descriptive relations. Algebraic structures with relations of dynamic and operative type are not contemplated.
Since universal quantification is not explicit, it is not possible to distinguish between generic and particular identity expressions.
It is not possible to express all laws and all relations as equations. For example, inverse elements in fields are defined for non-zero elements, but this constraint (condition) cannot be expressed in an equation.
It is not a general mathematical language. It is a language that can only be described as "equational". It does not really pursue the unification of mathematics. It only aims to provide a general framework of abstraction to all algebraic structures that allows to relate and compare them.
In short, universal algebra clarifies certain aspects of traditional algebraic structures, but universal algebra is just algebra, i.e., it is about elements of a set and operations between them. It is not a complete mathematical language. There are only 3 implicit semantics: set, internal n-ary operations, and universal quantification. More semantics are missing. For an algebra to qualify as "universal" it must contemplate the combinatorics of all kinds of mathematical entities.
Characteristics of MENTAL as a universal algebra
The properties common to all algebraic structures are not to be sought in particular structures, but in the mechanisms that generate these structures, that is, in the primary archetypes, in the degrees of freedom, in the source from which all algebraic structures arise by combinatorics.
We can point out the following characteristics of MENTAL as a universal algebra:
Universal quantification is explicit. It is realized by means of parameterized generic expressions in which even operators can be parameterized.
There is full orthogonality. Any primitive can be combined with the other primitives without restrictions. For example:
⟨( x+y+b = a )⟩/(a° = 5) // ev. ⟨( x+y+b = 5 )⟩
⟨x+y+1⟩+⟨x+y+1⟩ // ev. 2*⟨x+y+1⟩
It allows to include conditions, which the universal algebra does not contemplate.
Variables are defined in substitution expressions (immediate or delayed) when a name is specified on the left side. When, instead of a name, an expression is used, we have imaginary expressions. Two archetypal examples are the definition of the imaginary unit (i*i = −1) and the definition of infinitesimal (ε*ε = 0).
It provides a unifying approach, not just an algebraic one. The unification is based on the primitives of the language (its degrees of freedom) and its combinatorics. Universal algebra is not a general mathematical theory (or general paradigm), it is just a branch of mathematics. In contrast, MENTAL is a universal language and a universal paradigm.
Universal algebra uses axioms, but has no language. In MENTAL, the formal axioms are generic and are expressed by the semantic axioms (the universal semantic primitives). MENTAL is a conceptual algebra and a conceptual mathematics: a language based on a combination of primary concepts.
If we associate "algebra" with variable names and operations with them, MENTAL is a universal algebra because we can assign names to all kinds of expressions and operate with them. Names are of two types: immediate substitution and delayed substitution (or representation).
In MENTAL not only imaginary numbers are contemplated, but imaginary expressions in general and higher order imaginary expressions.
MENTAL allows you to define higher-order algebraic structures, virtual, imaginary, linked, interlaced, etc.
MENTAL is based on static and dynamic relations between expressions, as in Whitehead's philosophy. Expressions are processes because they are evaluated.
In MENTAL there is the concept of abstract space (the space where expressions "live") and also the concept of abstract time. Both concepts go together, like space-time in physics.
MENTAL works directly with general concepts, as in Noether's conceptual mathematics, but MENTAL concepts are of supreme simplicity, abstraction and generality. Noether worked with algebraic structures. MENTAL is based on primary archetypes.
In short, MENTAL is at a higher level of abstraction than universal algebra and the theory of categories, up to the limit of the maximum conceptual abstraction, which are the primary archetypes, the archetypes of consciousness. With MENTAL the process of abstraction of mathematics is culminated.
Addenda
More about quaternions
Quaternions are a type of hypercomplex number; they are of dimension 4. Octonions are hypercomplexes of dimension 8 and sedenions are of dimension 16.
Quaternions can be written as a complex number of order 2:
a + bi + cj + dk = a + bi + cj + dij = (a + bi) + (c + di)j (as k = ij)
The conjugate of a quaternion q is q' = a − bi − cj − dk.
Fulfilled: qq' = a2 + b2 + c2 + d2 = |q|, where |q| is the modulus of q.
The multiplicative inverse of a quaternion is q−1 = q'/(qq').
A quaternion q consists of a real or scalar part S(q) and an imaginary part V(q):
S(q) = a V(q) = bi + cj + dk
Hamilton called the imaginary part "vector". So vectors have their origin in quaternions.
Vector spaces
In general, linear algebra studies vector spaces, a set based on two operations: 1) addition of vectors; 2) product of a scalar by a vector. It also studies linear transformations, which are functions between vector spaces that satisfy the linearity conditions:
T(u+v) = T(u) + T(v)
T(r•u) = r•T(u)
Vectors are not necessarily a sequence of numbers. They can be elements of any set.
Vector spaces can be of finite or infinite dimension. The best known vector spaces are: vectors in Rn, matrices of m×n and polynomials of one variable.
Vector product, outer product and geometric product
The origin of vector calculus goes back to complex numbers and Hamilton's quaternions, with the definitions of scalar product and vector product of two vectors in 3D space.
For example, if we express the two vectors, v1 and v2 as a linear combination of 3 unit vectors perpendicular to each other (e1, e2 and e3), we have:
v1 = a1e1 + a2e2 + a3e3 v2 = b1e1 + b2e2 + b3e3
The scalar product is: ei•ej = 0 if < i>i≠j and ei•ei = 1. It is the projection of one unit vector onto the other.
The vector product (or cross product) is: e1×e2 = e3, e2×e3 = e1, e3×e1 = e2,
con ei×ej = −ej× ei y ei×ei = 0. It is a vector perpendicular to the two unit vectors and of modulus 1 (the surface they form).
Therefore,
Grassmann defined the outer product as a generalization of the 3D vector product for any dimension, and which is usually denoted as a∧b ( or wedge product). The outer product is associative, while the vector product (a∧b) is not. The outer product is anticommutative, like the vector product.
The outer product of two vectors a∧b is called a bivector or 2-vector and represents the oriented surface of the parallelogram formed by the two vectors. The outer product of 3 vectors a∧b∧c is the oriented volume formed by the three vectors. And so on for any dimension n.
The vector calculus, as we know it today, is the one introduced by Clifford with his geometric algebra, valid for any number of dimensions. Clifford's geometric product generalizes the scalar product and the vector product of two vectors.
Basic constructions of universal algebra
Assuming an Ω type is set, there are 3 basic constructions in universal algebra:
Homomorphism between two algebras A and B (of arity n).
Is a function h: A→B such that at each operation fA of < i>A corresponds to another operation fB of B:
h(fA(x1, ... , xn)) = fB(h(x1, ..., xn))
Subalgebra.
A subalgebra of A is a subset that is closed with respect to all operations of A.
Product.
A product of a set of algebraic structures is the Cartesian product of the sets with corresponding coordinate operations.
Group with operators
A group with operators is a group G provided with a Δ domain of distributive operators with respect to the internal law of G. That is, if α is an operator and x and y are elements of G and "*" is the inner operation, it is verified α(x*y) = αx*αy. In abbreviated form it is said that G is a Δ-group. When Δ-group includes only the identity operator, we have a normal group.
Bibliography
Al-Khwarizmi. Le commencement de l´Algebre. Librarie Scientifique et Technique Albert Blanchard, Paris, 2007.
Beachy, John A.; Blair, William D. Abstract Algebra. Waveland Press, 1996.
Beachy, John A. Abstract Algebra On Line. (Contiene definiciones y teoremas.)
Begmann, Cliff. Universal Algebra: Fundamentals and Selected Topics. Chapman and Hall/CRC, 2011.
Bergman, George. An Invitation to General Algebra and Universal Constructions. Disponible online.
Birkhoff, Garrett. On the structure of abstract algebras. Proc. Cambridge Philos. Soc. 31, 433-454, 1935. Disponible online.
Burris, S.; Sankappanavar, H.P. A course in Universal Algebra. Springer-Verlag, 1981. Disponible online. (Un texto clásico.)
Cohn, Paul M. Universal Algebra. Kluwer Academic Publishers, 1981.
Dávila Rascón, Guillermo. El desarrollo del álgebra moderna. Partes I, II y III. Disponible online.
Denecke, Klaus; Wismath, Shelly L. Universal Algebra and Applications in Theoretical Computer Science. Chapman & Hall/CRC, 2002.
