Množina: algebras

ETYM Late Lat. algebra, from Arabic al-jebr reduction of parts to a whole, or fractions to whole numbers, from jabara to bind together, consolidate; al-jebr w'almuqâbalah reduction and comparison (by equations): cf. French algèbre, Italian and Spanish algebra.
The mathematics of generalized arithmetical operations.
System of mathematical calculations applying to any set of nonnumerical symbols (usually letters), and the axioms and rules by which they are combined or operated upon; sometimes known as generalized arithmetic.
“Algebra” was originally the name given to the study of equations. In the 9th century, the Arab mathematician Mohammed ibn-Musa al-Khwarizmi used the term al-jabr for the process of adding equal quantities to both sides of an equation. When his treatise was later translated into Latin, al-jabr became “algebra” and the word was adopted as the name for the whole subject.
In ordinary algebra the same operations are carried on as in arithmetic, but, as the symbols are capable of a more generalized and extended meaning than the figures used in arithmetic, it facilitates calculation where the numerical values are not known, or are inconveniently large or small, or where it is desirable to keep them in an analyzed form.
The basics of algebra were familiar in ancient Babylonia (c. 18th century BC). Numerous tablets giving sets of problems and their answers, evidently classroom exercises, survive from that period. The subject was also considered by mathematicians in ancient Egypt, China, and India. A comprehensive treatise on the subject, entitled Arithmetica, was written in the 3rd century AD by Diophantus of Alexandria. In the 9th century, al-Khwarizmi drew on Diophantus’ work and on Hindu sources to produce his influential work Hisab al-jabr wa’l-muqabalah/Calculation by Restoration and Reduction.
The development of symbolism.
From ancient times until the Middle Ages, equation-solving depended on expressing everything in words or in geometric terms. It was not until the 16th century that the modern symbolism began to be developed (notably by François Viète) in response to the growing complexity of mathematical statements which were impossibly cumbersome when expressed in words. Further research in algebra was aided not only because the symbolism was a convenient “shorthand” but also because it revealed the similarities between different problems and pointed the way to the discovery of generally applicable methods and principles.
Quarternions and the idempotent law.
In the mid-19th century, algebra was raised to a completely new level of abstraction. In 1843, Sir William Rowan Hamilton discovered a three-dimensional extension of the number system, which he called “quaternions”, in which the commutative law of multiplication is not generally true; that is, ab ¹ ba for most quaternions a and b. In 1854 George Boole applied the symbolism of algebra to logic and found it fitted perfectly except that he had to introduce a “special law” that a2 = a for all a (called the idempotent law).
Discoveries like this led to the realization that there are many possible “algebraic structures”, which can be described as one or more operations acting on specified objects and satisfying certain laws. (Thus the number system has the operations of addition and multiplication acting on numbers and obeying the commutative, associative, and distributive laws.).
In modern terminology, an algebraic structure consists of a set, A, and one or more binary operations (that is, functions mapping A × A into A) which satisfy prescribed “axioms”. A typical example is a structure which had been studied from the 18th century onward and is known as a group. This structure had turned up in the study of the solvability of polynomial equations, but it also appears in numerous other problems (for example, in geometry), and even has applications in modern physics.
Modern algebra.
The objective of modern algebra is to study each possible structure in turn, in order to establish general rules for each structure which can be applied in any situation in which the structure occurs. Numerous structures have been studied, and since 1930 a greater level of generality has been achieved by the study of “universal algebra”, which concentrates on properties that are common to all types of algebraic structure.

Množina: algebra systems

Of or relating to algebra; SYN. algebraical.
1. Relating to, involving, or according to the laws of algebra
2. Involving only a finite number of repetitions of addition, subtraction, multiplication, division, extraction of roots, and raising to powers

Množina: algebraic equations

Množina: algebraic languages

An algorithmic language having statements that resemble algebraic expressions.

Množina: algebraic numbers

Root of an algebraic equation with rational coefficients.

Množina: algebraic semantics

Množina: algebraic specifications

Množina: algebraic structures

Of or relating to algebra