Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their nonabelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation \cdot that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The symbo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Welldefined Expression
In mathematics, a welldefined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if ''f'' takes real numbers as input, and if ''f''(0.5) does not equal ''f''(1/2) then ''f'' is not well defined (and thus not a function). The term ''well defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined. For example, if ''f''(''x'') = 1/''x'', then the fact that ''f''(0) is undefined does not mean that the ''f'' is ''not'' well defined – but that 0 is simply not in the domain of ''f''. Example Let A_0,A_1 be sets, let A = A_0 \cup A_1 and "define" f: A \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Multiplication Table
In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with baseten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9. History In premodern time The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Cayley Table
Named after the 19th century British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a groupsuch as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's centercan be discovered from its Cayley table. A simple example of a Cayley table is the one for the group under ordinary multiplication: History Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation ''θ'' ''n'' = 1". In that paper they were referred to simply as tables, and were merely illustrativethey came to be known as Cayley tables later on, in honour of their creator. Structure and layout Because many Cayley tables describe groups that are not abelian, the product ''ab'' with respect to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Finite Group
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album '' Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a nonfinite clause In linguistics, a nonfinite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Dover Publications
Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books in the public domain. The original published editions may be scarce or historically significant. Dover republishes these books, making them available at a significantly reduced cost. Classic reprints Dover reprints classic works of literature, classical sheet music, and publicdomain images from the 18th and 19th centuries. Dover also publishes an extensive collection of mathematical, scientific, and engineering texts. It often targets its reprints at a niche market, such as woodworking. Starting in 2015, the company branched out into graphic novel reprints, overseen by Dover acquisitions editor and former comics writer and editor Drew Ford. Most Dover reprints are photo facsimiles of the originals, retaining the original pagination and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Mineola, New York
Mineola is a village in and the county seat of Nassau County, on Long Island, in New York, United States. The population was 18,799 at the 2010 census. The name is derived from an Algonquin Chief, Miniolagamika, which means "pleasant village". The Incorporated Village of Mineola is located primarily in the Town of North Hempstead, with the exception being a small portion of its southern edge within the Town of Hempstead. especially see page 5 Old Country Road runs along the village's southern border. The area serviced by the Mineola Post Office extends farther south into the adjacent village of Garden City, where the Old Nassau County Courthouse is located. Offices of many Nassau County agencies are in both Mineola and Garden City. History The central, flat, grassy part of Long Island was originally known as the Hempstead Plains. In the 19th century, various communities were started in this area. One of those communities was called "Hempstead Branch," which would ultimatel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

David Buchsbaum
David Alvin Buchsbaum (November 6, 1929 – January 8, 2021) was a mathematician at Brandeis University who worked on commutative algebra, homological algebra, and representation theory. He proved the Auslander–Buchsbaum formula and the Auslander–Buchsbaum theorem. Career Buchsbaum earned his Ph.D. under Samuel Eilenberg in 1954 from Columbia University with thesis ''Exact Categories and Duality''. Among his doctoral students are Peter J. Freyd and Hema Srinivasan. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Maurice Auslander
Maurice Auslander (August 3, 1926 – November 18, 1994) was an American mathematician who worked on commutative algebra, homological algebra and the representation theory of Artin algebras (e.g. finitedimensional associative algebras over a field). He proved the Auslander–Buchsbaum theorem that regular local rings are factorial, the Auslander–Buchsbaum formula, and, in collaboration with Idun Reiten, introduced Auslander–Reiten theory and Auslander algebras. Born in Brooklyn, New York, Auslander received his bachelor's degree and his Ph.D. (1954) from Columbia University. He was a visiting scholar at the Institute for Advanced Study in 195657. He was a professor at Brandeis University from 1957 until his death in Trondheim, Norway aged 68. He was elected a Fellow of the American Academy of Arts and Sciences in 1971. Upon his death he was survived by his mother, his widow, a daughter, and a son. His widow Bernice L. Auslander (November 21, 1931  June 18, 2022) was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Partially Ordered Group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order "≤" that is ''translationinvariant''; in other words, "≤" has the property that, for all ''a'', ''b'', and ''g'' in ''G'', if ''a'' ≤ ''b'' then ''a'' + ''g'' ≤ ''b'' + ''g'' and ''g'' +'' a'' ≤ ''g'' +'' b''. An element ''x'' of ''G'' is called positive if 0 ≤ ''x''. The set of elements 0 ≤ ''x'' is often denoted with ''G''+, and is called the positive cone of ''G''. By translation invariance, we have ''a'' ≤ ''b'' if and only if 0 ≤ ''a'' + ''b''. So we can reduce the partial order to a monadic property: if and only if For the general group ''G'', the existence of a positive cone specifies an order on ''G''. A group ''G'' is a partially orderable group if and only if there exists a subset ''H'' (which is ''G''+) of ''G'' such that: * 0 ∈ ''H'' * if ''a'' ∈ ''H'' and ''b'' ∈ ''H'' then ''a'' + ''b'' ∈ ''H'' * if ''a'' ∈ ''H'' then ''x'' + ''a'' + ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Nearring
In mathematics, a nearring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Nearrings arise naturally from functions on groups. Definition A set ''N'' together with two binary operations + (called ''addition'') and ⋅ (called ''multiplication'') is called a (right) ''nearring'' if: * ''N'' is a group (not necessarily abelian) under addition; * multiplication is associative (so ''N'' is a semigroup under multiplication); and * multiplication ''on the right'' distributes over addition: for any ''x'', ''y'', ''z'' in ''N'', it holds that (''x'' + ''y'')⋅''z'' = (''x''⋅''z'') + (''y''⋅''z'').G. Pilz, (1982), "NearRings: What They Are and What They Are Good For" in ''Contemp. Math.'', 9, pp. 97–119. Amer. Math. Soc., Providence, R.I., 1981. Similarly, it is possible to define a ''left nearring'' by replacing the right distributive law by the corresponding left distributive law. Both right and left nearrings occur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 

Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module conc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] 