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Quasifield
In mathematics, a quasifield is an algebraic structure (Q,+,\cdot) where + and \cdot are binary operations on Q, much like a division ring, but with some weaker conditions. All division rings, and thus all fields, are quasifields. Definition A quasifield (Q,+,\cdot) is a structure, where + and \cdot \, are binary operations on Q, satisfying these axioms : * (Q,+) \, is a group * (Q_,\cdot) is a loop, where Q_ = Q \setminus \ \, * a \cdot (b+c)=a \cdot b+a \cdot c \quad\forall a,b,c \in Q (left distributivity) * a \cdot x=b \cdot x+c has exactly one solution \forall a,b,c \in Q, a\neq b Strictly speaking, this is the definition of a ''left'' quasifield. A ''right'' quasifield is similarly defined, but satisfies right distributivity instead. A quasifield satisfying both distributive laws is called a semifield, in the sense in which the term is used in projective geometry. Although not assumed, one can prove that the axioms imply that the additive group (Q,+) is abelian. Thus, wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation Plane
In mathematics, a translation plane is a projective plane which admits a certain group of symmetries (described below). Along with the Hughes planes and the Figueroa planes, translation planes are among the most well-studied of the known non-Desarguesian planes, and the vast majority of known non-Desarguesian planes are either translation planes, or can be obtained from a translation plane via successive iterations of dualization and/or derivation. In a projective plane, let represent a point, and represent a line. A '' central collineation'' with ''center'' and ''axis'' is a collineation fixing every point on and every line through . It is called an ''elation'' if is on , otherwise it is called a ''homology''. The central collineations with center and axis form a group. A line in a projective plane is a ''translation line'' if the group of all elations with axis acts transitively on the points of the affine plane obtained by removing from the plane , (the affine deri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hall Plane
In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order ''p''2''n'' for every prime ''p'' and every positive integer ''n'' provided ''p''2''n'' > 4. Algebraic construction via Hall systems The original construction of Hall planes was based on the Hall quasifield (also called a ''Hall system''), H of order p^ for ''p'' a prime. The creation of the plane from the quasifield follows the standard construction (see quasifield for details). To build a Hall quasifield, start with a Galois field, F = \operatorname(p^n) for ''p'' a prime and a quadratic irreducible polynomial f(x) = x^2 - rx - s over ''F''. Extend H=F\times F, a two-dimensional vector space over ''F'', to a quasifield by defining a multiplication on the vectors by (a,b)\circ (c,d) = (ac -bd^f(c), ad - bc + br) when d \neq 0 and (a,b) \circ (c,0) = (ac, bc) otherwise. Writing the elements of ''H'' in terms of a basis , that is, identifyi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Ternary Ring
In mathematics, an algebraic structure (R,T) consisting of a non-empty set R and a ternary mapping T \colon R^3 \to R \, may be called a ternary system. A planar ternary ring (PTR) or ternary field is special type of ternary system used by Marshall Hall to construct projective planes by means of coordinates. A planar ternary ring is not a ring in the traditional sense, but any field gives a planar ternary ring where the operation T is defined by T(a,b,c) = ab + c. Thus, we can think of a planar ternary ring as a generalization of a field where the ternary operation takes the place of both addition and multiplication. In effect, in computer architecture, this ternary operation is known, e.g., as the multiply–accumulate operation (MAC). There is wide variation in the terminology. Planar ternary rings or ternary fields as defined here have been called by other names in the literature, and the term "planar ternary ring" can mean a variant of the system defined here. The term "te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-field (mathematics)
In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. Definition A near-field is a set Q together with two binary operations, + (addition) and \cdot (multiplication), satisfying the following axioms: :A1: (Q, +) is an abelian group. :A2: (a \cdot b) \cdot c = a \cdot (b \cdot c) for all elements a, b, c of Q (The associative law for multiplication). :A3: (a + b) \cdot c = a \cdot c + b \cdot c for all elements a, b, c of Q (The right distributive law). :A4: Q contains an element 1 such that 1 \cdot a = a \cdot 1 = a for every element a of Q (Multiplicative identity). :A5: For every non-zero element a of Q there exists an element a^ such that a \cdot a^ = 1 = a^ \cdot a (Multiplicative inverse). Notes on the definition # The above is, strictl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semifield
In mathematics, a semifield is an algebraic structure with two binary operations, addition and multiplication, which is similar to a field, but with some axioms relaxed. Overview The term semifield has two conflicting meanings, both of which include fields as a special case. * In projective geometry and finite geometry ( MSC 51A, 51E, 12K10), a semifield is a nonassociative division ring with multiplicative identity element. More precisely, it is a nonassociative ring whose nonzero elements form a loop under multiplication. In other words, a semifield is a set ''S'' with two operations + (addition) and · (multiplication), such that ** (''S'',+) is an abelian group, ** multiplication is distributive on both the left and right, ** there exists a multiplicative identity element, and ** division is always possible: for every ''a'' and every nonzero ''b'' in ''S'', there exist unique ''x'' and ''y'' in ''S'' for which ''b''·''x'' = ''a'' and ''y''·''b'' = ''a''. : Note in p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moufang Plane
In geometry, a Moufang plane, named for Ruth Moufang, is a type of projective plane, more specifically a special type of translation plane. A translation plane is a projective plane that has a ''translation line'', that is, a line with the property that the group of automorphisms that fixes every point of the line acts transitively on the points of the plane not on the line. A translation plane is Moufang if every line of the plane is a translation line. Characterizations A Moufang plane can also be described as a projective plane in which the ''little Desargues theorem'' holds. This theorem states that a restricted form of Desargues' theorem holds for every line in the plane. For example, every Desarguesian plane is a Moufang plane. In algebraic terms, a projective plane over any alternative division ring is a Moufang plane, and this gives a 1:1 correspondence between isomorphism classes of alternative division rings and Moufang planes. As a consequence of the algebraic Arti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternative Division Ring
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. The associator Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by : ,y,z= (xy)z - x(yz). By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent toSchafer (1995) p. 27 : ,x,y= 0 : ,x,x= 0. Both of these identities together imply that : ,y,x= , x, x+ , y, x- , x+y, x+y= , x+y, -y= , x, -y- , y, y= 0 for all x and y. This is equivalent to the '' flexible identity''Schafer (1995) p. 28 :(xy)x = x(yx). The associator of an alt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. Life Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous. Early life Veblen was born in Decorah, Iowa. His parents were Andrew Anderson Veblen (1848–1932), Professor of Physics at the University of Iowa, and Kirsti (Hougen) Veblen (1851–1908). Veblen's uncle was Thorstein Veblen, noted economist and sociologist. Oswald went to school in Iowa City. He did his undergraduate studies at the University of Iowa, where he received an AB in 1898, and Harvard University, where he was awarded a second BA in 1900. For his graduate studies, he went to study mathematics at the University of Chicago, where he obtained a PhD in 1903. His dissertation, ''A System of Axioms for Geometry' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Quadrangle
In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six lines connecting the six pairs of points. Dually, a ''complete quadrilateral'' is a system of four lines, no three of which pass through the same point, and the six points of intersection of these lines. The complete quadrangle was called a tetrastigm by , and the complete quadrilateral was called a tetragram; those terms are occasionally still used. Diagonals The six lines of a complete quadrangle meet in pairs to form three additional points called the ''diagonal points'' of the quadrangle. Similarly, among the six points of a complete quadrilateral there are three pairs of points that are not already connected by lines; the line segments connecting these pairs are called ''diagonals''. For points and lines in the Euclidean plane, the dia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
