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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 for x, \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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggesting that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Real Numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used for either of these, the notation \R_ or \R^ for \left\ and \R_^ or \R^_ for \left\ has also been widely employed, is aligned with the practice in algebra of denoting the exclusion of the zero element with a star, and should be understandable to most practicing mathematicians. In a complex plane, \R_ is identified with the positive real axis, and is usually drawn as a horizontal ray. This ray is used as reference in the polar form of a complex number. The real positive axis corresponds to complex numbers z = , z, \mathrm^, with argument \varphi = 0. Properties The set \R_ is closed under addition, multiplication, and division. It inherits a topology from the real line and, thus, has the structure of a multiplicative topological group or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily the elementary arithmetic multiplication): , or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. As in the case of groups or magmas, the semigroup operation need not be commutative, so is not necessarily equal to ; a well-known example of an operation that is associative but non-commutative is matrix multiplication. If the semigroup operation is commutative, then the semigroup is called a ''commutative semigroup' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a field. We call ''D'' a division algebra if for any element ''a'' in ''D'' and any non-zero element ''b'' in ''D'' there exists precisely one element ''x'' in ''D'' with ''a'' = ''bx'' and precisely one element ''y'' in ''D'' such that . For associative algebras, the definition can be simplified as follows: a non-zero associative algebra over a field is a division algebra if and only if it has a multiplicative identity element 1 and every non-zero element ''a'' has a multiplicative inverse (i.e. an element ''x'' with ). Associative division algebras The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite- dimensional as a vector space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distributive lattices. The smallest semiring that is not a ring is the two-element Boolean algebra, for instance with logical disjunction \lor as addition. A motivating example that is neither a ring nor a lattice is the set of natural numbers \N (including zero) under ordinary addition and multiplication. Semirings are abundant because a suitable multiplication operation arises as the function composition of endomorphisms over any commutative monoid. Terminology Some authors define semirings without the requirement for there to be a 0 or 1. This makes the analogy between ring and on the one hand and and on the other hand work more smoothly. These authors often use rig for the concept defined here. This originated as a joke, suggestin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set (mathematics), set of all rational numbers is often referred to as "the rationals", and is closure (mathematics), closed under addition, subtraction, multiplication, and division (mathematics), division by a nonzero rational number. It is a field (mathematics), field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of numerical digit, digits (example: ), or eventually begins to repeating decimal, repeat the same finite sequence of digits over and over (example: ). This st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign (mathematics)
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set with an Binary operation, operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is Associative property, associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition, addition operation form a group. The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry, groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the object, and the transformations of a given type form a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: \begin (2 + 3) + 4 &= 2 + (3 + 4) = 9 \,\\ 2 \times (3 \times 4) &= (2 \times 3) \times 4 = 24 . \end Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with A Mathematical Theory of Communication, a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of neural networks and para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
