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Fuzzy Subalgebra
Fuzzy subalgebras theory is a chapter of fuzzy set theory. It is obtained from an interpretation in a multi-valued logic of axioms usually expressing the notion of subalgebra of a given algebraic structure. Definition Consider a first order language for algebraic structures with a monadic predicate symbol S. Then a ''fuzzy subalgebra'' is a fuzzy model of a theory containing, for any ''n''-ary operation h, the axioms \forall x_1, ..., \forall x_n (S(x_1) \land ..... \land S(x_n) \rightarrow S(h(x_1, ..., x_n)) and, for any constant c, S(c). The first axiom expresses the closure of S with respect to the operation h, and the second expresses the fact that c is an element in S. As an example, assume that the valuation structure is defined in ,1and denote by \odot the operation in ,1used to interpret the conjunction. Then a fuzzy subalgebra of an algebraic structure whose domain is D is defined by a fuzzy subset of D such that, for every d1,...,dn in D, if h is the interp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Set Theory
In mathematics, fuzzy sets (a.k.a. uncertain sets) are Set (mathematics), sets whose Element (mathematics), elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi Asker Zadeh, Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, ''L''-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , Decision making, decision-making , and Cluster analysis, clustering , are special cases of ''L''-relations when ''L'' is the unit interval [0, 1]. In classical set theory, the membership of elements in a set is assessed in binary terms according to a Principle of bivalence, bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of ''all'' algebraic structures. "Subalgebra" can refer to either case. Subalgebras for algebras over a ring or field A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras. Only for unital algebras is there a stronger notion, of unital subalgebra, for which it is also required that the unit of the subalgebra be the unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy. An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called ''scalar multiplication'' between elements of the field (called '' scalars''), and elements of the vector space (called '' vectors''). Abstract algebra is the name that is commonly given to the study of algebraic structures. The general theory of algebraic structures has been formalized in universal algebra. Category theory is another formalization that includes also other mathematical structures and functions between structures of the same type (homomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unary Operator
In mathematics, an unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function , where is a set. The function is a unary operation on . Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial ), functional notation (e.g. or ), and superscripts (e.g. transpose ). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent of the argument. Examples Unary negative and positive As unary operations have only one operand they are evaluated before other operations containing them. Here is an example using negation: :3 − −2 Here, the first '−' represents the binary subtraction operation, while the second '−' represents the unary negation of the 2 (or '−2' could be taken to mean the integer −2). Therefore, the expression is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Model
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer) (born 1939), Danish composer Jens Vilhelm Pedersen * ''Fuzzy'' (album), 1993 debut album by the Los Angeles rock group Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul'' album by Collective Soul * "Fuzzy", a song by Poppy from '' Poppy.Computer'' Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (born 1941), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), United States Navy rear admiral * Fuzzy Thurston (1933-2014), American National Football League player * Fuzzy Vandivier (1903−1983), American high school and col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation Structure
Valuation may refer to: Economics *Valuation (finance), the determination of the economic value of an asset or liability **Real estate appraisal, sometimes called ''property valuation'' (especially in British English), the appraisal of land or buildings *A distinction between real prices and ideal prices in Marxist theory. *The term valuation function is often used as a synonym to utility function. *The sociology of valuation also takes economic valuation practices as an object of study. * '' Valuation: Measuring and Managing the Value of Companies'' Mathematics *Valuation (algebra), a measure of multiplicity **p-adic valuation, a special case *Valuation (geometry), a generalization of finitely-additive measures *Valuation (logic), an operation on well-formed formulas with the semantics of evaluation * Valuation (measure theory), a tool for constructing outer measures Other uses * Valuation (ethics), the determination of the ethic or philosophic value of an object * For personal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate ''T'' (for "tall") and assign it the extension (for "Abraham Lincoln"). Note that all our interpretation does is assign the extension to the non-logical constant ''T'', and does not make a claim about whether ''T'' is to stand for tall and 'a' f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Cut
Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, an interval which includes its endpoints * Closed line segment, a line segment which includes its endpoints * Closed manifold, a compact manifold which has no boundary Other uses * Closed (poker), a betting round where no player will have the right to raise * ''Closed'' (album), a 2010 album by Bomb Factory * Closed GmbH, a German fashion brand * Closed class, in linguistics, a class of words or other entities which rarely changes See also * * Close (other) * Closed loop (other) * Closing (other) * Closure (other) * Open (other) Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neutral Element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures such as groups and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with. Definitions Let be a set equipped with a binary operation ∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (often denoted as 1). These need not be ordinary additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fuzzy Equivalence
Fuzzy or Fuzzies may refer to: Music * Fuzzy (band), a 1990s Boston indie pop band * Fuzzy (composer) (born 1939), Danish composer Jens Vilhelm Pedersen * ''Fuzzy'' (album), 1993 debut album by the Los Angeles rock group Grant Lee Buffalo * "Fuzzy", a song from the 2009 ''Collective Soul'' album by Collective Soul * "Fuzzy", a song by Poppy from '' Poppy.Computer'' Nickname * Faustina Agolley (born 1984), Australian television presenter, host of the Australian television show ''Video Hits'' * Fuzzy Haskins (born 1941), American singer and guitarist with the doo-wop group Parliament-Funkadelic * Fuzzy Hufft (1901−1973), American baseball player * Fuzzy Knight (1901−1976), American actor * Andrew Levane (1920−2012), American National Basketball Association player and coach * Robert Alfred Theobald (1884−1957), United States Navy rear admiral * Fuzzy Thurston (1933-2014), American National Football League player * Fuzzy Vandivier (1903−1983), American high school and col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Of Transformation
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Klir
George Jiří Klir (April 22, 1932 – May 27, 2016) was a Czech-American computer scientist and professor of systems sciences at Binghamton University in Binghamton, New York. Biography George Klir was born in 1932 in Prague, Czechoslovakia. In 1957 he received a M.S. degree in electrical engineering at the Czech Technical University in Prague. In the early 1960s he taught at the Institute of Computer Research in Prague. In 1964 he received a doctorate in computer science from the Czechoslovak Academy of Sciences. In the 1960s Klir went to Iraq to teach at the Baghdad University for two years. At the end he managed to immigrate to the U.S.''George Klir, pioneer in systems science, ready to retire'' , Watsons Review, Spring 2007. He started teaching computer science at UCLA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
