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Isotopy Of Algebras
Isotopy may refer to: Mathematics * Homotopy#Isotopy, a continuous path of homeomorphisms connecting two given homeomorphisms is an isotopy of the two given homeomorphisms in homotopy *Regular isotopy of a link diagram, an equivalence relation in knot theory *Ambient isotopy (or h-isotopy), two subsets of a fixed topological space are ambient isotopic if there is a homeomorphism, isotopic to the identity map of the ambient space, which carries one subset to the other *Isotopy of quasigroups. See Quasigroup#Homotopy and isotopy. * Isotopy of loops, a triple of maps with certain properties. *Isotopy of an algebra In mathematics, an isotopy from a possibly non-associative algebra ''A'' to another is a triple of bijective linear maps such that if then . This is similar to the definition of an isotopy of loops, except that it must also preserve the linear st ..., a triple of maps with certain properties. Semiotics * Isotopy, a repetition of a basic meaning-trait (seme); the dir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Isotopy
In the mathematical subject of knot theory, regular isotopy is the equivalence relation of link diagrams that is generated by using the 2nd and 3rd Reidemeister moves only. The notion of regular isotopy was introduced by Louis Kauffman (Kauffman 1990). It can be thought of as an isotopy of a ribbon pressed flat against the plane which keeps the ribbon flat. For diagrams in the plane this is a finer equivalence relation than ambient isotopy of framed links, since the 2nd and 3rd Reidemeister moves preserve the winding number In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ... of the diagram (Kauffman 1990, pp. 450ff.). However, for diagrams in the sphere (considered as the plane plus infinity), the two notions are equivalent, due to the extra freedom of passing a strand thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ambient Isotopy
In the mathematical subject of topology, an ambient isotopy, also called an ''h-isotopy'', is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let N and M be manifolds and g and h be embeddings of N in M. A continuous map :F:M \times ,1\rightarrow M is defined to be an ambient isotopy taking g to h if F_0 is the identity map, each map F_t is a homeomorphism from M to itself, and F_1 \circ g = h. This implies that the orientation must be preserved by ambient isotopies. For example, two knots that are mirror images of each other are, in general, not equivalent. See also * Isotopy * Regular homotopy *Regular isotopy References *M. A. Armstrong, ''Basic Topology'', Springer-Verlag Springer Science+Business M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure that resembles a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that the associative and identity element properties are optional. In fact, a nonempty associative quasigroup is a group. A quasigroup that has an identity element is called a loop. Definitions There are at least two structurally equivalent formal definitions of quasigroup: * One defines a quasigroup as a set with one binary operation. * The other, from universal algebra, defines a quasigroup as having three primitive operations. The homomorphic image of a quasigroup that is defined with a single binary operation, however, need not be a quasigroup, in contrast to a quasigroup as having three primitive operations. We begin with the first definition. Algebra A quasigroup is a non-empty set with a binary operation (that is, a magma, indicating that a quasigroup has to sat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotopy Of Loops
In the mathematical field of abstract algebra, isotopy is an equivalence relation used to classify the algebraic notion of loop. Isotopy for loops and quasigroups was introduced by , based on his slightly earlier definition of isotopy for algebras, which was in turn inspired by work of Steenrod. Isotopy of quasigroups Each quasigroup is isotopic to a loop. Let (Q,\cdot) and (P,\circ) be quasigroups. A quasigroup homotopy from ''Q'' to ''P'' is a triple of maps from ''Q'' to ''P'' such that :\alpha(x)\circ\beta(y) = \gamma(x\cdot y)\, for all ''x'', ''y'' in ''Q''. A quasigroup homomorphism is just a homotopy for which the three maps are equal. An isotopy is a homotopy for which each of the three maps is a bijection. Two quasigroups are isotopic if there is an isotopy between them. In terms of Latin squares, an isotopy is given by a permutation of rows ''α'', a permutation of columns ''β'', and a permutation on the underlying element set ''γ''. An autotopy is an isotop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotopy Of An Algebra
In mathematics, an isotopy from a possibly non-associative algebra ''A'' to another is a triple of bijective linear maps such that if then . This is similar to the definition of an isotopy of loops, except that it must also preserve the linear structure of the algebra. For this is the same as an isomorphism. The autotopy group of an algebra is the group of all isotopies to itself (sometimes called autotopies), which contains the group of automorphisms as a subgroup. Isotopy of algebras was introduced by , who was inspired by work of Steenrod. Some authors use a slightly different definition that an isotopy is a triple of bijective linear maps ''a'', ''b'', ''c'' such that if then . For alternative division algebras such as the octonions the two definitions of isotopy are equivalent, but in general they are not. Examples *If is an isomorphism then the triple is an isotopy. Conversely, if the algebras have identity elements 1 that are preserved by the maps ''a'' and ''b'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotopy (semiotics)
In a story, we detect an ''isotopy'' when there is a repetition of a basic meaning trait ( seme); such repetition, establishing some level of familiarity within the story, allows for a uniform reading/interpretation of it. An example of a sentence containing an isotopy is ''I drink some water''. The two words ''drink'' and ''water'' share a seme (a reference to liquids), and this gives homogeneity to the sentence. This concept, introduced by Greimas in 1966, had a major impact on the field of semiotics, and was redefined multiple times.Introduction'' to Greimas, aSigno Catherine Kerbrat-Orecchioni extended the concept to denote the repetition of not only semes, but also other semiotic units (like phonemes for isotopies as rhymes, rhythm for prosody, etc.). Umberto Eco showed the flaws of using the concept of "repetition", and replaced it with the concept of "direction", redefining isotopy as "the direction taken by an interpretation of the text". Redefinitions The concept was h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isotopic (other)
Isotopic may refer to: * In the physical sciences, to do with chemical isotopes * In mathematics, to do with a relation called isotopy; see Isotopy (other) Isotopy may refer to: Mathematics * Homotopy#Isotopy, a continuous path of homeomorphisms connecting two given homeomorphisms is an isotopy of the two given homeomorphisms in homotopy *Regular isotopy of a link diagram, an equivalence relation ... * In geometry, isotopic refers to facet-transitivity {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
