Loop (mathematics) | Wikipedia audio article

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Quasigroup


00:01:29 1 Definitions
00:02:59 1.1 Algebra
00:04:29 1.2 Universal algebra
00:05:59 2 Loops
00:07:29 3 Symmetries
00:08:13 3.1 Semisymmetry
00:09:43 3.2 Triality
00:11:13 3.3 Total symmetry
00:11:58 3.4 Total antisymmetry
00:12:43 4 Examples
00:15:42 5 Properties
00:16:27 5.1 Multiplication operators
00:17:12 5.2 Latin squares
00:18:42 5.3 Inverse properties
00:19:27 6 Morphisms
00:20:12 6.1 Homotopy and isotopy
00:21:42 6.2 Conjugation (parastrophe)
00:22:27 6.3 Isostrophe (paratopy)
00:23:12 7 Generalizations
00:24:41 7.1 Polyadic or multiary quasigroups
00:26:11 7.2 Right- and left-quasigroups
00:27:41 8 Number of small quasigroups and loops
00:29:11 9 See also
00:30:41 10 Notes
00:32:10 11 References
00:35:55 12 External links
00:37:25 Inverse properties
00:41:09 Morphisms
00:41:54 f(x)f(y). Quasigroup homomorphisms necessarily preserve left and right division, as well as identity elements (if they exist).
00:43:24 Homotopy and isotopy
00:44:54 Conjugation (parastrophe)
00:46:24 z) we can form five new operations: x o y := y ∗ x (the opposite operation), / and \, and their opposites. That makes a total of six quasigroup operations, which are called the conjugates or parastrophes of ∗. Any two of these operations are said to be "conjugate" or "parastrophic" to each other (and to themselves).
00:47:53 Isostrophe (paratopy)
00:48:38 If the set Q has two quasigroup operations, ∗ and ·, and one of them is isotopic to a conjugate of the other, the operations are said to be isostrophic to each other. There are also many other names for this relation of "isostrophe", e.g., paratopy.
00:50:08 Generalizations
00:50:53 === Polyadic or multiary quasigroups
00:51:38 An n-ary quasigroup is a set with an n-ary operation, (Q, f) with f: Qn → Q, such that the equation f(x1,...,xn)
00:53:08 Right- and left-quasigroups
00:54:37 x \ (x ∗ y).
00:56:07 Number of small quasigroups and loops
00:56:52 The number of isomorphism classes of small quasigroups (sequence A057991 in the OEIS) and loops (sequence A057771 in the OEIS) is given here:
00:57:37 See also



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SUMMARY
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In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they are not necessarily associative.
A quasigroup with an identity element is called a loop.