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This is an audio version of the Wikipedia Article:\nhttps://en.wikipedia.org/wiki/Quasigroup\n\n\n00:01:29 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
\n\n\nListening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.\n\nLearning by listening is a great way to:\n- increases imagination and understanding\n- improves your listening skills\n- improves your own spoken accent\n- learn while on the move\n- reduce eye strain\n\nNow learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.\n\nListen on Google Assistant through Extra Audio:\nhttps://assistant.google.com/services...\nOther Wikipedia audio articles at:\nhttps://www.youtube.com/results?searc...\nUpload your own Wikipedia articles through:\nhttps://github.com/nodef/wikipedia-tts\nSpeaking Rate: 0.8022371415319163\nVoice name: en-GB-Wavenet-D\n\n\n"I cannot teach anybody anything, I can only make them think."\n- Socrates\n\n\nSUMMARY\n=======\nIn 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.