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I am not a mathematician but confused about division algebra vs. algebra. I suspect that "division algebra" is a sub-category(literally, not a math concept) of general or abstract algebra because of following. "A commutative division algebra is called a field." Complex numbers do not have ordering. Quaternions are division algebra so multiplication is not commutative for them.

  1. Why do not we simply call them algebra instead of division algebra. When we call them division algebra and know that they do not have all properties for algebra(ordering, multiplication commutative) it seems weird that they are being used everywhere and apply to everything, for vectors etc. How they apply universally?

    1.1. May be i am missing something that their definition already match those things.

  2. On the contrary, so if they apply to all math and physics, could we say that division algebra already encompasses all the things in universe, so everything has all those properties already? So, division algebra underlie the laws of physics?

lockedscope
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    See Division algebra. There are two related meanings of "algebra": 1) the mathematical discipline (abstract algebra) that studies "structures", i.e. mathematical objects with specified properties. – Mauro ALLEGRANZA Sep 14 '20 at 10:17
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    A division algebra is a mathematical object or structure. It is not a field of study. Is is a special type of algebra (which is also a mathematical object, but it shares name with a field of study). The set of complex numbers, with its standard arithmetic operations, is an algebra (and a division algebra) over the real numbers, for instance. – Arthur Sep 14 '20 at 10:20
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    And 2) : the structures themselves, i.e. the mathematical objects. A Field is an example of mathematical structure. An Algebra over a field is another example. A Division algebra is a third one: "an algebra over a field in which division, except by zero, is always possible." – Mauro ALLEGRANZA Sep 14 '20 at 10:20
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    There's a lot of confusion in your post. It is not true that "Quaternions are division algebra so multiplication is not commutative for them." for example. There are commutative division algebras, and non-commutative division algebras. All fields are division algebras, all division algebras are algebras. Look up the various definitions. – lulu Sep 14 '20 at 10:23
  • @lulu i actually wanted to say quaternions are non-commutative division algebra. – lockedscope Sep 14 '20 at 10:30
  • So, @MauroALLEGRANZA, i see they are objects of algebra but how could we use them like they are encompassing everything. I cannot get that math intuition or sense(feeling)! may be but could we say that their definition has the boundaries matching the real world so we could use them universally. Or mathematicians already looked the nature and define these objects, structures so they match the universe by their definition. May be i should see them as a tool that is defined to be used in real world. So, division algebra is not a system/discipline that define real world but a mere object. – lockedscope Sep 14 '20 at 10:39
  • I started to catch the intuition when i see them as a tool. So, their name confused me like they encompass everything but lack of some properties which seems weird. So, could we say that they do not have every property for real world and mathematicians still need more to define real world. Actually i saw "Heisenberg Lie algebra" which i think takes things forward to define real world... Sorry for my insufficient information about math. – lockedscope Sep 14 '20 at 10:48
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    Not very clear... Try with a simpler example: a Matrix is a mathematical object whose properties are studied by a mathematical discipline called Linear algebra. – Mauro ALLEGRANZA Sep 14 '20 at 11:11
  • Oh @MauroALLEGRANZA, that is a better example. So, division algebra is like matrix here but the algebra in its name causes confusion for me, when using name "division algebra" becomes second nature then i could start to see it as a matrix. So, i need to study and deeply immerse myself. – lockedscope Sep 14 '20 at 11:16

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