My understanding is that addition and multiplication are operations required to satisfy field axioms and that one may construct a field without defining the operations of subtraction and division. On this Wikipedia page, subtraction and division are referred to as "inverse" operations, which I believe I understand in context. Is there a term other than "inverse" that may be used to characterize such operations not required to satisfy axioms that may be defined for a given algebraic structure?

Intuitively, I would guess at natural language terms such as "secondary operations," "subsequent operations," and "implicit operations," though I would like to know what, if any, terms are actually used or would be appropriate to use to refer to operations that aren't required for qualification as a given algebra but may nevertheless be defined for it.

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    $\begingroup$ Generally such operations are often called derived operations. They are studied abstractly in algebraic structures via the notion of clones. See also extension by definition. $\endgroup$ – Bill Dubuque Jun 9 at 16:30

Say I had a ring of matrices with addition and multiplication. There are loads of more complex operations we can define in terms of those starting ones. For example $[X,Y] = XY-YX$. In fact this definition makes sense for an arbitrary ring and what it means depends on what addition and multiplication mean. If I had to give a name for all such operations I would call them derived operations. But if you're only dealing with a few specific ones you should just give them names. The operation $[X,Y]$ is sometimes called the commutator or Lie bracket and it is derived from addition and multiplication.


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