Is an algebra or 'a algebra' the same thing as an algebraic structure? Or does it have a different meaning?
Thanks
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$\begingroup$Depends a bit on the context, but "an algebra" is often taken to mean a specific kind of algebraic structure, namely, a vector space with a multiplication operation (or a ring with a vector space structure). See, for example, this Wikipedia entry.
$\endgroup$ $\begingroup$A couple of years behind but I think it would still be of some pedagogical value.
It is helpful to distinguish between an "Algebra" and an "Algebraic structure", not that such a distinction actually exist but only to make it as clear as possible. ....................................................................................................................................................
Algebra is a very general concept and it is roughly defined as follows:
An algebra A is a set A together with one or more operation o(i) (i=natural numbers). A= {A, O(1), O(2), ....O(n)}
Ex: <A, + , *>
Ex: <{0,1}, |(Sheffer stroke) > this is basically the semantics of first-order logic ....................................................................................................................................................
Algebraic Structures are the familiar structures like groups, fields and rings that are usually taught in a general abstract algebra course. .....................................................................................................................................................
These are basically the same creatures, the difference makes sense as long as we insist on differentiating them based on their immediate significance for the modern mathematics.
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