Of course I added extra structure since I wanted to make a point about different kinds of algebraic structures which are all subsumed by the term "algebraic structure" or "algebra". And it's only possible to distinguish kinds of algebraic structures by differences in structure.
But adding extra structure in one example doesn't mean that I somehow exclude magmas from the definition. Here is the example again, extended to be include a component with no extra structure:
If I have a structure M with no structure but an operation, a structure S with an associative operation, and another structure G with an associative operation and a neutral element, I will say that M and S and G are different algebras, not "different magmas". Others looking at M or S or G will not ask "oh, what kind of magma do you have there", they will ask what kind of algebra.
Of course this extension by M doesn't change anything about the validity of the example. Magmas are just as included in the term "algebraic structure" as semigroups, groups, rings, and fields are.
> So you're saying people shorten the phrase "algebraic structure" to "algebra"; this hasn't been my experience.
Of course I added extra structure since I wanted to make a point about different kinds of algebraic structures which are all subsumed by the term "algebraic structure" or "algebra". And it's only possible to distinguish kinds of algebraic structures by differences in structure.
But adding extra structure in one example doesn't mean that I somehow exclude magmas from the definition. Here is the example again, extended to be include a component with no extra structure:
If I have a structure M with no structure but an operation, a structure S with an associative operation, and another structure G with an associative operation and a neutral element, I will say that M and S and G are different algebras, not "different magmas". Others looking at M or S or G will not ask "oh, what kind of magma do you have there", they will ask what kind of algebra.
Of course this extension by M doesn't change anything about the validity of the example. Magmas are just as included in the term "algebraic structure" as semigroups, groups, rings, and fields are.
> So you're saying people shorten the phrase "algebraic structure" to "algebra"; this hasn't been my experience.
<shrug> It has been mine. Wikipedia has lots of uses of the phrase "the algebra of": https://en.wikipedia.org/w/index.php?search=%22the+algebra+o..., always meaning something like "the algebraic structure of set X with operations f, g, and h".