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I remember groups, rings, monoids, lattices, etc. being taught in my undergraduate mathematics course.

I never really understood what they were for. After that lesson, we moved on to other lessons without looking back to this specific one.

So, what exactly are they for? What are its implications for mathematicians, and what does it mean for mathematics beginners?

Zaenille
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1 Answers1

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One reason to care about mathematical structures is they allow us to prove things "once and for all." For example, we can prove that if a poset has all bounded-above non-empty joins, then it has all bounded-below non-empty meets, and vice versa. Therefore, since $\mathbb{R}$ can be viewed as a poset with respect to the usual order relation, and since the completeness axiom tells us that $\mathbb{R}$ has all bounded-above non-empty joins (i.e. suprema), hence our theorem tells us that it has all bounded-below non-empty meets (i.e. infima). We don't have to "re-prove" this for the special case of $\mathbb{R}$.

There is at least one more reason to care about mathematical structures, namely the homomorphisms. At a basic level, homomorphisms are important because they preserve lots of goodies. For example, if:

  1. $X$ and $Y$ are monoids,
  2. $\varphi : X \rightarrow Y$ is a monoid homomorphism
  3. $x,x' \in X$ are elements

then $$xx' = e_X \;\implies\; f(x)f(x') = e_Y.$$

So monoid homomorphisms preserve inverses, and therefore, they preserve invertibility.

Taking a more sophisticated viewpoint, structures and their homomorphisms tend to form into categories. For example, there is a category $\mathbf{Grp}$ whose objects are groups and whose arrows are group homomorphisms. There is also usually a so-called forgetful functor to $\mathbf{Set}$ (see here). A major application of this occurs in abstract algebra, whereby we can describe the construction of free algebras as left-adjoint to the forgetful functor. This leads to the idea of presentations; e.g. we can present a group by generators and relations.

goblin GONE
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  • Can you also add a section to your answer that explains it in a way that non-mathematics majors would understand it? At the extent that could possibly be explained, at least. That would be a great addition. – Zaenille Oct 17 '14 at 04:25
  • @MarkGabriel, I think this would be hard to explain to someone without a good deal of math background. Perhaps the key would be to stick with the point made in the first paragraph, and to pick a familiar example, such as vector spaces. Also make sure to emphasize that at a basic level, a mathematical structure is just a bundle of data; its a set together with some other stuff. Bundling things together in this way sometimes makes it easier to say things. – goblin GONE Oct 17 '14 at 04:33