While I was writing the vectors post, when I commented about how math geeks always build algebras around things, I realized that I hadn't yet written a basics post explaining what we mean by algebra. And since it isn't really what most people think it is, it's definitely worth taking the time to look at.
Algebra is the mathematical study of a particular kind of structure: a structure created by taking a set of (usually numeric) values, and combining it with some operations operate on values of the set.
One of the simplest examples of a kind of algebra is a simple group. A group a pair (G,*) where G is a set of values, and "*" an an operation on values in G such that:
- Closure: For all a,b ∈ G, a*b∈G.
- Associativity: For all a,b,c ∈ G, a*(b*c)=(a*b)*c
- Identity: There is a value i∈G such that for all a∈G a*i=i*a=i.
- Existence of Inverses: For every value a∈G, there is an inverse a-1∈G, and for which a*a-1 = a-1*a = i.
(In a moment of brain-death, I originally typed "Commutativity" instead of "Closure" in property 1 of groups. Stupid mistake on my part, pointed out by two commenters within minutes of posting! Thanks to both of you!
For an example of a group, think of (Z,+) - that is, the integers and the addition operation. Adding any two integers together always results in an integer. 0 is an identity element for integers with addition - for any a, a+0=0+a=a. Addition is also commutative and associative over the integers. And finally, for any integer i, there's another integer -i such that i+-i=0.
It's a very simple structure - a set of values with one operation and four simple properties. But that's enough to capture the entire concept of symmetry. Exactly how it does that is well beyond the scope of a basics post, but some day I'll resurrect my group theory posts from blogger, and post them here.
There are numerous kinds of algebra. A partial list includes groups, rings, and fields; linear algebra (an algebra which looks at the properties of vectors and matrices); algebraic topology (an algebra which looks at the properties of topological spaces), and too many others to mention.
The point of algebras is that they capture essential properties of structure in sets in terms of the ways that those sets can be manipulated using closed operations. Groups captures the concept of symmetry, and what it means for a set to possess a kind of symmetry. Fields capture the structure of real numbers (and other similar sets), and
what kinds of properties they have. Linear algebra explores the world of matrices, and what kinds of properties matrices have, and how you can manipulate them.
Algebras do this - and they do it in a symbolic way. Notice that in the definition of integers as a group, the only integer I needed to name explicitly was 0: for any other value, I could just use a symbol to represent any value, and talk about what it meant to do something to that value.
What we're taught in high school as algebra is one very limited case of this. If you take the set of real numbers, along with addition and multiplication, and look at them as a field, then there's a set of things that you can do symbolically with any statement about the real numbers. That set of things is what's generally taught as algebra. But as I hope you've grasped from this brief post, algebra is much more than that one limited case.
In your definition of a group, I think you mean "Closure" instead of "Commutivity" for your first condition.
Without confusing the issue too much (this is a "Basics" post after all), I was wondering what the relationship is between algebras, fields, etc. and categories. If I've understood your posts about the latter it seems that they too are all about structure in the abstract too.
Beat me to it! I was a little surprised to see "commutativity", and even more surprised not to see "closure". Lots of interesting groups are not commutative (abelian).
Mark & Susan:
Thanks for the catch... I was thinking closure, but for some reason, I typed commutativity. Temporary brain death.
You're picking up on my own personal bias. I tend to view math overall as the study of abstract structure. When you dig down through almost any field of math, you find structure - and each field focuses on different kinds of structure. Algebra is a kind of symbolic structure involving operations over sets; topology is a kind of spatial structure involving the ideas of closeness and continuity; category theory is a kind of symbol structure over the idea of mappings.
Slight error in your definition of identity: "a*i=i*a=i" should read "a*i=i*a=a".
I'm really enjoying your basics series, Mark. Thanks.
I'd love to see the proof of those properties.
At a deep level, much of Mathematics is about 3 things:
So in your definition of an Algebra, you emphasize operations (changes) on structures.
Most people think only of "elementary algebra" or "intermediate algebra", where one uses variables to relate Quantities. Or of tracking down one specific (static) quantity: the Unknown, to its lair, be a series of maneuvers.
This thread, instead, addresses what is also known as "Abstract Algebra."
After "elementary algebra" or "intermediate algebra", some courses are called "College Algebra" or "Precalculus" in order to get the student to calculus: which Newton and Leibnitz invented to be able to handle Change.
In another thread, I'd asked for definitions of "simple math" versus "higher math."
My knee-jerk partial answer to my own question is that "higher math" combines Quantity, Structure, and Change in nontrivial ways.
Symbolic algebra as most people learn it at school has been around in some form or other for about 4000 years the approach that you use in this post is entirely a product of the last 200 years and really came to dominance within the last 100 years. The 20th century has seen a massive change in the way we view and use maths.
These basics are excellent; my comment is unworthy: your algebraic structures are defined via sets, but presumably any set theory would do. That is, the same groups would exist, as abstract structures, whatever the set theory was (different set theories would not give you different group theories, since the different groups would be isomorphic). And presumably mereology or plural logic, in place of sets, would also give you groups (whereas you have defined a group via sets). Does category theory solve this problem?
Point 3 in your definition is equivalent to saying that the set of such elements x in G that x*a=a*x=a is not empty.
Then, what that "i" in point 4 refers to?
To Slawekk: you can show, in one line (plus some hard thought) , that such an x is unique: there's one, and there's no more than one. This justifies treating it as a one-and-only, a special thing in the group.
To Tordek: they can't be proved 'cause they don't always hold: take G to be the points on the line, and take a*b to be the leftmost of the two. Resulting * is associative but has no identity i.
As long as we're nit-picking: Your description of "Identity" in general doesn't fit with your example dealing with addition: One is
and the other is
I assume the general definition should be a*i=i*a=a. As far as I can make out, under the other definition, there is no i for addition. But there is one for multiplication (0), which means multiplication has a property that addition lacks. What's that called?
konrad_arflane: In order to have a group, you need to have a '*' and an 'i'. In the example with the integers, your '*' is '+', and your 'i' is '0'.
You can think that, in the definition, '*' and 'i' are variables, which in the example get '+' and '0' as values.
When I was in school, I made the mistake of thinking simple structures like groups are really simple to learn. Complex structures like Fields and Vector Fields are harder to learn. Wrong! Groups can kill a student and I was not even admitted to the topics (advanced) Algebra course because I did not have good enough a grade... :-(
Oh well the lesson learned was the fewer definition it has, the more flexible it is and the more powerful it is. And that is all I have learned in my undergrad degree in Math. :-)
Mark W in Vancouver BC
Does category theory solve this problem?
You can certainly construct groups using category theory, if that's what you have in mind. Under this construction, a group is simply a category with one object, with all arrows invertible (each element of the group corresponds to an arrow).
Konrad: the term you are looking for is "zero" (really), or "absorbing element". They don't arise in groups, but they do arise in semigroups.
To Jonathan Lubin: Yes, I know what is the cardinality of the set of neutral elements in a monoid. I know it works out in the end and definition of group is stated in this form in many popular textbooks on algebra. It does not change the fact the the point 4 in the definition uses an object that is not defined by 3.
I think it is rather unfortunate that the existential quantifier is commonly read in English as "exists a" (singular) rather than plural "exist some". This leads to mistakes like this and also to frequent implicit (and usually unnecessary) use of axiom of choice in proofs (as in "by lemma 2 we know that for every number i there exists a set b(i) such that property P holds. Now take the union of sets b(i)...)
The set G (for the group definition) should be non-empty. The * is a "binary" map from (the cartesian product)GXG to G, i.e. a*b is the image of (a,b) under *. Hence, it is not necessary to use the word closure, since it is already included in the definition of the binary map.
> the set G (for the group definition) should be non-empty
G is non-empty by 3 in the definition
The * is a "binary" map from (the cartesian product)GXG to G, i.e. a*b is the image of (a,b) under *. Hence, it is not necessary to use the word closure, since it is already included in the definition of the binary map.
* is neither explicitly nor implicitly defined as a map G x G -> G prior to 1 (the only statement is that it takes 2 elements of G as inputs; up to this point, the output could land anywhere), so the closure property is definitely necessary in this presentation. And for the intended audience, probably clearer.
But statement 3 comes after 1 and 2.
Speaking about "*". What is an operation on values?. It should be defined clearly. Hence, it is a (binary) map GXG -> G.
If I remember from math class correctly, we defined a group to be a set with a binary operation * such that
1. Associativity (as stated by Mark CC)
2. There is an i such that a*i = a
3. For every a there is an a^-1 such that a*a^-1 = i
Demanding * to be a binary operation is just a different way of establishing the closure property, as commented by nikos and Davis.
Note that 2 does not include the statement i*a = a, and 3 does not include the statement a^-1*a = i. This is because these statements are able to be proven from the above ones, and we generally like to keep our axioms as minimal as possible.
Ugh, you do know that "simple" is a technical term? Not all groups are simple groups.
Statements like "Fields capture the structure of real numbers" are also not so good. A better statement would be something like, Fields are a generalisation of certain properties of the real numbers.
I realise that explaining this is difficult, but still.