First shot at Group Theory

Ok, so I've been tossing this around ye olde noggin for a few days now. I've been reading about elementary group theory lately and I've been fascinated by it's implications. I've never really cared too much about abstract mathematics since this. I think the reason why I'm loving it so far is that it isn't buried in obscure jargon (not yet, anyway) and it's core principles are based solely on logic. The very basics of the theory aren't buried in a ton of numbers and odd greek letters.

After reading about a few simple number groups, and a few more abstract logical ones, I've tried to come up with a group concerning the logic of something I see in every day life. I chose how I think I could model the process involved in governing the final temperature of the water as it hits you. I, most appropriately, got this idea while I was in the shower. I'm still trying to work through the details but I think I might have something.

First, how I think the temperature is governed. It's not that big of a deal if this part is way off... what's important is the logic it follows. The idea is if such a system existed, then I would have, in essence, modeled a real-life system in an alarmingly mathematical way.

You have two shower knobs, hot and cold. A few notes about the knobs. There is some number of "ticks" you can make before the knob has gone from fully closed to fully open, i.e. a discreet system. The knob may feel continous, but at the smallest level it could be billions of tiny little steps. Imagine a really loooooong staircase with tiny steps. Anyway, each knob has a certain number of turns. Each knob can only send a single temperature of water to the final spiget. By turning the knob the intensity of the water is increased or decreased. Positive numbers denote hot water intensities and negative numbers denote cold water intensities. If we had a pair of knobs with 3 settings, they would be enumerated like this: -3, -2, -1, 1, 2, 3.

That's just how the water works. I now need to decide my set. My set consist of all possible final temperatures, illustrated as the set of all possible pairs of hot and cold temperatures. I won't list them all, but some possible pairs would be (-2, -2), (-1, 1), (1, -4), etc. The operation would "turning the knob", akin to adding the components of two pairs (-1, 2) + (-1, 1) = (1, 3).

To illustrate the group operation, if I turn the hot knob a lot and the cold knob just a little, the final temperature will be pretty hot, but not quite the hottest it can be. Now, I need to check my 4 rules and this is where I'm a bit iffy.

I'll start with the ones I think I've figured out.

1) Closure under the group operation.

Any combination of knob turns will reside within the realm of possible temperatures (since there are finite turns of each knob).

2) Associativity:

The order in which you turn the knobs doesn't matter. Any hot cold followed by a cold turn is the same were it reversed. -1 + 2 = 2 + -1 = 1

3) Existance of the identity element:

If I simply "do not turn", then the temperature remains unchanged.

4) Existence of the inverse:

Any combination of each knob that's at the same intensity yields the same exact temperature (felt as warm). 3 + -3 = 0.

However, this is wrong. Here's the zinger! I've modeled this system, under all the logic, based on a set of numbers (-3, -2, -1, 1, 2, 3) with the operation of simple addition. If you examine this from a purely mathematical standpoint (dealing with numbers only), the first rule is broken. If you add a positive and negative number that are the same intensity, you get zero. -3 + -3 = 0. The number 0 is NOT part of the original set of numbers, so this proposed group breaks closure and is therefore not a group. What's awesome is that the logical way that I've defined my group (with shower turns, and knobs) also fails BECAUSE they are the same exact group! These two concepts, on the most general level, are exactly the same in that their group tables are identical! The reason they both fail as groups is because there are more possible elements than should be allowed! -3 + -3 = 0; if you never turn the knobs, the temperature is undefined. These two fake-groups are homomorphisms! (I think?)

There are logical fallacies in the other rules, but I won't go into them.

I think I've exhausted my mental power for now. I shall continue to try to find my group.

EDIT: When trying to visualize the "group table", imagine the multiplication tables you saw in elementary schools.