Notation question about min and max functions

[stevenb]

I have a general question about the notation of min and max functions. I'm wondering if there is an accepted notation, used by mathematicians, for the min and max functions, other than the usual min(x,y) and max(x,y) that I typically see.

To give similar examples for what I'm looking for.

absolute value: abs(x) is notated as |x|
round to lower integer: floor(x) is notated as $$\lfloor x \rfloor$$
round to upper integer: ceil(x) is notated as $$\lceil x \rceil$$

Lately, I've been using min and max often and it would be nice to have a shorthand notation. I could make up my own, but I don't want to use a notation that isn't an accepted standard. So far, my searching indicates that I'm stuck with min and max, but I figured this forum would be a good place to see if anybody is aware of any obscure notation that would be acceptable to mathematicians.

 

[micromass]

If x and y are real numbers, then the maximum is often denoted as $x\vee y$. The minimum is $x\wedge y$. In fact, this notation holds in situations more general than real numbers, but then they represent the supremum and the infimum.

The supremum of an arbitrary set (if it exists) can be denoted as $\bigvee X$.

 

[stevenb]

If x and y are real numbers, then the maximum is often denoted as $x\vee y$. The minimum is $x\wedge y$.

Thank you very much. That is interesting. There is some potential for those symbols to have other meanings, but for the context I'm working in, I think this can work well.

I'm wondering if it is proper for me to use this notation to denote functions, and if so, the proper way to write it.

For example if I have f(t)=max(0, g(t)), I'm defining the function f(t) to be equal to g(t) as long as g(t) >=0, and zero otherwise. This is more compact of a notation than a case definition.

I'm tempted to write the following.
$$f(t)=0\bigvee g(t)$$

Is this OK, or is it abusing the notation since g(t) is a real function and not a real number?

 

[micromass]

Both 0 and g(t) are real numbers, so there is no problem in writing $0\vee g(t)$.

Writing $0\vee g$ on the other hand is a little abuse of notation. But it should be clear if you know what you are doing.

 

[Devil Doc]

Just out of curiosity, what level algebra is this symbolism used for? This is the first time I've seen it. I am familiar with max and min, just not in this notation. Thanks.

 

[micromass]

Just out of curiosity, what level algebra is this symbolism used for? This is the first time I've seen it. I am familiar with max and min, just not in this notation. Thanks.

It is quite standard notation. It is the notation of lattice theory. It is often used in real analysis books and beyond.

 

[stevenb]

Both 0 and g(t) are real numbers, so there is no problem in writing $0\vee g(t)$

Excellent. This will work very well for me then.

So, looking at a few on-line references on lattice theory, I see that notation is used extensively, as you said. I even notice some nice looking theorems that might prove useful for me in what I'm doing.

Thank you very much for this helpful information.

 

[Devil Doc]

Thanks for the reply and information, micromass. I do appreciate it.