The operators min, max, if-then-else, and even |x| (absolute value)
are not considered "algebraic" by our ancestors, but that should
constitute no resistance.  In the present context, the ultimate goal
and meaning of "algebra" is:

1. to symbolically model a numerical situation we encounter
2. to calculate with and reason about the model by symbolic manipulations,
   so that it helps us solve problems and answer questions about the
   original situation

It is a fact of experience that the four operators above are
invaluable for #1.  The present rejection of these operators by some
people is, candidly, due to the apparent absence of relevant
manipulation methods (algebraic laws) for the sake of #2.  But this we
can fix.

Here are some algebraic laws for min and max.

  min(x,y) = min(y,x)
  min(min(x,y),z) = min(x,min(y,z))
  max(x,y) = max(y,x)
  max(max(x,y),z) = max(x,max(y,z))

Thus they are commutative and associative.  Because of this, and
because they are useful binary operators, they deserve infix notations
like addition and multiplication do.  I write x/\y for max(x,y), and
x\/y for min(x,y); this is only due to the ASCII restriction here, and
I would prefer using upward and downward arrows instead.  Here are
the above laws again, plus more laws, using the new notation:

  x\/y = y\/x
  (x\/y)\/z = x\/(y\/z)
  x/\y = y/\x
  (x/\y)/\z = x/\(y/\z)
  x\/x = x
  x/\x = x
  x\/(x/\y) = x
  x/\(x\/y) = x
  x\/(y/\z) = (x\/y)/\(x\/z)
  x/\(y\/z) = (x/\y)\/(x/\z)
  x<=y iff x\/y=x
  x<=y iff x/\y=y
  (x\/y)+z = (x+z)\/(y+z)
  (x/\y)+z = (x+z)/\(y+z)
  -(x\/y) = (-x)/\(-y)
  -(x/\y) = (-x)\/(-y)

The if-then-else operator can be said to come from computer
programming, but there is nothing wrong in incorporating it into
algebra and mathematics.  Physics, surveying, statistics, and
economics have all inspired additions to mathematics; so can computer
programming.

There is existing mathematical notation related to if-then-else.
We can define piecewise functions by writing something like

  f(x) = { blah   if x is prime
         { bleh   otherwise

But the if-then-else operator is more general.  It does not require
defining a function just so as to use it once.  I can write directly
"if x>y then x+y else x-5y"; I don't have to write indirectly "f(x,y)
where f(x,y) = ..."

Only two things distinguish if-then-else from other algebraic
operators: It is a tenary operator, and its first operand is a boolean
rather than a number.  But it is useful enough that we should include
and master it, rather than hide our heads in sand.

Here are some laws for if-then-else.

  if true then x else y = x
  if false then x else y = y
  f(if b then x else y) = if b then f(x) else f(y)
    e.g. z+(if b then x else y) = if b then z+x else z+y
  if b then x else x = x

The absolute value operator is familiar enough, so I won't repeat
its laws here.

A note on "making decisions": we make decisions all the time, even
with "algebra" as we knew it two centuries ago.  Whenever we see
x*y/y, we say to ourselves, "if y is 0, x*y/y is undefined; otherwise,
x*y/y is x."  The division operator requires a decision all the time.
There is nothing wrong with an algebraic operator prescribing a
decision.

A note on the construction sqrt(x^2) to mimic |x|, min/max, and
if-then-else: it is technically correct, it is clever, and it even
fits the bill of "algebraic expression" in its narrowest, most ancient
sense.  But it is obsfucation!  A good vocabulary is one that allows
you to say directly what you mean.  If I say "13 + (x-2) * (2.5 + 1.5
* (x-2)/sqrt((x-2)^2)), and by the way pretend 0/0=1", do you know
what I am trying to describe?  Don't you appreciate it if I just
plainly say "if x<2 then x+11 else 4x+5"?  The use of clever tricks to
communicate something supposedly straightforward does not underline
the cleverness of the writer; rather, it underlines the flaws of the
language and vocabulary used.  A good set of operators and notations
does not glorify the writer by way of intimidating the reader.