We usually only consider a set of four binary operators , , and when we discuss groups and fields but can we systematically extend these binary operators arbitrarily (for whatever reason)? For example, multiplication can be implemented as

and

provided that and

and can be negative of course as this just results in a potentially complex result. However what if we replace the addition operator with a multiplication operator and the subtraction operator with a division operator ?

So let's say we define two new operators "Up" and "Down (for no immediate purpose) as

provided that a, b \ne 0

and

provided that and

(note that and should not be confused with intersection and union for the purpose of this post).

In addition I would prefer to exclude negative numbers for now and only consider and for and and for .

It may be of interest to note that the extended set of binary operators

,, , has the following group/field properties

(commutative)

(associative)

(distributive, reasonably assuming Up would take precedence over multiplication)

(identity where is defined as an identity analogous to "" for previous binary operators)

(inverse where is analogous to with previous binary operators)

Further we can define a unitary operator "a to the Up power n" I guess as

As expected the binary operators have the same properties as

This removes our binary operator impoverishment and extends our binary wealth to

and our unitary wealth now expands to and

(Note similar results can be shown for and )

Now we may ask, "what use is any of this anyway". Admittedly I have no idea

. However, if we went back to simpler times and tried to explain multiplication to a farmer, it could be difficult to explain the usefulness of this new binary operator when addition was perfectly adequate for counting his flock. Equally when selling property subtraction would tell people how much money they had left, but why multiply?

I guess the first interest in multiplication may have been inspired from taxation - then finances are lost as a proportion - even the concept of a "fraction" required equally some concept of division.

However I just find some things intriguing in their own right. Perhaps some use will come later. I have played a bit with geometries and some interesting parallels drop out with sin, cos and tan. Also the equivalent of complex numbers can be found by asking for a solution to

what is where ?

(same concept as where we can show, from the defined range of that a for all as previously defined)

If we call this "solution" "" analogous to "" then an up-complex number can be defined as

analogous to of course

Finally, so as not to short change ourselves of even higher binary operators, we can systematically extend the set withing the existing definition for Up and Down,

etc

One thing I thought I might do with Up and Down is use them to construct very large (and small) numbers, as it seems to me that we are at a point were filling pages up with digits to describe a quantity is a bit cumbersome. The decibel system helps a bit I guess but generates lots of digits when the numbers get large. A number system based on "up powers with multiplication" instead of "multiplied powers with addition" would probably go a long way to solve this hassle.

Anyway, I am just posting these ideas in case anyone finds such things to be mildly curious