Non fungible numbers
Multiplication and addition that is no longer commutative
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Imagine a world where added numbers are identical but not equivalent.
So 4 apples + 3 oranges = 4 apples then 3 oranges
This is identical to 7 but not equivalent.
I am thinking of potential use in government accounting and algebraic programming with numbers.
Rather than all taxes going into a single taxable account and then spread out to different funding initiatives, you track the income of each area and track the costs of each taxraising and each funding. So roads should be self funding.
When I have a configuration for a web server I want to add to it a particular behaviour and have it slot into the right place automatically. So the data structure is a polynomial and I add a polynomial confifuration that "fits" the shape of the polynomial that is already there.
Usually in Kubernetes configuration there is few places that can accept my configuration fragment. I should be capable of "adding" to a particular YAML key. There is only small subset where it is valid. The polynomials aren't compatible except for certain areas.
さて、乗算または加算、あるいはその両方の下で、多くの非可換である数のリングを見つけたり、思いついたりすることができます。そして、なぜ乗算と加算だけなのか？たくさんの演算子があります。行列の環は、乗算などでは非可換です。
ですから、ネーミングを除けば、これは新しいことではないと思います。なぜ「可換」に「代替可能」という言葉を使うのですか？実際には、ビットコインで等号 "="を定義する方法ですが、計算の複雑さやマイニングに必要なエネルギー（またはその1 BTCを構成する各構成ブロック）の下では、1BTCは別の1BTCと等しくないはずですが、ビットコインコードによって定義された「同等性」は、それらを人為的に「同等」にします。実際、基本的な現実では、おそらく実際には何も同一ではなく、等式はプロセスループ（つまり、他のプログラムを逆転させるプログラム、状態ループの形成、または同値類として構築されます。 / wiki / Equivalence_class）操作中）。
つまり、プロセスは非可換であることが多いため、状態空間でパスを検索して処理する必要があります（たとえば、マシンまたはクラウドインフラストラクチャの状態空間は、同じ操作で常に元に戻せるとは限りませんが、異なる操作または演算子が適用されます）それら（=プログラム）に適用して、元の状態に戻って戻るには、同じ状態に戻るか進むかを指定する必要があります。
自然発生とは別に、環の特性を公理的に選択することにより、加算の意味を設定することもできます。たとえば、「加算」を数値ではなく数値の加算として定義すると、乗算と加算は可換ではないことがわかります。
4 3 = "43"、これは3 4="34"と同じではありません
4 * 3 = "3333"、これは等しくありません3 * 4 = "444"
結局のところ、数値は計算のツールであり、ツールは本質的に実用的です-私はそうだと思います-前進し、便利な場所で新しい数値システムを定義します。
Well, you can find or come up with many rings of numbers, that are non-commutative, either under multiplication, or addition, or both. And, why just multiplication and addition? We have lots of operators. Ring of matrices are non-commutative under multiplication, etc.
So, I guess this is not a new thing, except for naming. Why use the word "fungible" for "commutative"? In reality, it's how you define the equality sign "=", in bitcoin, 1 BTC should not be equal to another 1 BTC under computational complexity or energy needed to mine it (or each constituent block making up that 1 BTC), but the "equality" defined by bitcoin code artificially makes them be "equal". In fact, at the base reality, perhaps nothing is really identical, and equalities get constructed as process loops (i.e., programs, that reverse other programs, forming state loops, or equivalence classes under operations).
I mean, processes are often non-commutative, so you have to deal with them by searching paths in state spaces (let's say a state space of a machine or cloud infrastructure can not always be reversible by the same operation but different operations or operators applied to them (=programs) need to be applied to reach the same state backward vs forward) to go and come back to the original state.
Apart from natural occurrences, you can also set the meaning of addition by choosing ring properties axiomatically. For example, if we define "addition" as adding up numerals, rather than numbers, we get that multiplication and addition are not commutative:
4 + 3 = "43", which is not the same as 3 + 4 = "34"
4 * 3 = "3333", which is not equal 3 * 4 = "444"
Ultimately, numbers are tools of computation, and tools are pragmatic in nature -- I'd say yes -- go forward, define new number systems, where it is convenient.