Group theory is super-important. There is no modern science without groups. If we’d value group theory economically, it would be many trillions of dollars, probably hundreds. Without group theory, hardly is quantum mechanics, and therefore no transistors, and I would not be able to write this message in Internet.
I will argue that two kinds of objects of my theory: funcoids and ordered semigroup actions (in another variant, ordered precategory actions) are not less (and probably more) important than group theory.
Importance of Funcoids
The simplest of several equivalent ways to define funcoids is as a relation between sets, such that four axioms hold:
¬(I δ ∅), I ∪ J δ K ⇔ I δ K ∨J δ K,
¬(∅ δ I), K δ I ∪ J ⇔ K δ I ∨K δ J.
(That is hard to believe, but this system of axioms was started to be considered only in about 1998, by me when I was a first-year university student.)
Compare group axioms (considering elements and the inversion function):
- (a · b) · c = a · (b · c)
- e · a = a
- a · e = a
- a · a-1 = e
- a-1 · a = e
You see that groups have 5 axioms, more than 4 axioms of funcoid.
Moreover groups definition uses functions, whose definition in ZF theory is clunky. But funcoids definition uses only basic concepts of ZF.
So, we see that funcoids are defined simpler than groups.
Being defined in a simple way is not a warranty of importance. For example, if we exclude semigroups from mathematics, it would not break badly, like as it would break if we exclude groups. We need also that based on the concept a rich theory builds. You can see that funcoids are fertile just like groups by reading my 500 pages text focused mainly on funcoids.
There is also needed rich connection with the rest of mathematics, to declare an object important. And yes, funcoids are in numerous ways interconnected with general topology (see the same text) and therefore with all the mathematics.
So, we have a fact: funcoids are both simple and fertile like groups, they are richly interconnected with the rest of mathematics. Therefore, I am declaring that funcoids are not less important than groups.
But further, as you see from the same text, funcoids unlike groups have rather numerous cryptomorphisms. Mathematicians know (TODO: reference) that having several cryptomorphic definitions is a sign of importance of the defined concept. So, we have an additional sign of importance for funcoids. Maybe, therefore funcoids are more important than groups.
Importance of Ordered Semigroup Actions
Ordered Semigroup Actions or Actions of Ordered Semigroups (unbelievable, but the concept from these three words was discovered (by me) just in 2019, see the same text) are a generalization of funcoids. That is immediately sign of their importance.
Further: properties of funcoids are completely described as properties of elements of the appropriate ordered semigroup with action. So, aren’t ordered semigroup actions even more important than funcoids?
Moreover, all kinds of spaces in general topology are completely described as elements of ordered semigroup actions. We have space-in-general (well, only for general topology). Space-in-general was believed not to exist by math community, but it seems to exists, at least it exists in general topology.
Again: Ordered semigroup actions encompass all the general topology, and therefore all analysis and functional analysis. They are super-important.
Moreover, if we consider a little more general object, ordered precategory actions (or in a different wording actions of ordered precategories), we have a common framework for both spaces and functions between these spaces. This allows for example to define continuity in a simple algebraic way.
Importance for Mankind
So, all physics, all engineery, economics, etc. depend on my research. Participate in my research or donate money.
I mispublished my research on funcoids, and therefore I can’t publish funcoids and ordered semigroup actions again in peer review. I also discovered discontinuous analysis, and I have troubles publishing it (the same 500 pages text), too. What a dystopian world: discontinuous analysis is discovered but cannot be published!
So, I blocked development of science:
- no funcoids
- no ordered semigroup actions
- no discontinuous analysis
The world is stuck (just by me writing too long, 500 pages, scientific article that I therefore cannot publish). Or better to say that academia is choked by a too big piece of science, like a sparrow with a too big piece of bread. World ending with a buffer overflow error :-).
So, now the science is like a building being built having a half of foundation or a car going with a missing wheel.
So, put figuratively: I discovered that our car (science) has a missing wheel, I tried to attach the wheel, but the wheel mount is so much bad that it broke, too.
The sheep is dying, the passengers of the car are in a big danger: Now in epoch of climate change it’s especially important to invest into science, otherwise mankind will disappear. Support scientific research.
It like as if in biology we knew millions of species but cat, cow, and wheat were discovered just recently. Nobody noticed that they need to be included into the classification. And we would say like “well, that plant that usually grows on fields… we need to measure its characteristics again”. “How to get rid of mouses? anyone?”; “The amount of seeds produced by this thing that grows on fields and the amount of nitrates needed to grow that thing that growth on fields are definitely related fields of science, a professor noticed in his book, however we are not sure what is the exact relation between them.”
Famous mathematicians such as Timothy Gowers and Terrence Tao, despite of being comparably clever to me, need a psychiatrist:
These hundreds-dollars salaries people seriously thought that the “revolt of mathematicians” that they tried to lead is about them, not about qualified amateurs (the world “amateur” means a human who has not enough money to buy the right to receive a salary for his/her work) who really can’t pay for publication.
See also: The Last Survived Mathematician