An Hv - interview, i.e. weak

with Th. Vougiouklis

Interviewer N. Lygeros

 

 

January 2005, Xanthi, Greece

 

 

 

Lygeros N.

Could you please tell me about your first contact with the concept of hypergroup.

Vougiouklis Th.

In late 70s I was a fresh researcher at the Democritus University of Thrace working on various subjects including Lie Algebras and Astronomy. That time L. Konguetsof was appointed Professor of Mathematics in our Department and we all adapted our research interests accordingly. He came from Canada, he had spent a long time of study and teaching in France and Belgium and his object was the Didactic of Mathematics. However, as I was interested in Algebra, he told me that his Thèse D' Etat was in Algebra and on certain rings, or, more precisely, on structures with fewer axioms called annoides. These annoides are related to a structure on which he had worked called hypergroup. He had studied hypergroups with late Professor M. Krasner, his supervisor in France. It was then I heard about Krasner, a great mathematician, a multifarious personality with many research interests, one of which was the hypergroups. I liked this notion of hypergroup and Konguetsof showed me the book Algébre, Vol.I, 1963 by P. Dubreil. There on page 167, on a footnote, there was the definition of hypergroup in the sense of Frèdèric Marty. So, it was then I saw, for the first time, the name of F. Marty who, in an announcement in the 8th Congress of Scandinavian Mathematicians in 1934, had given this renowned definition. However, it was a long time till I was able to get hold of this paper. In this work there exists a motivation example for this structure, which is the quotient of a group by any subgroup which is not an invariant subgroup.

Lygeros N.

Therefore that was your first contact with hyperstructures.

Vougiouklis Th.

Exactly. There, I saw and understood why Marty used the reproduction axiom instead of the two axioms: the existence of neutral element and the existence of inverse element. He set the hypergroup free from the obligation to have neutral element and he gave the possibly most widely used definition. It was, I would like to stress, an inspired action. A brilliant definition!

Lygeros N.

Yes, if one wishes to generalize the axioms, he would have to put the neutral element and the reverse elements.

Vougiouklis Th.

In my opinion Marty, managed to do the greatest generalisation anybody would ever do, acting as a pure and clever researcher. He left space for future generalisations "between" his axioms and other hypergroups, as the regular hypergroups, join spaces etc. The reproduction axiom in the theory of groups is also presented as solutions of two equations, consequently, Marty got round that hitch, too.

Lygeros N.

That is to say that Marty fixed a total process and not a partial one. It was that time you manage to get hold of Martys paper then?

Vougiouklis Th.

No, not yet. It was not easy for someone to find a paper at that time, because there was not Internet in 1977-78. Even worse, it was not published in a Journal but in the Proceeding of a Congress. Later I found out more about Marty.

Lygeros N.

So you did not know the ideology of Marty and why he has chosen it to give this definition? You just saw a "dry" definition via the Dubreil.

Vougiouklis Th.

Yes, then I found out that we had the possibility of finding papers via a service called Lending Division and, for a small consideration, we could have photocopies of several papers, mainly from periodicals and journals.

Lygeros N.

Did Konguetsof not have the article? Had he seen it?

Vougiouklis Th.

I do not know whether he had seen the paper in the decade of 60s and if he had used it. Nevertheless, when I asked him, he did not have it. Konguetsof knew hyperstructures via Krasner, but the subject of hyperstructures was not the subject of his thesis. Even later, I found out that Krasner had worked on this object in early 40.

Lygeros N.

How did you begin your research?

Vougiouklis Th.

I started the research with two objectives. The first thing was to find who had dealt with hyperstructures, who was still working on the object, which problems had been solved and which still remained unsolved. Second, I had to look at the structure itself. Moreover, I had to find examples and to prove properties. You can see that this procedure included great risk because, somebody might have already found them and they had been studied in-depth. What I wanted was to enter in the body of this structure via that procedure.

Lygeros N.

However, did you have a concrete way of research?

Vougiouklis Th.

No, not at all. I wanted to make examples, to conquer the structure and afterwards possibly to work on them. The possibility of finding the clew of the hyperstructures was also the Mathematical Reviews and the Zentralblatt für Mathematik, where reviews of papers were likely to be found. Again, however, I did not know names and the fields where they could be written likely reviews. You see, the term hypergroup is also used elsewhere as in harmonic analysis, which, however does not have any relation with multivalued operations. Searching, I discovered H.S. Wall, 1937, R. Dacic, 1969, M. Koskas, 1970, Y. Sureau, 1977. I asked papers from some of them and I this is how I slowly started entering the community of hypergroupists. Later, towards the end of my thesis, I found out about P. Corsini, who had already founded a school of hypergroupists in Italy and who, interestingly, had begun the research on hyperstructures reading a paper of J. Mittas. You know, Mittas was here in Greece, in Thessaloniki but I did not know his object. Via Sureau I found out that Krasner was still alive and had co-operators in Athens and in Patras. Then I also came to know their research fields and what they had discovered and proved till then. The system I mainly worked in was as follows: I had a pre-fabricated letter that said "I request to send me the paper and any of your related papers". Most of the researchers responded with pleasure. Afterwards I investigated whether hyperstructures was their main research interest or they had just happened to have worked on them only casually.

Lygeros N.

So this is how the filing of the papers started.

Vougiouklis Th.

Yes. I gathered the papers and I classified them. I found the paper of Walls when I was about to finish my PhD thesis and it was there I saw some of the basic definitions of cyclicity. Fortunately not all of them! For example, there were not the definitions of the set of generators, the single power cyclicity etc, which terms, however, do not exist in the classical structures.

Lygeros N.

Does the term renaissant come from Mittas?

Vougiouklis Th.

The term renaissant was introduced by Mittas in 1984, so it appears 4 years after my PhD thesis. The renaissant is a special case of the single power cyclicity.

Lygeros N.

Yes, each element is a generator: it produces the hypergroup.

Vougiouklis Th.

I had, as Wall also did, the possibility of having generators or not, as well as to have generators with different periods. That time I introduced, for example, the single power cyclic hypergroups, as I named them, one power can cover the whole hypergroup. It is also possible to have generators with greater period from the order of the group. That time I had also made a list of hypergroups with two elements where certain new elements can already be identified.

Lygeros N.

We also observe that having the cyclicity, we also obtain commutativity. This is interesting because we have the mentality of Gröthendick according to which, when we want to prove something, we should generalise it, keeping the structure that we have it in as a case. Here it is proved, as a simple remark, that these things are not in effect simple and one can see them only in the list of the 8 non-isomorphic hypergroups with two elements.

Vougiouklis Th.

Here, I would like to report that my good example of this is what I introduced with the name P-hyperoperations. Already, certain theorems and terms had been presented in the list of 8 small hypergroups. Up to that moment a great number of researchers had been involved in almost all fields of research on hyperstructures. Characteristic examples include the P-hypergroups, P-hyperrings, P-hyperfields, P-hyper vector spaces, P-Hv-structures, P-lattices etc. Then, when we use the cyclicity in the P-hyperoperations, that is to say taking a group or a semigroup and a subset, we have an enormous crowd of classes of hypergroups that present terms and theorems which could not exist in the classic theory. In my thesis, however, I studied mainly the P-hypergroups where I have sets that contain only one element and the neutral element. Moreover, at that time, in Xanthi, at the Democritus University, we had a computer Univac and using this we found all the generators and their order in groups up to order 40. Using this result, there were verified certain theorems that I had already proved. Most important, however, is that certain other resulted, from which I was able to prove the most; however, some of them are still open problems to be researched. In these cases we know the period of elements which are generators but we can not prove the related theorems.

Lygeros N.

Who helped you with the computer scientific part and why up to 40?

Vougiouklis Th.

In the computer scientific part, the program was made by D. Diamandidis with my help in the presentation of the problems, of course, and it was 40 because there was enough paper only for such table. That was completely accidental!

Lygeros N.

Up to the development of your thesis, it appears that you were somehow isolated.

Vougiouklis Th.

Not somehow. Completely! I would say, and it is most important, that I did not have any confirmation of what I wrote.

Lygeros N.

Single element! Does a change of phase before and afterwards your thesis exist? How did you notify your thesis?

Vougiouklis Th.

Firstly, two of my papers, emanating from the thesis, were presented by Konguetsof in Czechoslovakia and were published there. That time there was also a new contact. In the audience, there was Professor Drbohlav who had a paper published on hyperstructures since 1956. He is another example of independent hypergroupist.

Lygeros N.

When did you find out about Corsini?

Vougiouklis Th.

I found out about Corsini after my thesis, via Mathematical Reviews. I wrote to him and I got information about his field of research and almost all about his school. I also sent him my papers. In early 80s, some Greek mathematicians knowing my dealing with hypergroups, put me in contact with in Krasner. Similarly, the President of the Greek Mathematical Society, S. Zervos, had invited K. Kuratowski who also gave a lecture in Xanthi and I came to know him, as well. When Zervos invited Krasner in Athens in 1981, he also invited me to meet Krasner and so it happened: I met Krasner and I explained my work to him. Krasner expessed a great interest in my work, especially for the last chapter of my thesis, which was about the associated relations of equivalences and various theorems of quotients in hyperstructures. There were also theorems, generalisations, isomorphisms between hypergroups. Mainly, there where theorems on isomorphisms using commutative diagrams.

Lygeros N.

Kuratowski was a great mathematician and, apart from topology, where he was one of the founders, he had also formulated fundamental theories in the theory graphs which I see used in hyperstructures recently.

Vougiouklis Th.

Lets return to Krasner! It was then I realised that Krasner was the one who, in the 40s had taken Martys hyperstructures, transported them, enriched them with a lot of conclusions and taught them to his students. The traces of Marty disappear in the Second World War and, apart from some researchers who have dealt with few papers in hyperstructures, Krasner is the one who has transferred them into our days. Knowing Corsinis School I found a host of mathematicians dealing with the object. I found a large number of papers I read, I came to know the interests of those researchers and their research results. Nevertheless, I am still in the phase of reading, not yet in co-operation with them. In 1981, there was an interruption of my research on hyperstructures because I took an educational leave for USA. Till then, I had not understood the difference between the term of hypergroup in the harmonic analysis from the term of hypergroup in the multivalued operations. So, I went to the Massachusetts Institute of Technology (MIT), invited by Professor S. Helgason who had been working in hypergroups, but in the sense used in harmonic analysis. During the one and half year time that I was in the USA, I worked in my old topic, the topic of Lie Algebras, mainly in the topic of Kac-Moody Infinite Dimensional Lie algebras. I co-operated with one of the founders of these algebras, V. Kac. What I had in mind at that time was to connect these objects which I finally managed to do, after 10 years, with some papers using the gradation of these algebras. More precisely, taking the gradation which exists in these Lie algebras, I introduced a special hyperoperation between the elements of these Lie algebras of infinite dimension and studied the properties of that hyperstructure.

Lygeros N.

Where there were many researchers who tried to combine hypergroups in the different regions as with harmonic analysis?

Vougiouklis Th.

Of course, as Spector, Campaigne and others, but this effort did not have duration. In the library of M.I.T., I began to search for hyperstructures independent from the channels of journals, but also search journals that I had not had any access to, during old days. I discovered a big number of papers and researchers in hyperstructures.

There, in the basement floor, the beams of the sun struggling to enter through a small skylight shed the light of knowledge as, to my utmost happiness, I saw, covered in dust, the volume of the proceeding of the first paper of Frederic Marty! It was a great discovery, the most exciting one.

I also discovered other papers including the Comptes Rendus of the French Academy of Science by Mark Krasner since the year 1940 etc

Lygeros N.

Was there the definition of the hyperring in the sense of Krasner?

Vougiouklis Th.

No, that hyperstructure was presented during a talk by Krasner in the University of Athens in 1956. He gave the definitions of a hyperfield and of a hyperring where the multiplication was an operation and released the addition making it into a hyperoperation. In this type of hyperring, that now we call additive hyperring, the hyperaddition gives a kind of hypergroups which are called canonical and they were studied mainly by Mittas and his school. Returning from USA I had a big bibliography but, more important, I was in contact with the scientists working in this field.

Lygeros N.

Then, did you come in contact with Mittas?

Vougiouklis Th.

Of course with Mittas in Thessaloniki as well as with his co-operators M. Serafimidou, C. Serafimidis, S. Ioulidis, who I had already met in the past but I did not know that they had been working in the same field. I had, therefore, two research objects: first, the Lie algebras of infinite dimension, for which, however, I could find any co-operators in Greece, as there were not any, and second the hypergroups, in which I was almost completely informed. The experience I had in the topic of Lie algebras helped me a lot in the theory of quotients. I came, that is to say, from the USA charged with quotients. Moreover, I had some knowledge on the representation theory of Lie algebras. For this reason I made an effort to transfer some of these objects in the topic of hyperstructures. I can now say hyperstructure as I escaped from hypergroups because the theory of representation requires hyperrings, hyperfields hyper-vector spaces, hypermatrices etc, objects that I was forced to define and to develop. I virtually tried to create the theory of all representations of hyperstructures and the involved hyperstructures, moreover within their general form. I discovered the generalised hyperrings and the generalised hypermatrices. I then found also some classes of hypergroups that can be represented by hypermatrices as the S-hypergroups and the very thin hypergroups.

Lygeros N.

How did you enter in the concept of the fundamental relations?

Vougiouklis Th.

In the effort of generalising the representations and in the search of new classes of hyperstructures, I always kept in mind that these should be connected with classic structures. I wanted the classic structures to be contained as sub-cases in the corresponding hyperstructures. Then I dealt with the fundamental relations and for a moment I believed that I discovered them, because I had not realised that they had been introduced by Koskas in 1970 and they had been studied by Corsinis school with a slightly different approach. I speak of relations and *. With regard to this subject, the following strange story happened: I was working far from the concept of hypergroups only. For this reason I was able to define the relations * in hyperrings and * in the hyper-vector spaces, I avoided to use that is reported as delta of Kronecker. I used Greek letters internationally as Koskas used the letter for the first time. On the other side, I made a proof for the * in a deductive rather than an inductive way and, furthermore, the proof was very short. In a similar way, I used short proofs for * and *, while traditionally, the inductive way required much longer proofs. Finally, I gave the name fundamental to them and this is the name widely used today. All these are presented and used as fundamental theorems in the theory of representations, as they finally appeared in my book in 1994. However, my first conclusions were in a paper in 1988 titled: How a hypergroup hides a group. I had presented them in a congress in Italy and there I saw that a great part of work on this topic had already been done by Corsinis school. The whole process proved once more that in research while proving and developing a subject, one follows a way other than the classical, he may be led to revolutionary changes. The reverse ways of definition of * fundamental relation, the * and *, now are always given as follows: they are the relations which are the minimal equivalence relations, so that the quotients by them are the corresponding algebraic (classic) structures.

Lygeros N.

If we compare the two ways of definitions and proofs, the way that you used is more direct, more explicit.

Vougiouklis Th.

So it appears to me, but, honestly, I can not compare these two ways because the other proof was the first one for the relation *. For me, my way is easy, after that I used this way of proof for the relations * and *, almost automatically. This study can be used up to the class of algebras.

Lygeros N.

Does the problem with hyperfields also appear here?

Vougiouklis Th.

It was not the target, but I began to deal with it. However, I was not able to define the general hyperfield. For this reason in my various lectures, mainly in Italy, I placed this problem as an unsolved problem, i.e. find the definition of the generalised hyperfield. It was very big problem to determine the unit element with respect to addition. I came back, therefore, to the problem of reproduction. That period researchers in Italy were dealing with the multiplicative hyperring, where only the multiplication is a hyperoperation. R. Rota, R. Procesi-Ciampi, G. Tallini, M. Scafati-Tallini, extended their results, up to our days, in hypervector spaces. The other Italian school, i.e. R. Migliorato, M. De Salvo, D. Freni, when they saw that the subject of * could be seen in another general way. They directed their research into other relations weaker than * and . The problem =* for hypergroups was still open but as it was proved by Freni in 1990, it was closed. However, the research was transferred into the other fundamental relations and more precisely into classes weaker than the . I would like to point out that in 1985, in a congress on hyperstructures, a pre-AHA I would say, all the hypergroupists gathered and we placed problems we had on the topic. It was a very decisive congress because we started to research in new sectors. During this congress, I was assigned to organize the following congress in Greece.

Lygeros N.

What post did you have that time in Greece?

Vougiouklis Th.

I was a lecturer. That period there was effort by some researchers to find connections of the hyperstructures with other topics. Thus, for example we have the congresses of Combinatorics where some hypergroupists participated and presented several papers. In Greece there were some teams of hypergroupists, for example, in Thessaloniki there were Mittas, Serafimidou, Serafimidis, Ioulidis, who were dealing with various fields such as canonical hypergroups, hypertreillis, etc; in Thrace: Vougiouklis, Kongeutsof, Spartalis dealing with generalised hyperrings or special hyperrings and hypergroups; in Patras: Stratigopoulos and in Athens Massouros, Pinotsis, Giatras etc. I do not know if I forget someone.

Lygeros N.

You forget the hyperfield!

Vougiouklis Th.

No I do not forget the hyperfield. This problem was still open, I was not able to define it. That time, however, a reverse question was being formed in my mind: which hyperstructures have fundamental relations?, that is to say I saw the problem in reverse. This is how the idea of the weak hyperstructures came. I remember very well that moment. I introduced and denote the weak associativity by WASS, the weak commutativity by COW etc. The basic point of the theory I created was that I replaced the equality of sets by the not empty intersection. The fundamental relations *, *, * still have structures as minimal quotients. Almost the entire fundamental theory is valid. These structures took my name, in a subscript, and they are called Hv-structures, although, in the beginning I had given a name by virtue, because, as quotients, they contain the classical structures. The definitions and the first conclusions were presented at the 4th AHA Congress in Xanthi in 1990. The first applications were presented by other researchers in the same congress. It opens an enormous field and I found myself in a chaos of hyperstructures. I did not know how to control them! This, perhaps, is a delusion, because when you make an opening as a researcher, you have also the ways to curb this structure. In the same congress, however, I also gave an important answer, at least for me. It was the definition of the generalised hyperfield. Here it is: each hyperring has a quotient with respect to the fundamental relation *, always a ring, if the quotient is a field, then the initial hyperstructure is defined as the generalized hyperfield. Apparently, that was another inversion. Consequently, I solved the open problem of the definition of the most general hyperfield which I stated several years earlier in the community of the hypergroupists.

Lygeros N.

This is a big innovation.

Vougiouklis Th.

Yes, at the same time I defined the smaller and greater Hv-structure and there exists the simplest and most useful theorem in this theory that says that bigger hyperstructures than those which are WASS and COW, are also WASS and COW, respectively. From that time and afterwards more than 200 papers in Hv-structures were published in international journals and proceeding. Moreover, they also have applications in other sectors of mathematics as well as in other, not purely mathematic topics, as, for example, in conchology.

Lygeros N.

Do you think that there exists a tendency to create some kind of file with articles of the above research topic?

Vougiouklis Th.

I put all the papers that I had in my file; independently I had used them or not, in the proceeding of the 4th AHA, in the booklet of summaries and later in the proceeding of the congress, published in 1991 by World Scientific.

Lygeros N.

I would now like to clarify the term very thin: how did you think about it, moreover how is it developed up to now?

Vougiouklis Th.

I discovered the very thin, in my effort to find examples on the topic. It was the time I was trying to understand the concept of the hyperstructure that time some ideas occurred to me. This idea is the step that leads from the usual structures to hyperstructures; actually, it is the verge between structures and hyperstructures. I introduced and named this turning point very thin. From this point on, we suppose that anyone has to deal with hyperstructures.

Lygeros N.

You made a transport in hyperstructures.

Vougiouklis Th.

In the beginning I defined the term very thin in hypergroups and for this reason my first results on questions were which hypergroups are very thin and how we can construct them. I proved the relative theorem, virtually a construction in which we have a structure of a group and then we attach an extra element in a concrete way. That time, I had not seen that such a construction could be used itself for enlarging hyperstructures, which in fact I did 15 years later.

Lygeros N.

Which team of researchers have been dealing with very thin?

Vougiouklis Th.

I had dealt with the finite case and the problem had almost been solved for me. However, in the infinity case, the problem was presented by a couple of researchers, Cornelia and Marin Gutan. They are Romanians but they have actually worked in France. In addition, some other researchers from Italy have been deal with the very thin.

Lygeros N.

Hence, this subject was completed in the sector of infinity.

Vougiouklis Th.

They have presented some very interesting results but I cannot say that I now know all their results. My idea was simply to present an algebraic hyperstructure. Later the subject of very thin was extended in other hyperstructures as hyperrings etc and they changed their way of use. However, I think that it is still an important topic with some interesting problems because it connects the classical algebraic theory with the hyperstructures. Note that C.Gutan has obtained her PhD in France working on this subject.

Lygeros N.

I remember that the theorem in finite hypergroups, where it acts as characteristic case, says that there are only two classes of very thin hypergroups.

Vougiouklis Th.

I do not remember the construction theorem. There is an additional element which is considered in one case but not in the other.

Lygeros N.

There is a characterization with which we know precisely the attributes of unique element. Remaining in the finite case and when we pass into hyperrings, there you had found five families and an open question exists whether these are the only very thin families. With regard to the first direction for the minimal and the finite case you deal with the small sets a little.

Vougiouklis Th.

It is reasonable, when somebody begins with an object to seek examples in small sets. The team of Italians are making efforts on the topic but the difficult attribute is the associativity, specifically in the finite case, which involves a big number of attempts while the infinite case is obtained with a general proof of some attribute.

Lygeros N.

In substance, the first classification of Vougiouklis is for order 2.

Vougiouklis Th.

When we have the WASS property and we take bigger hyperoperations then the WASS property is still valid

Lygeros N.

Have you defined the term minimal?

Vougiouklis Th.

Yes. The term minimal has a meaning mainly in Hv-structures. If these are WASS hyperstructures, as we usually do in hypergroups, we try to find hypergroups which generate them, which can produce them.

Lygeros N.

Let us come back to the P-hypergroups. For me the following is strange: in order to find the tables until 40 elements, you have used a computer. Why have not even used a computer in the case of hypergroups with neutral element? That is the easier case.

Vougiouklis Th.

We used computers in the P-hypergroups defined on groups with neutral element. Hence we used only one additional element. In the treatment in the computer I did not participate, because I did not know anything about computers. However, I had to participate in the process of controlling the results. I remember that we made various tricks in order to bring the computer to compute and to give correct results.

Lygeros N.

We are in 1980.

Vougiouklis Th.

No, it was before 1980. In 1980 they had been completed.

Lygeros N.

Did the computer specialist who helped you, also helped you in any other sector?

Vougiouklis Th.

When I finished my thesis I saw that that was a field that opened also other sectors.

Lygeros N.

How do you see the future in hyperstructures and in hypergroups? Which direction takes the research in this sector? We said for the background, for certain concepts that are important, fundamental relations, bigger, classification, infinity, very thin etc, what is the future in this sector?

Vougiouklis Th.

There are researchers dealing with the process of recognition of the fundamental structures. There are also certain researchers who tried to define hyperstructures and to explain their meaning in the other sectors of applications. A characteristic example is the topic of fuzzy sets.

Lygeros N.

Is this also valid for the Iranian school?

Vougiouklis Th.

The Iranian school have a lot to do with the direction of fuzzy. That is to say, they introduce hyperstructures of this type and they study them in depth. The big wave of Iranian researchers is due to the enormous crowd of applications of the fuzzy sets nowadays.

Lygeros N.

Who saw for the first time fuzzy as hyperstructures?

Vougiouklis Th.

Corsini was the one who made the first effort, but it appears the phenomenon that this moment, the 80% of efforts comes from Iranians who had formed the biggest teams in hyperstructures working in the fuzzy topic.

Lygeros N.

What is the reason such a thing happened?

Vougiouklis Th.

I assume that this is so because the founder of the fuzzy sets is the Iranian L. Zadeh, who was the first to define it in a paper in 1965. The Iranian school has the fuzzy as the mayor target.

Lygeros N.

Do any other research teams we have not discussed in this interview exist? French, Italian, Greek?

Vougiouklis Th.

When I saw your own work, it immediately reminded me of the presentation of papers of the French School where the subject is presented thoroughly and in depth in a strict and profound mathematic way. It is the way that the Bourbakis also work. It is not the same way in which today we present the results wishing to share them immediately with the mathematic community. That is why I have classified you in the French School, even if you use computers, which was not then used. This way of presentation is an entirely philosophical attitude. It is also connected a lot to the theory of categories and the theory of representations and in other sectors of mathematics where there are also efforts made of introducing a single theory or for the mathematics to be seen in an overall algebraic way.

Lygeros N.

As Carathèodory did with somas?

Vougiouklis Th.

Indeed, Carathèodory took this basic element, the somas, and began a series of papers. He created an algebraic theory that unfortunately he did not succeed to complete as he died in 1950. He had, however, had taken notes and his co-operators took these notes and they published the book 6 years after his death. I do not know if Carathèodory, when he created this work, knew the existence of certain other theories. The theory of hyperstructures did not exist really then or, at least it was not known. But I believe that if Carathèodory had known the theory of hyperstructures, if he had heard about this subject, it would have been much easier to put a triangle with different ways to have different results. This is a motivation to the researchers to take this idea of somas and to develop them in combination with hyperstructures.