Could you please tell me about your first contact with the concept of hypergroup.
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.
Therefore that was your first contact with hyperstructures.
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!
Yes, if one wishes to generalize the axioms, he would have to put the neutral element and the reverse elements.
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.
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 Marty’s paper then?
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.
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.
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.
Did Konguetsof not have the article? Had he seen it?
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
How did you begin your research?
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.
However, did you have a concrete way of research?
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,
So this is how the filing of the papers started.
Yes. I gathered the papers and I classified them. I found the paper of Wall’s 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.
Does the term renaissant come from Mittas?
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.
Yes, each element is a generator: it produces the hypergroup.
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.
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.
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.
Who helped you with the computer scientific part and why up to 40?
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!
Up to the development of your thesis, it appears that you were somehow isolated.
Not somehow. Completely! I would say, and it is most important, that I did not have any confirmation of what I wrote.
Single element! Does a change of phase before and afterwards your thesis exist? How did you notify your thesis?
Firstly, two of my
papers, emanating from the thesis, were presented by Konguetsof in
When did you find out about Corsini?
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 ’80’s, some Greek mathematicians knowing my
dealing with hypergroups, put me in contact with in
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.
Let’s return to
Where there were many researchers who tried to combine hypergroups in the different regions as with harmonic analysis?
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
I also discovered
other papers including the
Was there the
definition of the hyperring in the sense of
hyperstructure was presented during a talk by
Then, did you come in contact with Mittas?
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
How did you enter in the concept of the fundamental relations?
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 Corsini’s school with a
slightly different approach. I speak of relations â and â*. 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
ã* in hyperrings and å* in the hyper-vector spaces, I avoided to use ä that is reported as delta of Kronecker. I used Greek
letters internationally as Koskas used the letter â for the first time. On the other side, I made a
proof for the â* in a deductive rather than
an inductive way and, furthermore, the proof was very short. In a similar way,
I used short proofs for ã* and å*, 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
If we compare the two ways of definitions and proofs, the way that you used is more direct, more explicit.
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 â*. For me, my way is easy, after that I used this way of proof for the relations ã* and å*, almost automatically. This study can be used up to the class of algebras.
Does the problem with hyperfields also appear here?
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
What post did you
have that time in
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.
You forget the hyperfield!
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 â*, ã*, å* 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 ã*, 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.
This is a big innovation.
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.
Do you think that there exists a tendency to create some kind of file with articles of the above research topic?
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.
I would now like to clarify the term very thin: how did you think about it, moreover how is it developed up to now?
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.
You made a transport in hyperstructures.
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.
Which team of researchers have been dealing with very thin?
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,
Hence, this subject was completed in the sector of infinity.
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
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.
I do not remember the construction theorem. There is an additional element which is considered in one case but not in the other.
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.
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.
In substance, the first classification of Vougiouklis is for order 2.
When we have the WASS property and we take bigger hyperoperations then the WASS property is still valid …
Have you defined the term minimal?
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.
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.
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.
We are in 1980.
No, it was before 1980. In 1980 they had been completed.
Did the computer specialist who helped you, also helped you in any other sector?
When I finished my thesis I saw that that was a field that opened also other sectors.
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?
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.
Is this also valid for the Iranian school?
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.
Who saw for the first time fuzzy as hyperstructures?
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.
What is the reason such a thing happened?
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.
Do any other research teams we have not discussed in this interview exist? French, Italian, Greek?
When I saw your own
work, it immediately reminded me of the presentation of papers of the
As Carathèodory did with somas?
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.