** 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 8^{th} 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 Marty’s 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

*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,

*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 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.

*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-H _{v}-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

*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

*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 ’80’s, some Greek mathematicians knowing my
dealing with hypergroups, put me in contact with in

*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.*

Let’s return to

*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

*Lygeros** **N.*

Was there the
definition of the hyperring in the sense of

*Vougiouklis** **Th.*

No, that
hyperstructure was presented during a talk by *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

*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 *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 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 *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 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.

*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

*Lygeros** **N.*

What post did you
have that time in

*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 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 *H _{v}-structures*,
although, in the beginning I had given a name

*Lygeros** **N.*

This is a big
innovation.

*Vougiouklis** **Th.*

Yes, at the same
time I defined the *smaller and greater H _{v}-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 H

*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 4^{th} 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,

*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

*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 H_{v}-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

*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.