Internat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 725-728
725
METHODS OF CONSTRUCTING HYPERFIELDS
CH. G. MASSOUROS
54 Klious St. Cholargos
Ahtens, Greece
TK 155-61
(Received February I, 1983 and in revised form July 21, 1983)
ABSTRACT.
In this paper we introduce a class of hyperfields which contains non quotient
Thus we give a negative answer to the question of whether every hyperfield
hyperfields.
K
is isomorphic to a quotient
K
of a field
G
by some subgroup
of its multiplica-
rive group.
KEY WORDS AND PHRASES. Hypereomposition, hyperfield, quotient hyperfield, hyperring.
1980 AMS SUBJECT CLISSIFICATION CODE. 12K99.
1.
INTRODUCTION
A hyperfield is a triplet
(H, +, -)
composition (i.e. a mapping whose domain is
H), and
H
where
H
H
is a non-empty set,
is a hyper-
and whose range is the power set of
H (i.e. for every
an internal composition of
+
H, x-y
x,y
H).
These
two operations satisfy the following axioms:
I.
Multiplicative axiom
H
is an almost-group with regard to the multiplication.
S
semigroup
which is the union
G
where
An almost-group is a
is a group and 0 a bilaterally
we shall denote its neutral element and by 0 its ab-
By
S
absorbing element of
GU{0}
sorbing element.
II.
Additive axioms
x+y=y+x
(i)
(x + y) + z
(ii)
For every
(iii)
x’
III.
z
H
is written
E
x
+ y
x’
there exists one and only one
and called the opposite of
-x
y instead of
x
(iv)
+ (y + z)
x
x
x
H
such that
0 e x
+ x’.
x; moreover we shall write
+ (-y)
implies that
y e z
x
Distributive axiom
z
(x + y)
z
x
+
z
y
(x + y)
z
x
z
+ y
z
We remark that:
(a)
By usual convensions if
"-"
is an internal composition of a set
E
and
X,Y
are
CH.G. MASSOUROS
726
X-Y is the set of all
E,
subsets of
cases
Y
x
X
and
such that (x,y) XY
y
x
Y signifies the union
X
then
E
hypercomposition in
U(x,y
Y
and
is a
y
xxyX
{x}
will have the same meaning as
y
"-"
If
In both
{y}
X
re-
spectively.
By virtue of axioms ll.(iii) and ll.(iv) one has
(b)
x
Indeed; 0
x
longs to
so
+ 0.
x
x
x
Then 0
x
x
Also, because
(a + b)(c + d)
ac
+
+
+ 0
+ O.
H
x for each x
Let’s suppose
y
that some
So
x
be-
x
different than
which is absurd because of ll.(iii).
y
+ 0
x.
is a hypercomposition, the distributive axiom gives:
+
ad
(properties of hyperfields can be found in [3],
+ bd.
bc
[5]
If we replace the multiplicative axiom I by
(c)
I’.
O.
is a multiplicative semigroup having a bilaterally absorbing element
H
(H,+,-)
we obtain a more general structure
(see [4], [6])
M. Krasner in his study [I].
which is called hyperring.
The concept of the hyperfield has been introduced by
The hyperfield that occurs there was obtained by considering a quotient of the type
K
K
where
an equivalence relation defined from the
is a valued field and
0
field’s valuation.
2.
THE QUOTIENT HYPERFIELD.
F
Let
in
F
G
F
in
F
by defining the product
as subsets of
F
x,y
x
+ y
of
x,y
which are contained in the setwise sum of
If
F
F
is a ring and
F
x,y
F
G
a normal subgroup of
F
F
(:
by
F,{O}) (that
in
R
will be a hyperring called the quotient hyperring of
F
by
G
is a group, and
xG
Gx
F
G.
x
(G,-)
This
z
endowed with these two operations
for all
such that
G
to be the set of all classes
becomes a hyperfield ([2]) called quotient hyperfield of
R
to be the class which
(see [2]) that if we add
It can also be proved
the set that results is a union of classes modulo
enables us to define the sum
of
Then the
We shall denote by
A multiplication can be introduced
of two classes
y
x
F
form a partition of
the set of the classes of this partition.
results from their setwise product.
x,y
F’s multiplicative group.
a subgroup of
G
be a field and
multiplicative classes modulo
F
K(mod
This hyperfield was called the residual of
is a subset
then
The question that Krasner sets ([2])is whether there are other hyperfields besides the quotient hyperfields.
Indeed it is quite important to find out how rich the
class of all hyperfields is, for if there were only quotient hyperfields then a good
part of hyperfield theory could have been deduced directly from corresponding results
in field theory.
At first one can notice that the residual hyperfields are always quotient hyperNext we shall prove that every subhyperfield of a quotient hyperring is a
group. Indeed let R be a hyperring and
a subhyperfield of R
Then
fields
quotient of a field by a multiplicative
F
G
LEMMA i.
If
uG
is the unit of
,
F
then
uG
is a mu]tiplicative
group.
727
METHODS OF CONSTRUCTING HYPERFIELDS
Let
PROOF.
xuG
(u)uG
thus
xuG
uG
be an element of
uy
x
u2G
u(u’)G
cuG. uG
uGx
In the same way we can show that
uG
uG
Since
Gu
So the class xuG
uG
uG
we have:
uG
is a subset of
uG
and therefore
is
indeed a group.
[[FG]]
Now if we denote by
FG
[[FG]]
we observe that
LEMMA 2.
PROOF.
[[FG]] and
s’(uy)
Let
such that
G
PROPOSITION I.
R
and
Let
R
of
G
t’
R
[[FG]]
Thus
s’((uy)e)
(s’(u))e
se
[[FG]J
is the unit of
R
a normal multiplicative subgroup of
G
by
s’
then there exist
R
Then every subhyperfield of
is the
by some subgroup of this subfield’s multiplicative group.
R
[[FG]]
R
uG
and let
for every
is a hyperfield
We observe
be its unit.
[[FG]]
t
there
uG
[[FG]]
tG-t’G
such that
(t) (t’y’)
yt’y’ is the
such that
I_FG]]
[[FG]]
s
Thus we have:
e
G
be a ring,
Now since
uG
exists an element
y,7’
If
be a subhyperfield of
[[FG]]
F
G
that
F
Let
PROOF.
uG
and so
s
the quotient hyperring of
quotient of a subfield of
[[FG]] uG
s’(uy)
s
es
Similarly
which is generated by the set
is also the unit of
be the unit of
e
y
s
uG [[ FG]J
uG
The unit of
R
the subring of
where
e
uG, and therefore there exist
G
is both the unit of
e
inverse element of
t
and therefore
and the unit
[[FG]]
is a
field.
3.
NON QUOTIENT HYPERFIELDS.
We shall construct a class of hyperfields which contains hyperfields that are not
For this purpose we consider a commutative multiplica-
isomorphic to quotient ones.
tire almost-group
(H,.)
in which we introduce a hypercomposition
+
defined as
x
itself since
follows:
x
x
and
x
+ y
+ x
+ 0
{x,y}
if
x,y z 0
H\{x}
if
x
0
+
x
and
x e H
for every
x
y
x
0
Then:
PROPOSITION 2.
PROOF.
0 e x
+
(H,+,.)
The triplet
is a hyperfield.
At first one can notice that the opposite of
x
is
Next it is obvious that the hypercomposition is commutative and that the
x
associativity is valid when one of
x,y,z
is equal to 0.
Now if
x,y,z z 0
and
distinct, then we have:
x
+ (y + z)
x
+ {y,z}
A similar calculation shows
(x + y)U(x + z)
(x + y) + z
equal to each other; for example
x
+ (y + z)
+ (x + z)
(x + y) + z
x
and
x
y,
{x,y}U{x,z}
{x,y,z}
{x,y,z}
Finally if two of
x,y,z
are
we have:
-{x} U{x,z}
(x + x)U(x + z)
+ {x,z}
(z + z)U(w + z)=
(x + x) + z =[ H{x} + z
z
with w
{z} U{w,z}
H
H
x
Now as far as the axiom ll.iv, is concerned we have:
(i)
if
y
we must have
then
x
x
z
x
+ x
+ y
x
+
x
H{x}
which is in fact true.
This means that for every
z
H\{x}
CH.G. MASSOUROS
728
(ii)
x
y
If
z
+
0 then
x
{z,y}
y
{x,y}, thus
+ y
x
and if
z
x
If
y
z
or
Hy}
+ y
x E z
then
y
z
z
then
x
So this axiom is valid
as well.
Finally it is straight forward to verify the distributive axiom.
The class of the hyperfields constructed as above contains elements
THEOREM.
that do not belong to the class of quotient hyperfields.
We choose an almost group (H,.) with card
PROOF.
x
for every
H\{0}
E
Next we consider the hyperfield
3 and such that
H
(H, +, .)
x
which is con-
structed according to the above method and we suppose that this hyperfield is iso-
morphic to a quotient
xG. xG
(i)
(iii)
x2G
or
G
F
for
the following must be valid:
x e F
for every
-G
G
(ii)
G
F
hyperfield(..)Then
)G
G + G
2
=(x +2"
G
C
G
So we have
()
+ G
2
G
G
+ G, which
F.
But if
F
is not of charac-
+ i.
(x---------)
2
+(-1)
()
2
This contradicts (iii).
F
F
Now if the characteristic of
square; thus
F\G
can be written as a difference of two squares:
F
teristic 2 then every element of
x
G
must contain all the squares of
C
Because of (i)
+
G
from which
is 2 then the sum of two
contradicts again (iii).
So
H
squares is always a
cannot be isomorphic
to any quotient hyperfield.
REMARKS.
(a)
This theorem gives something more, that is, H
cannot be isomor-
phic even to a subhyperfield of a quotient hyperring.
(b)
Two other classes of hyperfields completely different can be found in [7].
The problem of those hyperfields’ isomorphism to quotient hyperfields is also dis-
cussed there but no final answer is given.
REFERENCES
I.
KRASNER, M.
2.
KRASNER, M.
3.
Sur les hyperanneaux
MITTAS, J.
(1973) 368-382.
4.
MITTAS, J.
5.
STRATIGOPOULCS, D.
0
Approximation des corps values complete de caracteristique p
par ceux de caracteristique 0. Colloque d’ Algebre Superieure (Bruxelles,
Decembre 1956), CBRM, Bruxelles 1957.
A class of hyperrings and hyperfields.
Sci. 6 (1983) 307-312.
Internat. J. Math. & Math.
et les hypercorps Mathematica Balkanica, t.
3.
Hyperanneaux et certains de leur proprietes C. R. Acad. Sc. Paris,
Serie A t. 269 (1969)623-626.
(1969)489-492.
6.
STRATIGOPOULOS, D.
7.
STRATIGOPOULOS, D.
..
Hyperanneaux, hypercorps, hypermodules, hyperspaces vectoriels
Acad. Sc. Paris, Serie A, t. 269
et leurs proprietes elementaires,
..
,
Hyperanneaux artiniens, centralisateur d’un hypermodule et
theoreme de densite,
t. 269 (1969) 889-891.
Acad. S__c. Paris, Seri______e
and
Balkanica, t. 12.
MASSOUROS, Ch. G.
On a class of fields Mathematica