ORDERED HYPERGROUPS

AS JOIN SPACES

SARKA HOSKOVA-MAYEROVA, JAN CHVALINA

email: sarka.hoskova@unob.czDepartment of Mathematics and Physics,

Faculty of Military Technology,University of Defence, Brno, Czech Republic

email: chvalina@feec.vutbr.czDepartment of Mathematics,

Faculty of Electrical Engineering and Communication,Brno University of Technology, Czech Republic

1

Hypergroups in the sense of Marty form the largest class of multivaluedsystems that satisfied group-like axiom.

Main motivations to study:

• non-commutative algebra,

• based on geometry.

The term ”join space”, which is a structure motivated by the geometricconsiderations, is related with the term ”hypergroup”.

Walter Prenowitz in his study of geometric structures in the forties of thetwentieth century focused mainly on the study of ordered and partially orderedgeometry, projective and spherical geometry.

2

He consistently used the fact that in the investigated structures ”join” mul-tioperation, which assigns an appropriate link to two different points, plays amain role.

• in the linear geometry a line segment is assigned to two different points,

• in spherical geometry a smaller great circle arc,

• in the projective geometry it is a line.

In the work of W. Prenowitz and J. Jantosciak ”Geometries and join spaces”was introduced and studied the concept of join space that covers a sufficientlygeneral theory of ordered linear, spherical and projective geometry, as well asabelian groups.

In the nineties of the 20th century Jan Chvalina introduced and investigatedthe ordered and quasi-ordered hypergroup.

We give a detailed proof of necessary and sufficient conditions for a hypegroupto be a join space.

3

The hypergroup (H, ∗) is called a quasi-order hypergroup if

(i) a ∈ a2 = a3 for any a ∈ H, (1)

(ii) a ∗ b = a2 ∪ b2 for any a, b ∈ H. (2)

The hypergroup (H, ∗) is called an order hypergroup if moreover

(iii) a2 = b2 implies a = b for any a, b ∈ H. (3)

The condition (3) is called the unique square root condition.

So:A hypergroup (H, ·) is a (quasi)-order hypergroupif and only if

there exists a (quasi)-ordering ρ on the set H such that for all (a, b) ∈ H×Hthere is a · b = ρ(a) ∪ ρ(b), where ρ(a) = x ∈ H, a ρ x.

4

A hypergroup (H, ·) is called a transposition hypergroup if it satisfies thetransposition axiom:

For all a, b, c, d ∈ H the relation

b\a ≈ c/d implies a · d ≈ b · c ,

where

– b\a = x ∈ H; a ∈ b · x (left extension),

– c/d = x ∈ H; c ∈ x · d (right extension)

and ≈ means that the sets are incident.

(The relation A ≈ B is read as A meets B, where A,B ⊆ H.)

A commutative transposition hypergroup (H, ∗) is called a join space.

5

Quite natural question occurs:

Under what conditions an order hypergroup is a join space?

Necessary and sufficient conditions gives the following theorem.

6

Let (X, ρ) be an ordered set, M ⊂ X. Denote:

Uρ(M) = x ∈ X : a ρ x for each a ∈M

Lρ(M) = x ∈ X : x ρ a for each a ∈M

The hypergroup (H, ) is called an upper quasi-order hypergroup (upper orderhypergroup) if a quasiordering (ordering) exists on H such that xy = [x)≤∪[y)≤,holds.

In previous were[m)≤ = x ∈ H;m ≤ x (principal upper end determined by m

and dually(m]≤ = x ∈ H;x ≤ m (principal lower end determined by m).

Theorem 1 Let (X, ρ) be an ordered set, (X, ), (X, ∗) be the upper, lower orderhypergroup of (X, ρ), respectively.

Then(X, ) is join space if and only if for any pair of ρ-incomparable elements a, b ∈ Xsuch that Lρ(a, b) 6= ∅ we have Uρ(a, b) 6= ∅.DuallyThen (X, ∗) is join space if and only if for any pair of ρ-incomparable elementsa, b ∈ X such that Uρ(a, b) 6= ∅ we have Lρ(a, b) 6= ∅.

7

Corollary 2 The upper (lower) order hypergroup of upper (lower) semilatticeis a join space.

Particularly,the upper (lower) order hypergroup of upper (lower) local finite tree is a joinspace.

Corollary 3 The upper and lower hypergroups of lattice are join spaces.

To be noted that in the theory of join spaces can be in an exclusive formdefined the term of convex set and linear (afine) set. (Prenowitz, Jantosciak)

8

ON CERTAIN

PROXIMITIES AND PREORDERS

ON THE JOIN SPACE OF

LINEAR FIRST-ORDER

PARTIAL DIFFERENTIAL

OPERATORS

SARKA HOSKOVA-MAYEROVA, JAN CHVALINA

9

Proximity spaces belonging to classical topological structures involving gener-alization of metric spaces and their uniformly continuous mappings are situatedout of interest of topologists since the time of the results due to E. M. Alfsenand J. E. Fenstad (1959) showing that these spaces can be considered as totallybounded uniform spaces.

Nevertheless, a proximity relation seems to be very useful tool for investiga-tion of weak hyperstructures in the sense of Vougiouklis monograph.

In particular, proximities on hyperstructures yield the way for a naturalgeneration of weak commutativity, weak associativity and weak distributivity inthe incidence of sets in corresponding identities can changed (or generalized infact) by nearness of the sets in question.

Moreover, the concept of similarity of various systems has its abstract math-ematical expression in terms of reflexive and symmetric relation on a set.

These relations are named tolerances and the use of these relations in connec-tion with other structures moves corresponding mathematical theories to usefulapplication.

In fact a proximity on a set is a tolerance on its powerset, so one can expectsome interesting connection between tolerances and proximities.

10

There are two principal approaches to proximity structures.The classical one–based on the construction of binary relation on the power

set of a set satisfying natural axioms–motivated by Smirnoff theory of proximityspaces.

The other approach consists in the axiomatization of the concept “to be far”,where the basic role plays a proximal neighbourhood of a set. This approachhas been developed in the rich theory of symtopogenous and topogenous struc-ture by Akos Czaszar and his collaborators.[13] In this theory the concept of apreorder and an order is playing an important role.

This shows that investigation of preordered and ordered hyperstructures isof a certain importance.

The importance of study of

• Equations expressing laws of conservation as the continuity equation,

• the motion equation,

• Maxwells’ equations of the electromagnetic field,

• linearized equations of acoustics,

• the equation of long-distance electrical line

• and many other equations used in physical investigations and in technicalapplications,

which are all linear partial differential equations of the first order, equationsmotivates our contribution.

11

Let Ω ⊆ Rn be an open connected subset (called a domain) of the n-dimensional euclidean space of n-tuples of reals. As usually, C1(Ω) stands forthe ring of all continuos functions of n-variables u : Ω → R with continuousfirst partial derivatives ∂u

∂xk, k = 1, 2, . . . , n. We will consider partial differential

operators of the form

D(a1, . . . , an, p) =n∑k=1

ak(x1, . . . , xn)∂

∂xk+ p(x1, . . . , xn) Id,

whereak ∈ C1(Ω) for k = 1, 2, . . . , n and p ∈ C1(Ω), p(x1, . . . , xn) > 0for any [x1, . . . , xn] ∈ Ω.

Denote by L1D(Ω) the set of all such operators which are associated to linearfirst-order homogeneous partial differential equations

n∑k=1

ak(x1, . . . , xn)∂u

∂xk+ p(x1, . . . , xn)u(x1, . . . , xn) = 0,

with ak, p ∈ C(Ω)

12

Define a binary operation “·” and a binary relation “≤” on the set L1D(Ω)by the rule

D(a1, . . . , an, p) ·D(b1, . . . , bn, q) = D(c1, . . . , cn, pq),

in short notation:D(~a, p) ·D(~b, q) = D(~c, pq)

where

ck(x1, . . . , xn) = ak(x1, . . . , xn) + p(x1, . . . , xn)bk(x1, . . . , xn),

andD(~a, p) ≤ D(~b, q)

wheneverp ≡ q and ak(x1, . . . , xn) ≤ bk(x1, . . . , xn)

for any [x1, . . . , xn] ∈ Ω and k = 1, 2, . . . , n.Evidently, the relation ≤ on L1D(Ω) is reflexive, antisymmetric and transitive

hence (L1D(Ω),≤) is ordered set.Moreover, (L1D(Ω), ·) is a non-commutative group in which any right trans-

lation and any left translation determined by arbitrary chosen operator D(~a, p) ∈L1D(Ω) is an isotone selfmap of (L1D(Ω),≤).

13

Consequently the following holds:Let Ω ⊆ Rn be a nonempty domain. Then the system of differential operators

(L1D(Ω), ·,≤) is an ordered (non-commutative) group.

14

Now applying a simple construction, we get the resulting hyperstructure.The following Ends Lemma will be useful in what follows.

Lemma 4 [5, 24][Ends Lemma] Let a triple (G, ·,≤) be a quasi-ordered semi-group. Define a hyperoperation

∗ : G×G→ P∗(G) by a ∗ b = [a · b)≤ = x ∈ G; a · b ≤ x

for all pairs of elements a, b ∈ G.

i) Then (G, ∗) is a semihypergroup which is commutative if the semigroup (G, ·)is commutative.

ii) Let (G, ∗) be the above defined semihypergroup. Then (G, ∗) is a hypergroupiff for any pair of elements a, b ∈ G there exists a pair of elements c, c′ ∈ Gwith a property a · c ≤ b, c′ · a ≤ b.

15

We construct such an action using partial differential operators of the firstorder, set of which is endowed with a suitable binary multiplication turning outthe set of operators into a non-commutative hypergroup.

Applying the Ends Lemma we get then a hypergroup of linear partial dif-ferential operators acting on the ring of all continuous functions of n-variablesu : Ω→ R with continuous partial derivatives of all orders.

16

Recall first, that a nonempty set H endowed with a binary hyperoperation

? : H ×H → P∗(H), (P∗(H) = P(H)− ∅)

is called a hypergroupoid.For any pair of elements a, b ∈ H there can be defined two fractions a/b =

x; a ∈ x ? b (right extension) and b\a = x; a ∈ b ? x (left extension). Ifx, y ⊆ H then we write X ≈ Y (read X meets Y ) whenever X, Y are incident,i.e., X ∩ Y 6= ∅.

17

Now an algebraic (non-commutative) join space termed also a transpositionhypergroup can be defined as an associative hypergroupoid (H, ?) satisfying thereproduction axiom

a ? H = H = H ? a for all a ∈ H

and the transposition axiom

b\a ≈ c/d implies a ? d ≈ b ? c for all a, b, c, d ∈ H.

A join space satisfying the equality a ? a = a for any a ∈ H is called asgeometrical; in the opposite case we speak about algebraic join space.

18

A partially order is a binary relation R on a set X which satisfies conditionsreflexivity, antisymmetry and tranzitivity. Sometimes we need to weaken thedefinition of partial order.

We say that a partial preordered is a relation which satisfies conditions re-flexivity and transitivity.

An algebraic system (G, ·,≤) is called a partially preordered (ordered) groupoidif (G, ·) is a groupoid and G,≤ is a partially preordered (ordered) set which sat-isfies monotone condition as follows:

if x ≤ y then a · x ≤ a · y and x · a ≤ y · a for every x, y, a ∈ G.

19

Defining a binary hyperoperation on L1D(Ω) by

D(~a, p) ?D(~b, q) = D(~c, s); D(~a, p) ·D(~b, q) ≤ D(~c, s),

ak, s ∈ C1(Ω), k = 1, 2, . . . , n= D(~c, pq); ak + pbk ≤ ck, ck ∈ C1(Ω), k = 1, 2, . . . , n

we obtain the following result:

Theorem 5 Let Ω ⊆ Rn be a domain. The hypergroupoid (L1D(Ω), ?) is anon-commutative algebraic join space.

20

Now, setting for any pair of subsets A,B ⊂ L1MD(Ω) that Ap(RM)B when-

everA 6= ∅ 6= B and D(a1, . . . , an, p) RM D(b1, . . . , bn, q) for some pair [D(a1, . . . , an, p),D(b1, . . . , bn, q)] ∈ A×B, we obtain the following theorem:

Theorem 6 [9] Let (L1MD(Ω), ?) be the join subspace of the join space (L1D(Ω), ?)

defined above. Then (L1D(Ω),p(RM)) is a proximity space such that for anyquadruple X, Y, U, V ⊂ L1

MD(Ω) of nonempty subsets with the property X p(RM)Y ,U p(RM)V we have

(X ? U) p(RM) (Y ? V ).

It is easy to see that any tolerance on a group compatible as for algebrasmust be transitive, hence it is a congruence. So, we can also treat semigroupsof operators with compatible tolerances and semihypergroups with proximitiesinduced by those.

Let M ⊂ Ω be a finite subset. Denote

L1MD(Ω) = D(~a, p) ∈ L1D(Ω); grad p|ξ = 0 for any ξ ∈M.

Evidently (L1MD(Ω), ·) is a subgroup of the group (L1D(Ω), ·).

We define a binary relation RM on the set of operators L1MD(Ω) by the

condition

D(~a, p) RM D(~b, q) whenever p = q and grad ak|ξ = grad bk|ξfor any ξ ∈M and k = 1, 2, . . . n.

Clearly, RM is an equivalence relation on the set L1MD(Ω).

Suppose D(c1, . . . , cn, s) ∈ L1MD(Ω) is an arbitrary operator.

Since

grad(ak + pck)|ξ = grad ak|ξ + grad p|ξck + p grad ck|ξ = grad bk|ξ + q grad ck|ξ= grad bk|ξ + grad q|ξck + q grad ck|ξ = grad(bk + qck)|ξ

21

for any ξ ∈M and k = 1, 2, . . . n, we have that

D(a1, . . . , an, p) RM D(b1, . . . , bn, q)

implies (D(~a, p) D(~c, s)

)RM

(D(~b, q) D(~c, s)

)and similarly (

D(~c, s) D(~a, p))

RM

(D(~c, s) D(~b, q)

).

Indeed

grad(ck + sak)|ξ = grad ck|ξ + grad s|ξak + s grad ak|ξ = grad ck|ξ + s grad bk|ξ= grad ck|ξ + grad s|ξbk + s grad bk|ξ = grad(ck + sbk)|ξ

ξ ∈M,k = 1, 2, . . . n. Further,

D−1(a1, . . . , an, p) = D(−a1

p, . . . ,−an

p,1

p

)and

D−1(b1, . . . , bn, q) = D(−b1q, . . . ,−bn

q,1

q

)

grad(−akp

)∣∣∣ξ

= − gradakp

∣∣∣ξ

= − grad(1

pak

)∣∣∣ξ

= − grad1

p

∣∣∣ξak −

1

pgrad ak

∣∣∣ξ

=1

p2grad p

∣∣∣ξak −

1

pgrad ak

∣∣∣ξ

= −1

pgrad ak

∣∣∣ξ

= −1

qgrad bk

∣∣∣ξ

=1

q2grad q

∣∣∣ξ− 1

pgrad bk

∣∣∣ξ

= grad−bkq

∣∣∣ξ

22

and 1p

= 1q, consequently

D(~a, p) RM D(~b, q) implies D−1(~a, p) RM D(~b, q).

Therefore the equivalence RM is a congruence on the group (L1MD(Ω), ·).

23

Denote by Fin(Ω) the lattice of all finite subset of the domain Ω. Using theEnds Lemma or using an union of the ends with respect to the ordering by setinclusion a hypergroup with the carrier Fin(Ω) can be created. For any non-empty set M ∈ Fin(Ω) we obtain the congruence RM on the group (L1D(Ω), ·)which was described above.

24

Recall the definition of a proximity in the sense of Cech monograph [3]:

Definition 7 A relation p on the family of all subsets of the set H is called aproximity on the set H if p satisfies the following conditions:

1. ∅nonpH

2. The relation p is symmetric, i.e., A,B ⊂ H, ApB implies B pA .

3. For any pair of subset A,B ⊂ H, A ∩B 6= ∅ implies ApB.

4. If A,B,C are subsets of H then (A∪B) pC if and only if either ApC orB pC.

Let us define (as in [20]) a multiautomaton:

Definition 8 Let S be a nonempty set, (H,) be a hypergroupoid and δ : S ×H → S be a mapping satisfying the condition

δ(δ(s, a), b

)∈ δ(s, a b) (GMAC)

for any triple (s, a, b) ∈ S ×H ×H, where δ(s, a b) = δ(s, x);x ∈ a b.Then the triple M = (S,H, δ) is called multiautomaton with the state set

S and the input hypergroupoid (H,). The mapping δ : S × H → S is calledtransition function or next-state function of the multiautomaton M.

We shall consider smooth functions, i.e., u ∈ C∞(Ω).Let P(a1, . . . , an, p) : C∞(Ω)→ C∞(Ω) be a fixed chosen operator,

P(a1, . . . , an, p)u =n∑k=1

ak(x1, . . . , xn)∂u

∂xk+ p(x1, . . . , xn)u(x1, . . . , xn).

25

Denote by Ct(P) the set of all differential operators D ∈ L1D(Ω) commutingwith the operator P, i.e.,

Ct(P) = D ∈ L1D(Ω); P ·D = D ·P.

Since the identity operator Id belongs to Ct(P), this set endowed with theunique operation “·” is a monoid which is called the centralizer of the operatorP within the group (L1D(Ω), ·).

Lemma 9 ([20]) Operators D1 = D(a1, . . . , an, p), D2 = D(b1, . . . , bn, q) fromthe group L1D(Ω) are commuting if and if for any k = 1, 2, . . . , n and any point[x1, . . . , xn] ∈ Ω there holds∣∣∣∣ 1− p(x1, . . . , xn) 1− q(x1, . . . , xn)

ak(x1, . . . , xn) bk(x1, . . . , xn)

∣∣∣∣ = 0.

Now, for any pair Dα,Dβ ∈ Ct(P) define a hyperoperation “” as follows:

: Ct(P)× Ct(P)→ P∗(Ct(P)

)by

DαDβ = Pn ·Dβ ·Dα;n ∈ N.

Consider the binary relation ρP ⊂ Ct(P)× Ct(P) defined by

Dα ρP Dβ if and only if Dβ = Pn ·Dα

for some n ∈ N0. We get without any effort that(Ct(P), ·, ρP

)is a quasi-ordered

monoid.Further, DαDβ = ρP(Dβ ·Dα) = [Dβ ·Dα)ρP and by Ends Lemma 4 we

obtain that(Ct(P),) is a hypergroup (non-commutative, in general).

26

As usually(Ct(P)

)+with the operation of concatenation means the free

semigroup of finite nonempty words formed by operators from the set Ct(P).Denote

SP =

(P ·D1 · · · · ·Dn)(f); f ∈ C∞(Ω),D1 · · · · ·Dn ∈(Ct(P)

)+and M(SP) the triple

(SP, (Ct(P),), δP

), where the action or transition function

δP : SP × Ct(P)→ SP

is defined by the rule

δP((P ·D1 · · · · ·Dn)(f),Dα

)= (P ·Dα ·D1 · · · · ·Dn)(f)

for any function f ∈ C∞(Ω) and any operator Dα ∈ Ct(P). The transitionfunction δP satisfies the Generalized Mixed Associativity Condition.

Indeed, suppose f ∈ C∞(Ω), Dα,Dβ,D1,D2 ∈ Ct(P) are arbitrary elements.We have

δP

(δP((P ·D1 · · · · ·Dn)(f),Dα

),Dβ

)= δP

((P ·Dα ·D1 · · · · ·Dn)(f),Dβ

)=((P ·Dβ ·Dα ·D1 · · · · ·Dn)(f)

)∈

(Pn+1 ·Dβ ·Dα ·D1 · · · · ·Dn)(f), n ∈ N0

= δP

((P ·D1 · · · · ·Dn)(f),Pn ·Dα ·Dβ

)= δP

((P ·D1 · · · · ·Dn)(f),DαDβ

),

so GMAC is satisfied, i.e., the triple M(SP) =(SP, (Ct(P),), δP

)is a multiau-

tomaton.With respect to the definition of connectivity of an automaton we give the

following definition:

27

Definition 10 1. Let A = (S,G, δ) be a multiautomaton with an input semihy-pergroup G. If ∅ 6= T ⊂ S and δ(t, g) ∈ T for any pair [t, g] ∈ T × G thenthe triad B = (T,G, δT ) (where δT = δ|(TxG)) is called a submultiautomatonof the multiautomaton A = (S,G, δ).

2. A submultiautomaton B = (T,G, δT ) of the multiautomaton A = (S,G, δ) issaid to be separated if δ(S r T,G) ∩ T = ∅. A nonempty multiautomatonis said to be connected (in the sense of [1]) if it has no separated propersubmultiautomaton.

28

References

[1] Bavel, Z., Grzymata-Busse, Soohong, K.:On the connectivity of the productof automata, Annales Societatis Mathematicae Polonae, Serie IV: Funda-menta Informaticae VII.2 (1984), 225–266.

[2] Antampoufis, N., Spartalis, S., Vougiouklis, Th.: Fundamental relationsin special extensions. In. 8th International Congress on AHA and Appl.,Samothraki, Greece (2003), 81–90.

[3] Cech, E.:Topological Spaces, revised by Z. Frolık and M. Katetov, Academia,Prague, 1966.

[4] Chajda, I.: Algebraic Theory of Tolerance Relations, The Palacky Univer-sity, Olomouc, 1991.

[5] Chvalina, J.: Functional Graphs, Quasi-ordered Sets and Commutative Hy-pergroups, MU Brno, Czech, 1995.

[6] Chvalina, J.: Commutative hypergroups in the sense of Marthy, in. Proc.Summer School , Hornı Lipova, Czech Republic (1994), 19–30.

[7] Chvalina, J., Chvalinova, L.: State hypergroups of automata, Acta Math.et Inf. Univ. Ostraviensis 4 (1996), 105–120.

[8] Chvalina, J., Chvalinova, L.: Modelling of join spaces by nth order linearordinary differential operators, in: Proc. Aplimat, Bratislava (2005), 279–285.

[9] Chvalina, J., Hoskova, S.: Join space of first-order linear partial differen-tial operators with compatible proximity induced by a congruence on their

29

group, in: Proc. Mathematical and Computer Modelling in Science andEngineering, Prague (2003), 166–170.

[10] Chvalina, J., Hoskova, S.: Transformation Tolerance Hypergroups, ThaiJournal of Mathematics, Vo. 4, Nr. 1., (2006), 63–72.

[11] Corsini, P.:Prolegomena of Hypergroup Theory, Aviani Editore, (Tricesimo,1993).

[12] Corsini, P., Leoreanu. V. :Applications of Hyperstructure Theory, KluwerAcademic Publishers, (Dordrecht, Hardbound, 2003).

[13] Czaszar, A.

[14] Davvaz, B., Leoreanu Fotea, V. :Hyperring theory and Applications.Hadronic Press, Palm Barbor, Fl. U.S.A, 2008.

[15] Davvaz, B.:Polygroup Theory and Related Systems. World Scientific, 2013.

[16] Heidari, D., Davvaz, B.:On ordered hyperstructures. U.P.B. Sci. Bull. SeriesA, Vol. 73, Iss. 2, 2011, 85- 96.

[17] Hoskova, S, Chvalina, J., Rackova, P.:

[18] Hoskova, S., Chvalina, J.: Discrete transformation hypergroups and trans-formation hypergroups with phase tolerance space. Discrete Math. 308, 4133–4143, 2008.

[19] Hoskova, S.: Discrete Transformation Hypergroups, in: Proc. Aplimat,Bratislava (2005), 275–279.

30

[20] Hoskova, S., Chvalina, J.: Multiautomata formaed by first order partialdifferential operators, Journal of Applied Mathematics, Vo.1, N. 1, 2008

[21] Hormander, L.: Analysis of Linear Partial Differential Operators I, II,Berlin, Springer I (1983).

[22] Jantosciak, J.: Transposition hypergroups: Noncommutative join spaces, J.Algebra 187(1997), 97–119.

[23] Massouros, G. G., Massouros, CH. G., Mittas I. D.: Fortified Join hyper-groups, Annales mathmatiques Blaise Pascal, 3 No. 2 (1996), 155–169.

[24] Novak, M.:Some basic properties of EL-hyperstructures. European J. ofCombinatorics 34, 446–459, 2013.

[25] Vougiouklis, T.: Hyperstructures and their Representations, Hadronic PressMonographs in Mathematics, Palm Harbor Florida 1994.

31