RATIO MATHEMATICA 23 (2012), 65–80 ISSN: 1592-7415

EL–hyperstructures: an overview

Michal Novák
Faculty of Electrical Engineering and Communication,

Brno University of Technology, Brno, Czech Republic

novakm@feec.vutbr.cz

Abstract

This paper gives a current overview of theoretical background of a
special class of hyperstructures constructed from quasi / partially or-
dered (semi) groups using a construction known as the ”Ends lemma”.
The paper is a collection of both older and new results presented at
AHA 2011.

Key words: group, hyperstructure, partial ordering, quasi order-
ing, semigroup.

MSC2010: 20N20.

1 Introduction

This paper is a written version of a lecture given at the 11th International
Conference on Algebraic Hyperstructures and Applications in October 2011.
Together with results new at that time I presented some older results pub-
lished in journals or proceedings of rather local impact thus unknown to the
international hyperstructure community. The new results presented at the
conference were meanwhile published as [20]. Therefore, this article instead
of presenting new unpublished results gives an overview of what has so far
been achieved in the area of EL–hyperstructures. For proofs of respective
theorems as well as for examples explaining their meaning cf either [20] or
works indicated throughout the paper. Where not stated otherwise all defi-
nitions of hyperstructure concepts or properties of hyperstructures are used
in the sense of the standard book [3].



Michal Novák

2 The ”Ends lemma”

The EL–hyperstructures are hyperstructures constructed from quasi /
partially ordered (semi)groups using the ”Ends lemma”, which has the form
of the following theorems from [4]. The notation [a)≤ used below stands
for the set {x ∈ H; a ≤ x}, where the properties of H are specified in the
respective theorems (typically (H, ·,≤) is a quasi-ordered / partially ordered
set / semigroup / group).

Lemma 2.1 ([4], Theorem 1.3, p. 146) Let (S, ·,≤) be a partially ordered
semigroup. Binary hyperoperation ∗ : S ×S →P′(S) defined by

a∗ b = [a · b)≤ (1)

is associative. The semi-hypergroup (S,∗) is commutative if and only if the
semigroup (S, ·) is commutative.

In accordance with other papers regarding this topic, the hyperstructure
(S,∗) constructed in this way will further on be called the associated hy-
perstructure to the single-valued structure (S, ·) or an ”Ends lemma”–based
hyperstructure, or an EL–hyperstructure for short. Instead of S the carrier
set will from Lemma 2.3 onward be denoted by H.

Lemma 2.2 ([4], Theorem 1.4, p. 147) Let (S, ·,≤) be a partially ordered
semigroup. The following conditions are equivalent:

10 For any pair (a,b) ∈ S2 there exists a pair (c,c′) ∈ S2 such that b · c ≤ a
and c′ · b ≤ a

20 The associated semi-hypergroup (S,∗) is a hypergroup.

Remark 2.1 If (S, ·,≤) is a partially ordered group, then if we take c =
b−1 ·a and c′ = a · b−1, then condition 10 is valid. Therefore, if (S, ·,≤) is a
partially ordered group, then its associated hyperstructure is a hypergroup.

Remark 2.2 The wording of the above lemmas is the exact translation of
theorems from [4]. The respective proofs, however, do not change in any
way, if we regard quasi-ordered structures instead of partially ordered ones
as the anti-symmetry of the relation ≤ is not needed (with the exception of
the ⇐ implication of the part on commutativity, which does not hold in this
case). The often quoted version of the ”Ends lemma” is therefore the version
assuming quasi–ordered structures.

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EL–hyperstructures: an overview

The ”Ends lemma” was later extended (cf e.g. [23]). Notice that if (H, ·)
is commutative, then (H,∗) is a join space. Also notice that unlike in the
original ”Ends lemma” the underlying single-valued structure in the following
theorem is a group (not a semigroup).

Lemma 2.3 ([23], Theorem 4) Let (H, ·,≤) be a quasi-ordered group and
(H,∗) be the associated hypergroupoid. Then (H,∗) is the transposition hy-
pergroup.

Initially, the typical use of the ”Ends lemma” was creating hyperstruc-
tures and proving or deriving their properties at random without any (or
with a very limited) theoretical background. This model is used in e.g.
[6, 8, 13, 21, 23]. In order to overcome this inconvenience, theoretical back-
ground of the ”Ends lemma” is being developed.

3 Extending the lemma, identities and in-

verses

After the ”Ends lemma” extension, i.e. Lemma 2.3, was proved, there
arose a question of whether one can go any further to stronger hyperstruc-
tures such as canonical hypergroups, strongly canonical hypergroups, etc. A
positive answer to this question would mean that numerous ring-like analo-
gies of EL–hyperstructures could be studied extensively. Unfortunately, the
answer – obtained in [18] – turned out to be negative.

Theorem 3.1 Let (H, ·,≤) be a non-trivial quasi-ordered group, where the
relation ≤ is not the identity relation, and let (H,∗) be its associated trans-
position hypergroup. Then:

1. (H,∗) does not have a scalar identity.

2. Regardless of commutativity, (H,∗) cannot be a canonical hypergoup.

Naturally, if we regard the definition of a canonical hypergroup, 2 imme-
diately follows from 1.

Once it was established that looking for scalar identities in EL–hyper-
structures based on groups is of no point, at least the issue of identities
was explored. In [18] the following simple results concerning identities were
obtained.

Theorem 3.2 Let (H,∗) be the semi-hypergroup associated to a quasi-ordered
semigroup (H, ·,≤) with the identity u.

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Michal Novák

1. An element e ∈ H is an identity of (H,∗) if and only if e ≤ u.

2. If (H, ·) is a group, then the identity of (H, ·) is an identity of (H,∗).

Again, 2 is naturally an immediate corollary of 1 yet we set it aside due
to uniqueness of the single-valued group identity.

Lemma 3.1 Let (H,∗) be the join space associated to a quasi-ordered com-
mutative group (H, ·,≤). If an element e ∈ H is an identity of (H,∗), then
e ≤ e−1.

Since the concept of an inverse in a hyperstructure is defined using the
concept of an identity, the issue of inverses was touched upon in the same
paper and the following result was obtained for the set i(a) of inverses of an
arbitrary element a ∈ H in (H,∗).

Theorem 3.3 Let (H,∗) be the transposition hypergroup associated to a
quasi-ordered group (H, ·,≤). Then for an arbitrary a ∈ H there holds

i(a) = {a′ ∈ H; a′ ≤ a−1} = (a−1]≤,

where a−1 is the inverse of a in (H, ·).

Corollary 3.1 Let (H,∗) be the transposition hypergroup associated to a
quasi-ordered group (H, ·,≤). Then (H,∗) is regular.

4 Ring-like hyperstructures

Since it turned out that once we start with groups the ”Ends lemma”
cannot effectively be used to construct canonical hypergroups, the scope for
use of this idea in the area of ring-like hyperstructures narrowed. Recall that
there is a great variety of definitions of ring-like hyperstructures, yet many
of them including the most often used one – that of Krasner hyperring – is
built on canonical hypergroups. However, the idea of limits of the ”Ends
lemma” in the area of hyperstructures with two (hyper)operations is still
worth exploring. In [19] (published before the author was able to get [12])
three possible extensions are suggested and explored:

1. Let (H, +) and (H, ·) be two single-valued structures. We can define a
hyperoperation using one of the operations + or · by e.g. a∗b = [a+b)≤
– thus we get an EL–hyperstructure (H,∗). The hyperstructure will
then be a triplet (H,∗, ·) where ∗ is a hyperoperation based on the
single-valued operation +.

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EL–hyperstructures: an overview

2. Let (H, +) and (H, ·) be two single-valued structures. We can define
two hyperoperations, each based on one single-valued operation, i.e.
for an arbitrary pair (a,b) ∈ H2 we can define a ∗ b = [a + b)≤ and
a ◦ b = [a · b)≤. Thus we get a triplet (H,∗,◦), where ∗ and ◦ are
hyperoperations.

3. However, we can also start with a single single-valued structure (H, ·)
and using it define a hyperoperation ∗ by a ∗ b = [a · b)≤. The hy-
perstructure will then be a triplet (H,∗, ·) where ∗ is a hyperoperation
based on the single-valued operation ·.

If we have a triplet (H, +, ·), where symbols + and · may stand for
both single-valued and multivalued operation, then for an arbitrary triplet
(a,b,c) ∈ H3 we may either ask that a ·(b + c) = a ·b + a ·c and (a + b) ·c =
a·c+b·c or we may ask that inclusions holds instead of equalities.1 Each of the
three ways to create ring-like hyperstructures was explored with respect to
both of these types of distributivity and the following results were obtained
in [19]. Notice the variety of conditions imposed on the respective struc-
tures (group / semigroup, quasi-ordered / partially ordered, single-valued /
multivalued).

Definition 4.1 [cf [28], p. 21, included as plain text] (R, +, .) is a hyperring
in the general sense if (R, +) is a hypergroup2, (·) is associative hyperoper-
ation and the distributive law3 x(y + z) ⊆ xy + xz, (x + y)z ⊆ xz + yz is
satisfied for every x,y,z of R. Additive hyperring is the one of which only
(+) is a hyperoperation, multiplicative hyperring is the one of which only (·)
is a hyperoperation. [. . .] (R, +, ·) will be called semihyperring if (+), (·) are
associative hyperoperations, where (·) is distributive with respect to (+). The
rest of definitions are analogous. If the equality in the distributive law is
valid, then the hyperring is called strong or good.

Theorem 4.1 Let (H, +, ·) be a ring such that (H, +) is a group, (H, ·) a
semigroup and ≤ quasi-ordering on H such that (H, +,≤) is a quasi-ordered
group and (H, ·,≤) is a quasi-ordered semigroup. Further, for an arbitrary
pair of elements (a,b) ∈ H2 define a∗b = [a + b)≤ and a◦b = [a ·b)≤. Then
(H,∗,◦) is a hyperring in the general sense.

1The former distributivity is sometimes called good distributivity while the latter is
often called weak distributivity.

2Vougiouklis uses the term hypergroup of Marty.
3Vougiouklis uses the sign ⊂ in the sense of ⊆.

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Michal Novák

Theorem 4.2 Let (H, +) be a semigroup, (H, ·) a group and ≤ quasi-ordering
on H such that (H, +,≤) is a quasi-ordered semigroup and (H, ·,≤) is a
quasi-ordered group. Further, for an arbitrary pair of elements (a,b) ∈ H2
define a ∗ b = [a + b)≤ and a ◦ b = [a · b)≤. Finally, let · distribute over +
from both left and right. Then (H,∗,◦) is a good semihyperring in the sense
of Definition 4.1.

Theorem 4.3 Let (H, +, ·) be a ring such that (H, +) is a group with neutral
element 0, (H\{0}, ·) a group and ≤ quasi-ordering on H such that (H, +,≤
) and (H, ·,≤) are quasi-ordered groups. Further, for an arbitrary pair of
elements (a,b) ∈ H2 define a∗b = [a+b)≤ and a◦b = [a·b)≤. Then (H,∗,◦)
is a good hyperring in the general sense.

Theorem 4.4 Let (H, +) be a group and (H, ·,≤) a quasi-ordered semigroup
and for an arbitrary pair of elements (a,b) ∈ H2 define a◦b = [a·b)≤. Further,
let (H, +, ·) be such that the operation · distributes over the operation + from
both left and right. Then (H, +,◦) is a good multiplicative hyperring.

Definition 4.2 [cf [14], Definition 2.1 and Remark] A hyperalgebra (R, +, ·)
is called a semihyperring if and only if

(i) (R, +) is a semihypergroup;

(ii) (R, ·) is a semigroup;

(iii) ∀a,b,c ∈ R, a · (b + c) = a · b + a · c and (b + c) ·a = b ·a + c ·a

If we replace (iii) by

∀a,b,c ∈ R,a · (b + c) ⊆ a · b + a · c and (b + c) ·a ⊆ b ·a + c ·a

we say that R is a weak distributive semihyperring. A semihyperring is called
with zero element, if there exists a unique element 0 ∈ R such that 0 + x =
x = x + 0 and 0 ·x = 0 = x · 0 for all x ∈ R. [. . .] A semihyperring is called
a hyperring provided (R, +) is a canonical hypergroup.

Theorem 4.5 Let (H, ·,≤) be a quasi-ordered semigroup such that · is a
commutative idempotent operation. Further, for an arbitrary pair of elements
(a,b) ∈ H2 define a ◦ b = [a · b)≤. Then (H,◦, ·) is a weak distributive
semihyperring.

Unfortunately, from Theorem 3.1 there follows that Krasner hyperrings
cannot be constructed using the ”Ends lemma” if the underlying single-
valued structure (H, +) is a group. However, there are weaker structures

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such as e.g. hyperringoids, which are defined as semihyperrings in the sense
of Definition 4.2 where (R, +) is a join space4, for the construction of which
the ”Ends lemma” might still be used.

The assumptions of the following theorem seem rather complicated. The
reason is simple: the requirement ”(H, +) is a group” results in trivialities.
Notice that condition 1 is the condition used in Lemma 2.1 – the one which
secures that (H,∗) is a hypergroup.

Theorem 4.6 Let (H, +) be a commutative semigroup, (H, ·) a group and
≤ quasi-ordering on H such that

1. to every pair of elements (a,b) ∈ H2 such that a ≤ b there exists a pair
of elements (c,c′) ∈ H2 such that b + c ≤ a, c′ + b ≤ a,

2. (H, +,≤) is a quasi-ordered semigroup and

3. (H, ·,≤) is a quasi-ordered group.

Moreover, for an arbitrary pair of elements (a,b) ∈ H2 define a∗b = [a+b)≤.
Finally, suppose that · distributes over + from both left and right. Then

if (H,∗) satisfies the transposition axiom, then (H,∗, ·) is a hyperringoid.

Corollary 4.1 If in Theorem 4.6 we suppose that (H, +,≤) is a quasi-
ordered semigroup without any further assumptions, then (H,∗, ·) is a semi-
hyperring in the sense of Definition 4.2.

Theorem 4.7 Let (H, +,≤) be a non-trivial quasi-ordered group with neu-
tral element 0 such that ≤ is not the identity relation, (H \{0}, ·) a group.
Moreover, for an arbitrary pair of elements (a,b) ∈ H2 define a∗b = [a+b)≤.
Finally, suppose that · distributes over + from both left and right. Then
(H,∗, ·) is a weak distributive hyperringoid.

5 The issue of a subhyperstructure

In order to proceed to the study of properties of EL–hyperstructures,
one must clarify the concept of a subhyperstructure of an EL–hyperstructure

4This definition is used in [3], Chapter 6. On contrary Massouros brothers in [17] call
this hyperstructure a join hyperringoid, while they call hyperringoid a hyperstructure such
that (R, +) is a hypergroup only. Further on in Theorem 4.6 the definition used in [3] is
regarded.

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Michal Novák

since there are two possible approaches to it. If we regard a quasi-ordered
semigroup (H, ·,≤) and define a hyperoperation ∗ on H by

a∗ b = [a · b)≤ = {x ∈ H; a · b ≤ x} (2)

for an arbitrary pair of elements (a,b) ∈ H2, we may in a subsemigroup (G, ·)
of the semigroup (H, ·) set either

a∗G b = [a · b)≤G = {x ∈ G; a · b ≤ x} (3)

or

a∗H b = [a · b)≤H = {x ∈ H; a · b ≤ x} (4)

and thus create either (G,∗G) or (G,∗H ). None of these concepts is obvi-
ously the only possible and ”correct” one since the definition of both can be
justified. In a way, the idea of ∗H conforms to the idea of the ”Ends lemma”
better. Thus in [22], which discussed the issue of subhyperstructures of EL–
hyperstructures, it was this concept that was favoured. The concept of the
upper set was introduced.5

Definition 5.1 Let (H, ·,≤) be a partially ordered semigroup and let G be a
nonempty subset of H.

1. If for an arbitrary element g ∈ G there holds [g)≤ ⊆ G, we call G an
upper end of H.

2. If there exists an element g ∈ G such that there exists an element
x ∈ H \ G such that g ≤ x (i.e. x ∈ [g)≤)6, we say that G is not an
upper end of H because of the element x.

Among other results concerning hyperoperation ∗H (4) the following was
proved.

Theorem 5.1 Let (H,∗) be the semihypergroup associated to a quasi-ordered
semigroup (H, ·,≤). Suppose that G is an upper end of H. If (G, ·) is a
subgroup of (H, ·), then (G,∗) is a subhypergroup of (H,∗).

5The concept itself is naturally not a new invention; the definition was only tailored
for use in the ”Ends lemma” context.

6We could – probably more properly since x 6∈ G – write g < x and x ∈ [g)≤ \{g} yet
in the definition we keep the ≤ notation of the Ends lemma for consistency reasons.

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EL–hyperstructures: an overview

Proposition 5.1 Let (H,∗) be the semihypergroup associated to a partially
ordered semigroup (H, ·,≤) and G ⊆ H nonempty. If (G,∗) is a subhyper-
group of (H,∗), then (G, ·) is a subsemigroup of (H, ·) and G is an upper end
of H such that for any pair (a,b) ∈ G2 there exists a pair (c,c′) ∈ G2 such
that b · c ≤ a and c′ · b ≤ a.

Theorem 5.2 Let (H,∗) be the semihypergroup associated to a partially or-
dered semigroup (H, ·,≤). Further, let G ⊆ H be non-empty and such that
(G, ·) is a subgroupoid of (H, ·), and let the relation ≤G be a restriction of ≤
on G, i.e. ≤G=≤ ∩(G × G).7 Finally – if it exists – denote u the identity
of (H, ·) and define a new hyperoperation ∗G : G×G → P∗(G) for arbitrary
elements a,b ∈ G by (3), i.e. by

a∗G b = [a · b)≤G = {x ∈ G; a · b ≤G x}.

Then

1. (G, ·) is a semigroup if and only if (G,∗G) is a semihypergroup.

2. (G, ·) is a monoid if and only if (G,∗G) is a semihypergroup and u ∈ G.

3. If (G, ·) is a group, then (G,∗) is a transposition hypergroup.

4. If (G,∗) is a hypergroup, then (G, ·) is a semigroup such that for any
pair (a,b) ∈ G2 there exists a pair (c,c′) ∈ G2 such that b · c ≤ a and
c′ · b ≤ a.

6 Properties of EL–hyperstructures

and their subhyperstructures

Results in this section were presented at AHA 2011 and later included
in [20].

Theorem 6.1 Let (H,∗) be the hypergroup associated to a quasi-ordered
group (H, ·,≤) and (G, ·) its subgroup such that G is an upper end of H. Then
(G,∗), where ∗ is defined for an arbitrary pair (a,b) ∈ H2 as a∗b = [a·b)≤H ,
is invertible and closed in H.

7The exact quote from [22] at this place reads ”for arbitrary elements a, b ∈ G let
a ≤ b ⇒ a ≤G b”.

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Michal Novák

Theorem 6.2 Let (H,∗) be the hypergroup associated to a quasi-ordered
group (H, ·,≤) and (G,∗) its arbitrary subhypergroup associated to a subgroup
(G, ·) of (H, ·), where G is an upper end of H (i.e. as defined in Theorem 5.1
using hyperoperation ∗H ). Denote u the identity of (H, ·). Then

1. G is ultraclosed if and only if for any h ∈ H such that h ≤ u it follows
that h ∈ G.

2. If G 6= H and if (H, ·,≤) has the smallest element, then (G,∗) is not
ultraclosed.

3. If (H, ·) or (H,∗) is commutative, then G is a complete part of H if
and only if for every h ∈ H such that h ≤ u there is h ∈ G.

Theorem 6.3 Let (H,∗) be the associated hypergroup of a quasi-ordered
group (H, ·,≤) and (G,∗) its arbitrary subsemihypergroup associated to a
subsemigroup (G, ·) of (H, ·).8 If for arbitrary x ∈ H and g ∈ G there holds
x ·g ·x−1 ∈ G, then (G,∗) is normal.

Corollary 6.1 Let (H,∗) be the hypergroup associated to a quasi-ordered
group (H, ·,≤) and (G, ·) its normal subgroup such that G is an upper end of
H. Then (G,∗), where ∗ is defined as a∗ b = [a · b)≤H , is reflexive.

Theorem 6.4 Let (H,∗) be the hypergroup associated to a quasi-ordered
group (H, ·,≤). Then (H,∗) is reversible.

As far as regularity of EL–hyperstructures is concerned, cf Theorem 3.3
and its corollary. In the following theorem notice that by a subhypergroup
we mean a subhyperstructure defined by hyperoperation ∗H (4). This is
important to consider since the definition of inner irreducibility relies on
subhyperstructures as a commutative hypergroup (H,∗) is called inner irre-
ducible if for any pair of its subhypergroups G1,G2 such that G1 ∗G2 = H
there holds G1 ∩G2 6= ∅.

Theorem 6.5 Let (H,∗) be the associated hypergroup of a partially ordered
commutative group (H, ·,≤).

1. If for every x ∈ H such that x,x−1 are incomparable with respect to ≤
there is either [x)≤ ∩ [x−1)≤ 6= ∅ or (x]≤ ∩ (x−1]≤ 6= ∅, then (H,∗) is
inner irreducible.

8In this context the fact whether we define a subhyperstructure by means of ∗H (4) or
∗G (3) is not important.

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EL–hyperstructures: an overview

2. If (H,≤) is a linear ordered set or if (H,≤) has the smallest or the
greatest element, then (H,∗) is inner irreducible.

Naturally, 2 is a corollary of 1 since in linear ordered sets all elements are
comparable.

7 The issue of origins of a hypergroup

The ”Ends lemma” describes a way to construct semihypergroups from
quasi-ordered semigroups and hypergroups from semigroups with a special
property. We know that groups are such structures that this property holds
trivially. This means that we know that using the ”Ends lemma” we may
create a hypergroup from a group.

Thus one can ask: if have a hypergroup (itself or as a subhypergroup
of a larger structure) created in the ”Ends lemma” fashion, is there a way
to determine whether its underlying single-valued structure is a group or a
semigroup? Answering this question is not academic only as proofs of some
of the above theorems have to answer this question in a rather complicated
way. This issue is also connected to the issue of the converse of the ”Ends
lemma”, which was already necessary to complete some proofs of theorems
on subhyperstructures. The proof of the following theorem can be found
in [22].

Theorem 7.1 Let (H, ·) be a non-trivial groupoid and ≤ a binary partial
ordering on H such that for an arbitrary pair of elements (a,b) ∈ H2, a ≤ b,
and for arbitrary c ∈ H there holds c·a ≤ c·b and a·c ≤ b·c. Further define a
hyperoperation ∗ : H×H →P∗(H) for an arbitrary pair of elements (a,b) ∈
H2 by a∗ b = [a · b)≤ = {x ∈ H; a · b ≤ x}. Then if the hyperoperation ∗ is
associative, then the single-valued operation · is associative too. Furthermore,
if there exists an element e ∈ H such that for every a ∈ H there holds
a ∗ e = e ∗ a = [a)≤, then this element e is the identity of the semigroup
(H, ·).

Notice that if the relation ≤ is not antisymmetric, the above theorem is
not true. This is caused by the fact that only for antisymmetric relations ≤
there holds that [a)≤ = [b)≤ implies that a = b. Indeed, suppose a simple
two element set M = {a,b} where the relation ≤ is defined as a ≤ a,a ≤
b,b ≤ a,b ≤ b. This reflexive and transitive relation ≤ is obviously not
antisymmetric and there holds [a)≤ = [b)≤ yet a 6= b.

A simple way of distinguishing between semigroups and groups is the
study of idempotent elements, i.e. elements which (if we ignore the identity)

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Michal Novák

exist in semigroups only. In [20] a few basic results concerning idempotent
elements are included.

Theorem 7.2 Let (H,∗) be the semihypergroup associated to a quasi–ordered
semigroup (H, ·,≤). For an arbitrary element a ∈ H there holds

a∗a = {a}⇔ a is an idempotent and simultaneously a maximal element of
(H, ·,≤).

Corollary 7.1 Let (H,∗) be the associated hypergroup of such a quasi-ordered
semigroup (H, ·,≤) that at least two distinct elements a,b ∈ H are in rela-
tion ≤ (i.e. ≤ is not trivial). If there exists an element a ∈ H such that
a∗a = {a}, then (H, ·) is not a group.

Corollary 7.2 Let (H,∗) be the hypergroup associated to a quasi-ordered
semigroup (H, ·,≤) and (G,∗) a subhypergroup of (H,∗).9

1. Denote u the identity of (H, ·). If u is the maximal element of (G,≤)
and at least two distinct elements a,b ∈ G are in relation ≤, then (G, ·)
is a subsemigroup of (H, ·) yet not a subgroup of (H, ·).

2. If for two distinct elements a,b ∈ H there holds a∗a = {a}, b∗b = {b},
then H does not have the greatest element. Also – obviously – (H, ·) is
not a group.

8 ”Ends lemma” in a broader context

Naturally, the ”Ends lemma” is not a revolutionary stand alone con-
cept. The study of relation of hyperstructures and ordered sets or binary
relations is included in [3] as chapter 3. This part of the ”canonical” book
on hyperstructures was inspired by works of Chvalina (especially [5]) and
Rosenberg (especially [24]). Results concerning the relation of hyperstruc-
tures and ordered sets have also been included in [12], another ”canonical”
book on hyperstructure theory. Among older works concerning relation of
ordered sets and hyperstructures there is e.g. [2], which studies the relation
in general giving a number of possible ways to create hyperstructures from
ordered sets, and [29], in which a concept in a way similar to the ”Ends
lemma” may be found. Recent works related to the concept discussed in this
article include e.g. works of Cristea and Ştefănescu which deal with n–ary
relations on hypergroups (e.g. [9, 10]) or study of fundamental relations on
hypergroupoids associated with binary relations (such as [11]), or works of
e.g. Spartalis or Massouros (such as [16, 25, 26]).

9The statement is valid for subhypergroups based on both ∗H (4) or ∗G(3).

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EL–hyperstructures: an overview

9 Open issues

There are many loose ends that wait to be tied. Most importantly, the
full potential of the property included in Lemma 2.1 must be explored. This
is closely connected to the problem of reversing the ”Ends lemma”, which
has been partly answered by Theorem 7.1, and to the problem of telling the
origins of the hypergroup for which the idea of idempotent elements is only
a first and insufficient attempt. Clarifying this issue would also help in the
study of ring-like EL–hyperstructures and in the study of such properties of
EL–hyperstructures that rely on the concept of a subhypergroup.

Naturally, it might be very useful to set the issue of EL–hyperstructures in
a broader perspective of hyperstructures constructed from binary operations
of single-valued (semi)groups.

References

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