/compile/output.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 2, 2019, 252-269
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Correction to the article: “An introduction to the

theory of hyperlattices”

Niovi Kehayopulu

Abstract. The definition of hyperlattices introduced in Mathematica Balkanica, 1977 by Kon-
stantinidou and Mittas in their paper “An introduction to the theory of hyperlattices” should
be corrected. As a result, the definition of distributive and modular hyperlattices introduced by
Konstantinidou should be also corrected. In the present paper we correct these definitions and
give some examples to show that these corrected forms work.

2010 Mathematics Subject Classifications: 06B75

Key Words and Phrases: hyperlattice, distributive, modular

1. Introduction

According to the bibliography, the concept of hyperlattices has been introduced by
Konstantinidou and Mittas in Math. Balkanica [4]. The aim of this note is to correct the
definition of hyperlattice introduced in [4], and the definitions of distributive and modular
hyperlattices based on it, introduced in [5, 6], and give some examples to show that these
corrected forms work.

For the sake of completeness, we first mention the definition of lattices just to see, step
by step, how we could generalize it to a hyperlattice.

A lattice is a nonempty set L with two binary operations “∧” and “∨” on L such that
(1) a ∧ a = a and a ∨ a = a
(2) a ∧ b = b ∧ a and a ∨ b = b ∨ a
(3) (a ∧ b) ∧ c = a ∧ (b ∧ c) and (a ∨ b) ∨ c = a ∨ (b ∨ c)
(4) a ∧ (a ∨ b) = a and a ∨ (a ∧ b) = a.

According to [4–7], the definition of a hyperlattice is given as follows:

Definition 1.1 [4] A hyperlattice is a nonempty set L with an hyperoperation “∨” (that
is a mapping that assigns to each couple a, b of elements of L a nonempty subset of L)
and an operation “∧” on L such that

(1) a ∈ a ∨ a a ∧ a = a

DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3415

Email address: nkehayop@math.uoa.gr

http://www.ejpam.com 252 c© 2019 EJPAM All rights reserved.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 253

(2) a ∨ b = b ∨ a a ∧ b = b ∧ a
(3) (a ∨ b) ∨ c = a ∨ (b ∨ c) (a ∧ b) ∧ c = a ∧ (b ∧ c)

(4) a ∈
[

a ∨ (a ∧ b)
]

∩
[

a ∧ (a ∨ b)
]

(5) a ∈ a ∨ b ⇒ a ∧ b = b.

In [5], immediately after this definition, the following is written: “The set H = {0, 1}
with a hyperoperation 0 ∨ 0 = 0, 0 ∨ 1 = 1 ∨ 0 = 1, 1 ∨ 1 = H and an operation
0 ∧ 0 = 1 ∧ 0 = 0 ∧ 1 = 0, 1 ∧ 1 = 1 is a hyperlattice. Respectively, for 0 ∨ 0 = H,
0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1 and 0 ∧ 0 = 1 ∧ 0 = 0 ∧ 1 = 0, 1 ∧ 1 = 1.”

There are two examples here. As they are similar, we deal with the first one. We write
a instead of 0 and b instead of 1. We also correct it somehow by writing a ∨ a = {a},
a ∨ b = b ∨ a = {b} as cannot be a ∨ a = a, a ∨ b = b, b ∨ a = b. There is no mention in
the paper of identifying the {a} by a. Thus this example in [5] is the following example.

Example 1.2 The set H = {a, b} with the operation and the hyperoperation defined by
Table 1 is a hyperlattice.

Table 1: The operation and the hyperoperation of the Example 1.2.

∧ a b

a a a

b a b

(a)

∨ a b

a {a} {b}

b {b} {a, b}

(b)

To show that this is a hyperlattice, we must show that a ∈ a∧ (a∨ b) and (a∨ b)∨ c =
a ∨ (b ∨ c) for all a, b, c ∈ H. According to Table 1, we have a ∧ (a ∨ b) = a ∧ {b}, but the
a∧{b} has no sense. The associativity of the hyperoperation “∨” has no sense as well. So
the Table 1 cannot define a hyperlattice.

According [4; Remark 1(a)], the Definition 1.1 implies that each lattice is a hyperlattice.
This, stated without proof in [4], cannot be proved by the Definition 1.1 as this definition
has no sense. As we see in Proposition 2.5 of the present paper, this is true.

The modular and distributive hyperlattices have been defined in [5] and [6], respec-
tively, as follows: “A hyperlattice L is said to be modular if a ≤ b ⇒ a∨(c∧b) = (a∨c)∧b
for any c ∈ H (as in the case of lattices).

A hyperlattice L is said to be distributive if a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) for every
a, b, c ∈ L.”

This is the Remark 1 in [5]: “Obviously, in every modular hyperlattice we have a ∨
(b ∧ a) = (a ∨ b) ∧ a.” This being obvious for lattices, cannot be proved by the definition
of modular hyperlattices given in [5]. As we will see later in section 4, in a modular



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 254

hyperlattice we have {v∧a | v ∈ a∨ b} = a∨ (b∧a); and if for the set {v∧a | v ∈ a∨ b} we
use the notation (a ∨ b) ∧ a, then we can say that in a modular hyperlattice the property
a ∨ (b ∧ a) = (a ∨ b) ∧ a holds.

After this remark, there is the Example 2(a) in [5]. It is no clear what “the obvious
operation” in this example means. In the Example 2(b), the b ∧ b and c ∧ c are missing.

Later Konstantinidou and Serafimidis changed the property (5) of Definition 1.1 and
wrote “a ∈ a∨ b ⇔ a∧ b = b” instead of “a ∈ a∨ b ⇒ a∧ b = b” [7] (the “⇐-part” being a
consequence of Definition 1.1, should be omitted from the definition). They also defined
the concepts of ∧-distributive and ∨-distributive hyperlattices as the hyperlattices in which
the properties (a∨b)∧c ⊆ (a∧c)∨(b∧c) and (a∧b)∨c ⊆ (a∨c)∧(b∨c), respectively hold.
Some authors consider only the properties (1)–(4) as the definition of the hyperlattice and
study hyperlattices having the property (5) as an additional property. Trying to transfer
the definition of lattices to hyperlattices it is much better not to include condition (5) in
the definition of hyperlattices. In the present paper we will do so.

In the definition of hyperlattices the part related to “∧” is the same as in the theory of
lattices, so it has the same properties as in lattices. Concerning the “∨”: We cannot write
a ∈ a∧ (a∨ b) as a is an element, a∨ b is a set and “∧” is an operation between elements.
Also we cannot write a∨(b∨c) = (a∨b)∨c as b∨c, a∨b are sets, c, a elements and “∨” and
operation between elements (the so called “hyperoperation”). That is expressions of the
form a ∈ a∧(a∨b), a∨(b∨c) and (a∨b)∨c have no sense. For the same reason the concepts
of distributive and modular hyperlattices [5, 6] and the concepts of ∧-distributive and ∨-
distributive hyperlattices considered in [7] have no sense as well. What we have already
said is about the definition of hyperlattice introduced by Konstantinidou and Mittas in
[4] and used in [5–7] as well. Later some authors working on the subject, just after the
Definition 1.1, they added: “Let A,B ⊆ L. Then define A ∨B =

⋃

{a ∨ b | a ∈ A, b ∈ B}
and A∧B = {a∧ b | a ∈ A, b ∈ B}” (see, for example [1, 2]). But this, written in a wrong
place (and not only), make the definition still unreadable.

2. Hyperlattices

To pass from lattices to hyperlattices, we only have to transfer the properties (a∨b)∨c =
a∨ (b∨ c), a∧ (a∨ b) = a and a∨ (a∧ b) = a. The property (a∨ b)∨ c = a∨ (b∨ c) can be
naturally transferred as follows: x ∈ u ∨ c for some x ∈ a ∨ b if and only if x ∈ a ∨ v for
some v ∈ b ∨ c. The a ∧ (a ∨ b) = a can be transferred as follows: there exists u ∈ a ∨ b
such that a∧ u = a; or for every u ∈ a∨ b, a∧ u = a. Finally, the property a∨ (a∧ b) = a
could be transferred as a ∈ a∨ (a∧ b); if x ∈ a∨ (a∧ b), then x = a or both. As we see, the
concept of a lattice can be extended not only in one way. To keep the existing definition
in the bibliography, the concept of a lattice can be naturally transferred to a hyperlattice
by the definition below. We denote by P∗(L) the set of (all) nonempty subsets of L.

Definition 2.1 Let L be a nonempty set,
∧ : L× L → L | (a, b) → a ∧ b an operation on L and
∨ : L× L → P∗(L) | (a, b) → a ∨ b a hyperoperation on L.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 255

We say that (L,∧,∨) is a hyperlattice if, for every a, b, c ∈ L, the following assertions are
satisfied:

(1) a ∧ a = a and a ∈ a ∨ a

(2) a ∧ b = b ∧ a and a ∨ b = b ∨ a

(3) (a ∧ b) ∧ c = (a ∧ b) ∧ c; and (a ∨ b) ∨ c = a ∨ (b ∨ c) in the sense that x ∈ u ∨ c for
some u ∈ a ∨ b if and only if x ∈ a ∨ v for some v ∈ b ∨ c.

(4) a ∧ (a ∨ b) = a in the sense that there exists u ∈ a ∨ b such that a ∧ u = a; and
a ∈ a ∨ (a ∧ b).

Remark 2.2 (A) The property (a ∨ b) ∨ c = a ∨ (b ∨ c) is clearly equivalent to
⋃

u∈a∨b

u ∨ c =
⋃

v∈b∨c

a ∨ v.

(B) There exists u ∈ a ∨ b such that a ∧ u = a if and only if a ∈ {a ∧ u | u ∈ a ∨ b}.

If in Definition 2.1 we add the property a ∈ a ∨ b ⇒ a ∧ b = b, then this definition is
equivalent to Definition 1.1 but only if, for any nonempty subsets A and B of L, we define
the A∨B and A∧B (there is no such a definition in [4–7]), preferable before the definition
or in a correct way if it is after that (I mean, not as in [1–2]); and emphasize the fact that
the element a should be identified by the singleton {a} if and when is convenient and no
confusion is possible.

Remark 2.3 (see, for example [1]) In a hyperlattice, a∧ b = b implies a ∈ a∨ b. [Indeed,
by Definition 2.1(4), we have a ∈ a∨ (a∧ b) = a∨ b]. As we see later, the converse of this
statement does not hold in general.

Example 2.4 The set L = {a, b, c} with the operation “∧” and the hyperoperation “∨”
given by Table 2 is a hyperlattice.

Table 2: The hyperlattice of the Example 2.4.

∧ a b c

a a a a

b a b b

c a b c

(a)

∨ a b c

a {a, b, c} {b, c} {c}

b {b, c} {b} {c}

c {c} {c} {c}

(b)



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 256

Proposition 2.5 Every lattice (L,∧,∨) is a hyperlattice.

Proof We consider the operation ∧ : L× L → L | (a, b) → a ∧ b and the hyperoperation
“
.
∨” on L defined by

.
∨ : L× L → P∗(L) | (a, b) → a

.
∨ b := {a ∨ b}.

Then (L,∧,
.
∨) is a hyperlattice. Indeed: The operation

.
∨ is well defined, the properties

(1) and (2) of Definition 2.1 are satisfied, the operation “
.
∨” is associative, indeed

⋃

u∈a
.

∨ b

u
.
∨ c =

⋃

u=a∨b

u
.
∨ c = (a ∨ b)

.
∨ c = {(a ∨ b) ∨ c},

⋃

v∈b
.

∨ c

a
.
∨ v =

⋃

v=b∨c

a
.
∨ v = a

.
∨(b ∨ c) = {a ∨ (b ∨ c)} and

since (a ∨ b) ∨ c = a ∨ (b ∨ c), we have
⋃

u∈a
.

∨ b

u
.
∨ c =

⋃

v∈b
.

∨ c

a
.
∨ v.

If a, b ∈ L then, for the element u := a ∨ b ∈ a
.
∨ b, we have a ∧ u = a ∧ (a ∨ b) = a; and

a ∈ {a} = {a ∨ (a ∧ b)} = a
.
∨(a ∧ b).

Second proof We consider the operation ∧ : L× L → L | (a, b) → a ∧ b
and the hyperoperation “

.
∨” on L defined by

.
∨ : L× L → P∗(L) | (a, b) → a

.
∨ b := {a, b, a ∨ b}.

Then (L,∧,
.
∨) is a hyperlattice. Indeed: The hyperoperation “

.
∨” is well defined and the

following assertions are satisfied:

(1) a ∈ a
.
∨ a, since a

.
∨ a := {a, a, a ∨ a} = {a}.

(2) It is clear.
(3) We have

⋃

u∈a
.

∨ b

u
.
∨ c =

⋃

u∈{a,b,a∨b}

u
.
∨ c = (a

.
∨ c) ∨ (b

.
∨ c) ∨

(

(a ∨ b)
.
∨ c

)

= {a, c, a ∨ c, b, c, b ∨ c, a ∨ b, c, (a ∨ b) ∨ c}.

⋃

v∈b
.

∨ c

a
.
∨ v =

⋃

v∈{b,c,b∨c}

a
.
∨ v = (a

.
∨ b) ∨ (a

.
∨ c) ∨

(

a
.
∨(b ∨ c)

)

= {a, b, a ∨ b, a, c, a ∨ c, a, b ∨ c, a ∨ (b ∨ c)}.

Since L is a lattice, we have (a ∨ b) ∨ c = a ∨ (b ∨ c) and so
⋃

u∈a
.

∨ b

u
.
∨ c =

⋃

v∈b
.

∨ c

a
.
∨ v.

(4) There exists u ∈ a
.
∨ b such that a∧u = a. Indeed, for the element u := a∨b ∈ a

.
∨ b,

we have a∧u = a∧ (a∨ b) = a; and a ∈ a
.
∨(a∧ b) since a

.
∨(a∧ b) = {a, a∧ b, a∨ (a∧ b)} =

{a, a ∧ b}. �

We apply Proposition 2.5 to the following example.

Example 2.6 We consider the lattice L defined by Figure 1.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 257

a

c

d

b

Figure 1: The lattice of the Example 2.6.

According to the first proof of Proposition 2.5, the set L with the operation and the
hyperoperation of Table 3 is a hyperlattice.

Table 3: The hyperlattice of the Example 2.6
that corresponds to the first proof of Proposition 2.5.

∧ a b c d

a a a a a

b a b a b

c a a c c

d a b c d

(a)

∨ a b c d

a {a} {a} {a} {a}

b {b} {b} {d} {d}

c {c} {d} {c} {d}

d {d} {d} {d} {d}

(b)

According to the second proof of Proposition 2.5, the same lattice with the operation and
the hyperoperation of Table 4 is a hyperlattice. In addition, Table 4 provides us with an
example of a hyperlattice for which the converse statement in Remark 2.3 does not hold.
Indeed, c ∈ c ∨ b but c ∧ b 6= b.

Table 4: The hyperlattice of the Example 2.6
that corresponds to the second proof of Proposition 2.5.

∧ a b c d

a a a a a

b a b a b

c a a c c

d a b c d



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 258

(a)

∨ a b c d

a {a} {a, b} {a, c} {a, d}

b {a, b} {b} {b, c, d} {b, d}

c {a, c} {b, c, d} {c} {c, d}

d {a, d} {b, d} {c, d} {d}

(b)

Proposition 2.7 In a hyperlattice the following are equivalent:

(1) For every u ∈ a ∨ b we have a ∧ u = a.

(2) {a ∧ x | x ∈ a ∨ b} = {a}.

Proof (1) =⇒ (2). If t ∈ {a ∧ x | x ∈ a ∨ b}, then t = a ∧ x for some x ∈ a ∨ b. Since
x ∈ a∨ b, by (1), we have a∧x = a, thus we get t = a. On the other hand, since a ∈ a∨ a
and a = a ∧ a, we have a ∈ {a ∧ x | x ∈ a ∨ b}.
(2) =⇒ (1). If u ∈ a ∨ b, then a ∧ u ∈ {a ∧ x | x ∈ a ∨ b} = {a} and so a ∧ u = a. �

3. Distributive hyperlattices

A lattice L is called distributive if, for any a, b, c ∈ L, we have a∧(b∨c) = (a∧b)∨(a∧c).
This concept can be naturally transferred to hyperlattices as follows:

Definition 3.1(1) A hyperlattice (L,∧,∨) is called distributive if the following assertions
are satisfied:

(1) if u ∈ b ∨ c, then a ∧ u ∈ (a ∧ b) ∨ (a ∧ c) and

(2) if u ∈ (a ∧ b) ∨ (a ∧ c), then there exists v ∈ b ∨ c such that u = a ∧ v.

In other words, u ∈ (a ∧ b) ∨ (a ∧ c) if and only if there exists v ∈ b ∨ c such that
u = a ∧ v. That is, if {a ∧ v | v ∈ b ∨ c} = (a ∧ b) ∨ (a ∧ c) for every a, b, c ∈ L.

The equivalent definition a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) of distributive lattices can be
transferred to hyperlattices in Definition 3.1(2). According to the problem we face we use
either the first or the second definition.

Definition 3.1(2) A hyperlattice (L,∧,∨) is called distributive if the following assertions
are satisfied:

(1) if u ∈ a ∨ (b ∧ c), then there exist v ∈ a ∨ b and w ∈ a ∨ c such that u = v ∧ w and

(2) if u ∈ a ∨ b and v ∈ a ∨ c, then u ∧ v ∈ a ∨ (b ∧ c).



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 259

In other words, u ∈ a ∨ (b ∧ c) if and only if there exist v ∈ a ∨ b and w ∈ a ∨ c such
that u = v ∧ w. That is, if {v ∧ w | v ∈ a ∨ b, w ∈ a ∨ c} = a ∨ (b ∧ c) for all a, b, c ∈ L.

Example 3.2 (see also [1; Lemma 3.20]) The hyperlattice of Table 5 is not distributive
in the sense of Definition 3.1(2). This is because for the element c ∈ b ∨ (a ∧ b), there
are no v ∈ b ∨ a and w ∈ b ∨ b such that c = v ∧ w. It is not distributive in the sense
of Definition 3.1(1) as well, since c ∈ (b ∧ a) ∨ (b ∧ c) but there is no v ∈ a ∨ c such that
c = b ∧ v.

Table 5: The hyperlattice of the Example 3.2.

∧ a b c

a a a a

b a b b

c a b c

(a)

∨ a b c

a {a} {b, c} {c}

b {b, c} {b} {c}

c {c} {c} {c}

(b)

Example 3.3 (see also [3; Example 1.8]) The hyperlattice defined by Table 6 is a dis-
tributive hyperlattice in the sense of Definition 3.1(1) since for all a, b, c ∈ L, we have
{a ∧ v | v ∈ b ∨ c} = (a ∧ b) ∨ (a ∧ c); but it is not distributive in the sense of Definition
3.1(2) as c ∈ d ∨ b, d ∈ d ∨ c but c ∧ d /∈ d ∨ (b ∧ c).

Table 6: The hyperlattice of the Example 3.3.

∧ a b c d

a a a a a

b a b a b

c a a c c

d a b c d

(a)

∨ a b c d

a {a} {b} {c} {d}

b {b} {a, b} {d} {c, d}

c {c} {d} {a, c} {b, d}

d {d} {c, d} {b, d} {a, b, c, d}

(b)



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 260

Proposition 3.4 If (L,∧,∨) is a distributive lattice, then the hyperlattice (L,∧,
.
∨), where

.
∨ : (a, b) → a

.
∨ b ∈ {a ∨ b} considered in the first part of Proposition 2.5 is a distributive

hyperlattice in the sense of Definition 3.1(1) and in the sense of Definition 3.1(2).

Proof The hyperlattice (L,∧,
.
∨) is distributive in the sense of Definition 3.1(1). In fact:

if u ∈ b
.
∨ c, then u = b ∨ c and a ∧ u = a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) ∈ (a ∧ b)

.
∨(a ∧ c).

If u ∈ (a∧ b)
.
∨(a∧ c), then u = (a∧ b)∨ (a∧ c) and, for the element v := b∨ c ∈ b

.
∨ c,

we have u = a ∧ v. It is distributive in the sense of Definition 3.1(2) as well. Indeed:
if u ∈ a

.
∨(b ∧ c), then u = a ∨ (b ∧ c) and, for the elements v := a ∨ b ∈ a

.
∨ b and

w := a ∨ c ∈ a
.
∨ c, we have u = a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) = v ∧ w. If u ∈ a

.
∨ b and

v ∈ a
.
∨ c, then u = a ∨ b, v = a ∨ c and then u ∧ v = (a ∨ b) ∧ (a ∨ c) = a ∨ (b ∧ c). �

Proposition 3.5 If (L,∧,∨) is a distributive lattice, then the hyperlattice (L,∧,
.
∨),

where
.
∨ : (a, b) → a

.
∨ b ∈ {a, b, a ∨ b} considered in the second part of Proposition 2.5

is a distributive hyperlattice in the sense of Definition 3.1(1). It is not a distributive
hyperlattice in the sense of Definition 3.1(2) in general, but it satisfies the property (1) of
Definition 3.1(2).

Proof Let u ∈ b
.
∨ c. Then a ∧ u ∈ (a ∧ b)

.
∨(a ∧ c). Indeed: if u = b, then a ∧ u =

a ∧ b ∈ (a ∧ b)
.
∨(a ∧ c); if u = c, then a ∧ u = a ∧ c ∈ (a ∧ b)

.
∨(a ∧ c); if u = b ∨ c, then

a ∧ u = a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) ∈ (a ∧ b)
.
∨(a ∧ c).

If u ∈ (a ∧ b)
.
∨(a ∧ c), then there exists v ∈ b

.
∨ c such that u = a ∧ v. Indeed: if

u = a ∧ b then, for the element v := b ∈ b
.
∨ c, we have u = a ∧ v; if u = a ∧ c then, for the

element v := c ∈ b
.
∨ c, we have u = a ∧ v; if u = (a ∧ b) ∨ (a ∧ c), then for the element

v := b ∨ c ∈ b
.
∨ c, we have u = (a ∧ b) ∨ (a ∧ c) = a ∧ (b ∨ c) = a ∧ v.

The hyperlattice L satisfies the property (1) of Definition 3.1(2); that is, if u ∈ a
.
∨(b∧c),

then there exist v ∈ a
.
∨ b and w ∈ a

.
∨ c such that u = v ∧ w. Indeed: if u = a then, for

the elements v := a ∈ a
.
∨ b and w := a ∈ a

.
∨ c, we have u = v ∧ w; if u = b ∧ c then,

for the elements v := b ∈ a
.
∨ b and w := c ∈ a

.
∨ c, we have u = v ∧ w; if u = a ∨ (b ∧ c)

then, for the elements v := a ∨ b ∈ a
.
∨ b and w := a ∨ c ∈ a

.
∨ c, we have u = a ∨ (b ∧ c) =

(a ∨ b) ∧ (a ∨ c) = v ∧ w.

We prove the rest of the proposition by the following example.

Let us consider the distributive lattice of Figure 2.

The hyperlattice L with the operation ∧ and the hyperoperation
.
∨ defined in the second

proof of Proposition 2.5 is given by Table 7 and, according to what we already said, it is
a distributive hyperlattice in the sense of Definition 3.1(1) and satisfies condition (1) of
Definition 3.1(2). But it is not distributive in the sense of Definition 3.1(2) as c ∈ c

.
∨ a,

b ∈ c
.
∨ b but c ∧ b /∈ c

.
∨(a ∧ b).



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 261

a

b

c d

e

Figure 2: The distributive hyperlattice of Proposition 3.5.

Table 7: The hyperlattice in Proposition 3.5.

∧ a b c d e

a a a a a a

b a b b b b

c a b c b c

d a b b d d

e a b c d e

(a)

.
∨ a b c d e

a {a} {a, b} {a, c} {a, d} {a, e}

b {a, b} {b} {b, c} {b, d} {b, e}

c {a, c} {b, c} {c} {c, d, e} {c, e}

d {a, d} {b, d} {c, d, e} {d} {d, e}

e {a, e} {b, e} {c, e} {d, e} {e}

(b)

�

Example 3.6 Let us consider the no distributive lattice of Figure 3.
The hyperlattice (L,∧,

.
∨) that corresponds to Figure 3 using the second proof of Proposi-

tion 2.5 is given by Table 8. This is not distributive hyperlattice in the sense of Definition
3.1(1) as e ∈ b

.
∨ c and d∧e /∈ (d∧b)

.
∨(d∧c), that is condition (1) of Definition 3.1(1) does

not hold; and not distributive hyperlattice in the sense of Definition 3.1(2) as e ∈ d
.
∨ b,

e ∈ d
.
∨ c and e∧ e /∈ d

.
∨(b∧ c) = {d, a}, that is condition (2) of Definition 2.8(2) does not

hold.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 262

a

b c d

e

Figure 3: The no distributive lattice of the Example 3.6.

Table 8: The hyperlattice of the Example 3.6.

∧ a b c d e

a a a a a a

b a b a a b

c a a c a c

d a a a d d

e a b c d e

(a)

.
∨ a b c d e

a {a} {a, b} {a, c} {a, d} {a, e}

b {a, b} {b} {b, c, e} {b, d, e} {b, e}

c {a, c} {b, c, e} {c} {c, d, e} {c, e}

d {a, d} {b, d, e} {c, d, e} {d} {d, e}

e {a, e} {b, e} {c, e} {d, e} {e}

(b)

Now we will give another proof of Proposition 2.5 in the next proposition.

Proposition 3.7 Let (L,∧,∨) be a lattice and “
.
∨” the hyperoperation on L defined by:

.
∨ : L× L → P∗(L) | (a, b) → a

.
∨ b := {t ∈ L | t ≤ a ∨ b}.

Then (L,∧,
.
∨) is a hyperlattice.

Proof Let x ∈ u
.
∨ c for some u ∈ a

.
∨ b. Then x ≤ u∨ c and u ≤ a∨ b, so x ≤ (a∨ b)∨ c =

a ∨ (b ∨ c). For the element v := b ∨ c ∈ b
.
∨ c, we have x ∈ a

.
∨ v, so condition (3) of

Definition 2.1 is satisfied. Let now a, b ∈ L. For the element u := a ∨ b ∈ a
.
∨ b, we have

a∧ u = a∧ (a∨ b) = a; and since a ≤ a∨ (a∧ b), we have a ∈ a
.
∨(a∧ b) and condition (4)

of Definition 2.1 also holds. �

Proposition 3.8 If (L,∧,∨) is a distributive lattice, then the hyperlattice (L,∧,
.
∨), where

a
.
∨ b := {t ∈ L | t ≤ a∨b} considered in Proposition 3.7 satisfies condition (1) of Definition

3.1(1) and condition (2) of Definition 3.1(2).



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 263

Proof Let u ∈ b
.
∨ c. Then a ∧ u ∈ (a ∧ b)

.
∨(a ∧ c). Indeed: Since u ≤ b ∨ c, we have

a ∧ u ≤ a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), so a ∧ u ∈ (a ∧ b)
.
∨(a ∧ c) and condition (1) of

Definition 3.1(1) is satisfied. Let now u ∈ a
.
∨ b and v ∈ a

.
∨ c. Then u ∧ v ∈ a

.
∨(b ∧ c).

Indeed, since u ≤ a ∨ b and v ≤ a ∨ c, we have u ∧ v ≤ (a ∨ b) ∧ (a ∨ c) = a ∨ (b ∧ c), then
u ∧ v ∈ a

.
∨(b ∧ c) and condition (2) of Definition 3.1(2) also holds. �

The question is: Given a distributive lattice (L,∧,∨) and the hyperlattice (L,∧,
.
∨),

where a
.
∨ b := {t ∈ L | t ≤ a ∨ b}, under what conditions the hyperlattice (L,∧,

.
∨) is a

distributive hyperlattice in the sense of Definitions 3.1(1) and 3.1(2)? As answer is given
in the rest of this section.

Proposition 3.9 Let (L,∧,∨) be a distributive lattice and (L,∧,
.
∨) the hyperlattice with

the hyperoperation a
.
∨ b := {t ∈ L | t ≤ a ∨ b} satisfying the property

u ∈ (a ∧ b)
.
∨(a ∧ c) and v ∈ b

.
∨ c imply a ∧ v ≤ u (3.1)

Then (L,∧,
.
∨) satisfies condition (2) of Definition 3.1(1).

Proof Let u ∈ (a∧ b)
.
∨(a∧ c). Then u ≤ (a∧ b)∨ (a∧ c) = a∧ (b∨ c). We put v := b∨ c

and we have u ≤ a ∧ v. On the other hand, since u ∈ (a ∧ b)
.
∨(a ∧ c) and v ∈ b

.
∨ c, by

(3.1), we have a ∧ v ≤ u. Hence we get u = a ∧ v and the hyperlattice (L,∧,
.
∨) satisfies

condition (2) of Definition 3.1(1). �

By Propositions 3.8 and 3.9 we have the following corollary.

Corollary 3.10 Let (L,∧,∨) be a distributive lattice and (L,∧,
.
∨) the hyperlattice defined

by a
.
∨ b := {t ∈ L | t ≤ a∨b} and having the property (3.1). Then (L,∧,

.
∨) is a distributive

hyperlattice in the sense of Definition 3.1(1).

Proposition 3.11 Let (L,∧,∨) be a distributive lattice and (L,∧,
.
∨) the hyperlattice with

the hyperoperation a
.
∨ b := {t ∈ L | t ≤ a ∨ b} satisfying the property

u ∈ a
.
∨(b ∧ c), v ∈ a

.
∨ b and w ∈ a

.
∨ c imply u ≥ v ∧ w (3.2)

Then (L,∧,
.
∨) satisfies condition (1) of Definition 3.1(2).

Proof Let u ∈ a
.
∨(b∧ c) Then u ≤ a∨ (b∧ c) = (a∨ b)∧ (a∨ c). We put v := a∨ b ∈ a

.
∨ b

and w := a ∨ c ∈ a
.
∨ c and we have u ≤ v ∧ w. On the other hand, by (3.2) we have

u ≥ v ∧ w, then u = v ∧ w and so condition (1) of Definition 3.1(2) is satisfied. �

By Propositions 3.8 and 3.11 we have the following corollary.

Corollary 3.12 Let (L,∧,∨) be a distributive lattice and (L,∧,
.
∨) the hyperlattice defined

by a
.
∨ b := {t ∈ L | t ≤ a∨b} and having the property (3.2). Then (L,∧,

.
∨) is a distributive

hyperlattice in the sense of Definition 3.1(2).

Problem 3.13 Write a program to show that the hyperlattice considered in Proposition
3.8 does not satisfy condition (2) of Definition 3.1(1) and condition (1) of Definition 3.1(2)
in general.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 264

4. Modular hyperlattices

A lattice L is called modular if, for any a, b, c ∈ L, a ≤ c implies a∨ (b∧c) = (a∨b)∧c.
This concept can be naturally transferred to a hyperlattice by the following definition.

Definition 4.1 A hyperlattice L is called modular if the following assertions are satisfied:

(1) if a ∧ c = a and u ∈ a ∨ (b ∧ c), then there exists v ∈ a ∨ b such that
u = v ∧ c and

(2) if a ∧ c = a and u ∈ a ∨ b, then u ∧ c ∈ a ∨ (b ∧ c).

In other words, if a∧ c = a, then we have u ∈ a∨ (b∧ c) if and only if there exists v ∈ a∨ b
such that u = v ∧ c. That is, if a ∧ c = a implies {v ∧ c | v ∈ a ∨ b} = a ∨ (b ∧ c).

Since a ∧ a = a, in a modular hyperlattice we have {v ∧ a | v ∈ a ∨ b} = a ∨ (b ∧ a).

Remark 4.2 While to a distributive lattice correspond two definitions of a distributive
hyperlattice, to a modular lattice corresponds only one. Indeed, if we get the equivalent
definition of a modular lattice (the dual definition) which is a ≥ c implies a ∧ (b ∨ c) =
(a ∧ b) ∨ c, this can be transferred to hyperlattices as follows:

(1) if a ∧ c = c and u ∈ b ∨ c, then a ∧ u ∈ (a ∧ b) ∨ c and

(2) if a ∧ c = c and u ∈ (a ∧ b) ∨ c, then there exists v ∈ b ∨ c such that u ∈ a ∧ v.

In other words, if a ∧ c = c implies {a ∧ u | u ∈ b ∨ c} = (a ∧ b) ∨ c. As we see, this is the
same with the definition of modular hyperlattice given by Definition 4.1 (by interchanging
a and c the two definitions coincide).

Example 4.3 We consider the no modular lattice of Figure 4.

a

b

c

d

e

Figure 4: The no modular lattice of the example 4.3.

The hyperlattice L with the operation ∧ and the hyperoperation
.
∨ defined in the

second proof of Proposition 2.5 is given by Table 9 and it is not modular as c∧ d = c and
e ∈ c

.
∨ b but e ∧ d /∈ c

.
∨(b ∧ d); that is condition (2) of Definition 4.1 does not hold.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 265

Table 9: The no modular hyperlattice of the Example 4.3.

∧ a b c d e

a a a a a a

b a b a a b

c a a c c e

d a a c d d

e a b c d e

(a)

.
∨ a b c d e

a {a} {a, b} {a, c} {a, d} {a, e}

b {a, b} {b} {b, c, e} {b, d, e} {b, e}

c {a, c} {b, c, e} {c} {c, d} {c, e}

d {a, d} {b, d, e} {c, d} {d} {d, e}

e {a, e} {b, e} {c, e} {d, e} {e}

(b)

Proposition 4.4 If L is a distributive hyperlattice in the sense of Definition 3.1(1), then
it is modular.

Proof Let a∧c = a and u ∈ a∨ (b∧c). Then u ∈ (a∧c)∨ (b∧c) = (c∧a)∨ (c∧b) and, by
the second property of Definition 3.1(1), there exists v ∈ a∨b such that u = c∧v (= v∧c).
Let now a∧ c = a and u ∈ a∨ b. Since u ∈ a∨ b, by the first property of Definition 3.1(1),
we have c∧ u ∈ (c∧ a)∨ (c∧ b). Since a∧ c = a, we have u∧ c ∈ a∨ (b∧ c) and the proof
is complete. �

According to Proposition 4.4, the distributive hyperlattice of Table 6 (Example 3.3) is
modular.

Proposition 4.5 Let (L,∧,∨) be a modular lattice. Then

(1) the hyperlattice (L,∧,
.
∨), where

.
∨ : (a, b) → a

.
∨ b := {a ∨ b} considered in the first

part of Proposition 2.5 is modular;

(2) the hyperlattice (L,∧,
.
∨), where

.
∨ : (a, b) → a

.
∨ b := {a, b, a ∨ b} considered in the

second part of Proposition 2.5 is also modular.

Proof (1) Let a ∧ c = a and u ∈ a
.
∨(b ∧ c). Since a ≤ c, u = a ∨ (b ∧ c) and (L,∧,∨) is

modular, for the element v := a∨b ∈ a
.
∨ b of L, we have u = a∨(b∧c) = (a∨b)∧c = v∧c,

so condition (1) of Definition 4.1 is satisfied. Let now a∧ c = a and u ∈ a
.
∨ b. Since a ≤ c,

u = a ∨ b and (L,∧,∨) is modular, we have u ∧ c = (a ∨ b) ∧ c = (a ∨ b) ∧ c, then
u ∧ c ∈ a

.
∨(b ∧ c) and condition (2) of Definition 4.1 also holds.



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 266

(2) Let a ∧ c = a and u ∈ a
.
∨(b ∧ c). If u = a then, for the element v := a ∈ a

.
∨ b of L,

we have u = v ∧ c; if u = b ∧ c then, for the element v := b ∈ a
.
∨ b, we have u = v ∧ c; if

u = a∨(b∧c) then, for the element v := a∨b ∈ a
.
∨ b, we have u = a∨(b∧c) = (a∨b)∧c = v∧c

and condition (1) of Definition 4.1 holds. Let now a ∧ c = a and u ∈ a
.
∨ b. If u = a, then

u∧ c = a∧ c = a ∈ a
.
∨(b∧ c); if u = b, then u∧ c = b∧ c ∈ a

.
∨(b∧ c). Finally, let u ∈ a∨ b.

Then we have u ∧ c = (a ∨ b) ∧ c. On the other hand, since a ≤ c and L is modular, we
have (a ∨ b) ∧ c = a ∨ (b ∧ c). Thus we have u ∧ c = a ∨ (b ∧ c) ∈ a

.
∨(b ∧ c). �

We apply Proposition 4.5 to the following example

Example 4.6 We consider the modular lattice L of Figure 5. The hyperlattice that
corresponds to L via the second proof of Proposition 2.5 is given by Table 10 and, according
to Proposition 4.5, this is a modular hyperlattice.

a

b

c

d

e

f

Figure 5: The modular lattice of the Example 4.6.

Table 10: The modular hyperlattice of the Example 4.6.

∧ a b c d e f

a a a a a a a

b a b b a b b

c a b c a b c

d a a a d d d

e a a b d e e

f a b c d e f

(a)



Niovi Kehayopulu / Eur. J. Pure Appl. Math, 12 (2) (2019), 252-269 267

.
∨ a b c d e f

a {a} {a, b} {a, c} {a, d} {a, e} {a, f}

b {a, b} {b} {b, c} {b, d, e} {b, e} {b, f}

c {a, c} {b, c} {c} {c, d, f} {c, e, f} {c, f}

d {a, d} {b, d, e} {c, d, f} {d} {d, e} {d, f}

e {a, e} {b, e} {c, e, f} {d, e} {e} {e, f}

f {a, f} {b, f} {c, f} {d, f} {e, f} {f}

(b)

Proposition 4.7 If (L,∧,∨) is a modular lattice, then the hyperlattice (L,∧,
.
∨), with the

hyperoperation a
.
∨ b := {t ∈ L | t ≤ a ∨ b} satisfies condition (2) of Definition 4.1.

Proof Let a ∧ c = a and u ∈ a
.
∨ b. Then u ∧ c ∈ a

.
∨(b ∧ c). Indeed: Since u ≤ a ∨ b, we

have u∧c ≤ (a∨b)∧c. Since a ≤ c and (L,∧,∨) is modular, we have (a∨b)∧c = a∨(b∧c).
Thus we get u ∧ c ≤ a ∨ (b ∧ c) and so u ∧ c ∈ a

.
∨(b ∧ c). �

We apply Proposition 4.7 to the following example.

Example 4.8 We consider the modular lattice of Figure 5 (Example 4.6). By Proposition
4.7, the hyperlattice defined by Table 11 satisfies condition (2) of Definition 4.1.

Table 11: The hyperlattice of the Example 4.8.

∧ a b c d e f

a a a a a a a

b a b b a b b

c a b c a b c

d a a a d d d

e a a b d e e

f a b c d e f

(a)

.
∨ a b c d e f

a {a} {a, b} {a, b, c} {a, d} {a, b, d, e} S

b {a, b} {a, b} {a, b, c} {a, b, d, e} {a, b, d, e} S

c {a, b, c} {a, b, c} {a, b, c} S S S

d {a, d} {a, b, d, e} S {a, d} {a, b, d, e} S

e {a, b, d, e} {a, b, d, e} S {a, b, d, e} {a, b, d, e} S

f S S S S S S

(b)



REFERENCES 268

The question is: Are there modular lattices for which the hypersemigroup defined in
Proposition 4.7 satisfies condition (1) of Definition 4.1? As answer is given by the following
proposition.

Proposition 4.9 Let (L,∧,∨) be a modular lattice and (L,∧,
.
∨) the hyperlattice with the

hypeoperation a
.
∨ b := {t ∈ L | t ≤ a ∨ b} satisfying the property

v ∈ a
.
∨ b and u ∈ a

.
∨(b ∧ c) imply v ∧ c ≤ u (4.1)

Then (L,∧,
.
∨) satisfies condition (1) of Definition 4.1.

Proof Let a ∧ c = a and u ∈ a
.
∨(b ∧ c). Then we have u ≤ a ∨ (b ∧ c) = (a ∨ b) ∧ c and,

for the element v := a ∨ b, we have u ≤ v ∧ c. On the other hand, since v ∈ a
.
∨ b and

u ∈ a
.
∨(b∧ c), by (4.1), we have v ∧ c ≤ u. Hence we obtain u = v ∧ c and property (1) of

Definition 4.1 is satisfied. �

By Propositions 4.7 and 4.9 we have the following

Corollary 4.10 Let (L,∧,∨) be a modular lattice and “
.
∨” the hyperoperation on L defined

by a
.
∨ b := {t ∈ L | t ≤ a ∨ b} and having the property (4.1). Then (L,∧,

.
∨) is a modular

hyperlattice.

Remark 4.11 The hyperlattice of Table 11 does not satisfy condition (4.1); indeed, we
have b ∈ d

.
∨ b and d ∈ d

.
∨(b ∧ c) but b ∧ c 6≤ d.

However it seems to be modular; write a program to check it.

Problem 4.12 Write a program to show that the hyperlattice considered in Proposition
4.7 does not satisfy condition (1) of Definition 4.1 in general.

Problem 4.13 Write a program to check if the hyperlattices of the examples of the paper
satisfy the relation a ∈ a ∨ b ⇒ a ∧ b = b.

I would like to thank the two anonymous referees for their time to read the paper
carefully, the useful discussions through the editor we had, and their prompt reply.

References

[1] Ameri, R., Amiri-Bideshki, M., Saeid, A.B., Hoskova-Mayerova, S.: Prime filters of
hyperlattices. An. St. Univ. Ovidius Constanta 24(2), 15–26 (2016)

[2] Ameri R, Amiri-Bideshki M., S. Hoskova-Mayerova S., Saeid A.B.: Distributive and
Dual Distributive Elements in Hyperlattices. An. St. Univ. Ovidius Constanta 25(3),
25–36 (2017)

[3] Koguep, B.B.N., Nkuimi, C., Lele, C.: On fuzzy ideals of hyperlattice. Int. J. Algebra
2(13–16), 739–750 (2008)

[4] Konstantinidou, M., Mittas, J.: An introduction to the theory of hyperlattices. Math.
Balkanica 7, 187–193 (1977)



REFERENCES 269

[5] Konstantinidou-Serafimidou, M.: Modular hyperlattices. Prakt. Akad. Athenon 53
(1978), 202–218 (1979)

[6] Konstantinidou-Serafimidou, M.: Distributive and complemented hyperlattices.
Prakt. Akad. Athenon 56 (1981), 339–360 (1982)

[7] Konstantinidou, M., Serafimidis, M.: Un théorèm d’isomorphisme concernant des
classes particulières d’hypertreillis. Honorary Volume dedicated to J.D. Mittas, Aris-
totle Univ. of Thessaloniki, Faculty of Engineering, Dept. Math. 263–271 (1999–2000)