ON CONGRUENCES AND HOMOMORPHISMS ON SOME

NON-DETERMINISTIC ALGEBRAS

I. P. Cabrera, P. Cordero, G. Guti

errez, J. Mart

ınez and M. Ojeda-Aciego

Dept. Applied Mathematics, University of M

alaga, Spain

Keywords:

L-fuzzy sets, Non-deterministic algebra, Congruence, Homomorphism.

Abstract:

Starting with the underlying motivation of developing a general theory of L-fuzzy sets where L is a multilat-

tice (a particular case of non-deterministic algebra), we study the relationship between the crisp notions of

congruence, homomorphism and substructure on some non-deterministic algebras which have been used in

the literature, i.e. hypergroups, and join spaces. Moreover, we provide suitable extensions of these notions to

the fuzzy case.

1 INTRODUCTION

This paper follows the trend of developing fuzzy

versions of crisp concepts in mathematics. Specif-

ically, we focus on congruence relations, substruc-

tures and homomorphisms in the framework of sev-

eral hyperstructures, which are non-deterministic al-

gebraic structures in the sense that the operations are

not single-valued but set-valued.

The study of congruence relations and homomor-

phisms between given hyperstructures plays an im-

portant role in the general theory of algebraic fuzzy

systems. The underlying idea here is to apply meth-

ods of the algebraic theory of ordinary congruences

and homomorphisms between classical structures in

studying suitable extensions for speciﬁc hyperstruc-

tures, such as hypergroups, and join spaces.

One can ﬁnd a number of extensions of classical

algebraic structures to a fuzzy framework in the liter-

ature, all of them based more or less in similar ideas.

However, the fuzzy extension of the notion of func-

tion has been studied from several standpoints, and

this fact complicates the choice of the most suitable

deﬁnition of fuzzy homomorphism: the most conve-

nient deﬁnition seems to depend on particular details

of the underlying algebraic structure under consider-

ation.

The study of congruences is important both from

a theoretical standpoint and for its applications in the

ﬁeld of logic-based approaches to uncertainty. Re-

garding applications, the notion of congruence is in-

timately related to the foundations of fuzzy reasoning

and its relationships with other logics of uncertainty.

More focused on the theoretical aspects of Computer

Science, some authors (B

elohl

avek, 2002),(Petkovi

2006) have pointed out the relation between congru-

ences, fuzzy automata and determinism.

More on the practical side, applications of the con-

cept of congruence can be seen in the World Wide

Web. Concerning web applications, some authors

have argued on the convenience of using Answer Set

Programming (ASP) in the Semantic Web. Congru-

ence relations have been used in the study of modu-

larization of ASP as a way to structure and ease the

program development process. Speciﬁcally, compo-

sition of modules has been formalized in(Oikarinen

and Janhunen, 2008; Janhunen et al., 2007) in terms

of equivalence relations which are proper congruence

relations.

The previous paragraphs have shown the useful-

ness of the theory of (crisp) congruences regarding

practical applications. At this point, it is important to

recall that the problem of providing suitable fuzziﬁ-

cations of crisp concepts is an important topic which

has attracted the attention of a number of researchers.

Since the inception of fuzzy sets and fuzzy logic,

there have been approaches to consider underlying

sets of truth-values more general than the unit inter-

val; for instance, consider the L-fuzzy sets introduced

by Goguen in (Goguen, 1967), where L is a complete

lattice.

There are more general structures than a com-

plete lattice which could host a suitable extension

of the notion of fuzzy set, for instance, the multi-

Cabrera I., Cordero P., Gutiérrez G., Martínez J. and Ojeda-Aciego M. (2009).

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 59-67

DOI: 10.5220/0002313300590067

 SciTePress

lattices and other general hyperstructures. The con-

cepts of ordered and algebraic multilattice were in-

troduced in (Benado, 1954): a multilattice is an alge-

braic structure in which the restrictions imposed on a

(complete) lattice, namely, the “existence of least up-

per bounds and greatest lower bounds” are relaxed to

the“existence of minimal upper bounds and maximal

lower bounds”. An alternative algebraic deﬁnition of

multilattice was proposed in (Medina et al., 2005),

which is more closely related to that of lattice, allow-

ing for natural deﬁnitions of related structures such

that multisemilattices and, in addition, is better suited

for applications. For instance, a general approach to

fuzzy logic programming based on a multilattice as

underlying set of truth-values was presented in (Med-

ina et al., 2007).

A number of papers have been published on the

lattice of fuzzy congruences on different classical al-

gebraic structures, and even in some hyperstructures,

for instance (Cordero et al., 2008) studies congru-

ences on a multilattice. In this paper, we will focus

on congruences and homomorphisms in the more gen-

eral setting of hyperstructures (Corsini and Leoreanu,

2003).

The structure of the paper is as follows: in Sec-

tion 2, we introduce the deﬁnitions of some hyper-

structures, together with some notational convention

to be used in the rest of the paper; then, in Sec-

tion 3, we concentrate on the deﬁnition of homo-

morphism between nd-groupoids and how it preserves

the different subhyperstructures. Later, in Section 4,

the relation between the introduced notion of nd-

homomorphism and (crisp) congruence on a hyper-

structure is investigated. It is in Section 5 where the

notion of fuzzy homomorphism is deﬁned and the

canonical decomposition theorem is presented. Fi-

nally, we present some conclusions and prospects for

future work.

2 PRELIMINARY DEFINITIONS

Firstly, let us introduce the preliminary concepts used

in this paper:

Deﬁnition 1. A hypergroupoid is a pair (A,·) con-

sisting of a non-empty set A together with a hyperop-

eration · : A ×A → 2

r ∅.

Traditionally, authors working with hyperstruc-

tures have considered the natural restriction of the

images of the operation to be non-empty sets,

for instance, the structures of hypergroup or join

space (Corsini and Leoreanu, 2003).

Deﬁnition 2. A hypergroupoid (A,·) is said to be a

hypergroup if the following properties hold:

• Associativity: (ab)c = a(bc) for all a,b,c ∈ A.

• Reproductivity: aA = Aa = A for all a ∈ A. That

is, for all a, b ∈ A, the equations ax = b y xa = b

have solutions.

(A,·) is said to be a join space if it satisﬁes associa-

tivity, reproductivity and, moreover, the two following

properties hold:

• Commutativity: ab = ba for all a,b,c ∈ A.

• Transposition Property: for all a, b, c, d ∈ A,

a/b ∩ c/d 6= ∅ implies ad ∩ bc 6= ∅ where a/b =

{x | a ∈ xb}.

It is remarkable that in the context of multilattices

it is admissible to consider the empty set in the range

of the hyperoperation, hence interest arises in devel-

oping a suitable extension of the concept of hyper-

groupoid, the so-called nd-groupoid. Despite of the

small change regarding the empty set, it is noticeable

that the resulting theory differs substantially.

Deﬁnition 3. An nd-groupoid is a pair (A,·) con-

sisting of a non-empty set A together with an nd-

operation · : A × A → 2

Remark 4. As usual, if a ∈ A and X ⊆ A then aX will

denote {ax | x ∈ X} and Xa will denote {xa | x ∈ X}.

In particular, a∅ = ∅a = ∅.

Note, as well, that in the rest of the paper we will

frequently write singletons without braces.

3 ON THE DEFINITION OF

ND-GROUPOID

HOMOMORPHISM

This section introduces the extension of the existing

results about homomorphisms to the more general

framework of nd-grupoids. Firstly, we begin by dis-

cussing the different versions of the concept of ho-

momorphism on hypergroupoids (also called multi-

groupoids) appearing in the literature. They are usu-

ally associated to particular classes of hypergroupoids

such as those of hypergroups and join spaces.

Some authors that deal with these and other hy-

perstructures use the following deﬁnitions of homo-

morphism (Corsini, 2003).

Deﬁnition 5. Let (A,·) and (B,·) be hypergroupoids.

A map h: A → B is said to be:

• Benado-homomorphism if h(ab) ⊆ h(a)h(b), for

all a,b ∈ A.

• Algebraic-homomorphism if h(ab) = h(a)h(b),

for all a,b ∈ A.

IJCCI 2009 - International Joint Conference on Computational Intelligence

Recall that this deﬁnition extends without modiﬁ-

cation to the framework of nd-groupoids.

Regarding the terminology, we depart here a bit

from the usual one. The ﬁrst one was the original def-

inition by Benado (Benado, 1954), which has been

used in several recent papers (Davvaz, 2000; Gen-

tile, 2006; Corsini, 2003). However, it is noticeable

that, ﬁnally, the authors concentrate mostly on the

equality-based deﬁnition. This choice is partly due to

the excessive generality of Benado’s deﬁnition, which

limits the possibility of obtaining interesting theoreti-

cal results. The terminology used in those papers is to

call homomorphism to Benado’s ones and call good

(or strong) homomorphism to algebraic ones.

We have adopted the term algebraic instead of

good or strong because this type of homomorphism

immediate allows the lifting of classical homomor-

phisms to the so-called powerset extension. Ob-

viously, the advantage of using algebraic homo-

morphisms is that one can transfer properties from

the powerset to the nd-groupoid very easily, so

that the presentation of multivalued (namely, non-

deterministic) concepts is greatly simpliﬁed.

Theorem 6. Let (A,·) and (B,·) be nd-groupoids and

h: A → B be a Benado-homomorphism. Let (2

,·),

,·) and H : 2

→ 2

be the usual power extension

of h, i.e.

X ·Y =

[

x∈X

y∈Y

xy and H(X) = {h(x) | x ∈ X}

Then, h is an algebraic-homomorphism if and only if

H is a homomorphism.

Proof. Straightforward. 

It should be noticed that the deﬁnition of algebraic

homomorphism, when applied to the lifted powerset

version, collapses to the classical algebraic results on

the powerset. Hence, the condition limits too much

the non-deterministic interpretation of the concept of

morphism in that does not provide any added value to

the theory.

Moreover, the term homomorphism should induce

the properties of the initial hypergroupoid on the im-

age set. It can be easily checked that this is the case

for algebraic-homomorphisms but, in general, it is not

true for Benado-homomorphisms as the following ex-

amples show.

Example 7. Let (A,·) and (B,·) be the hyper-

groupoids A = {a,b,c} and B = {a,b,c,d} with the

hyperoperations deﬁned as in the tables. The inclu-

sion map i from A to B is a Benado-homomorphism.

Let us consider the restriction of the operation in B to

the codomain i(A) = A.

(A,·)

· a b c

a a b c

b b b c

c c c c

(B,·)

· a b c d

a a b b,c A

b b b a,c A

c c,d c,d a,c A

d A A A A

(i(A),·)

· a b c

a a b b,c

b b b a,c

c c c a,c

The initial hypergrupoid (A,·) is commutative, idem-

potent and associative. However (i(A),·) is neither

commutative nor idempotent nor associative (for in-

stance, note that (ab)c = {a,c} 6= a(bc) = {a,b,c}).



When searching for a suitable deﬁnition which

allows to transfer properties to the codomain, it

is noticeable the following interesting property of

multisemilattices homomorphism in the case of the

operations are actually hyperoperations (not nd-

operations).

Theorem 8. Let (A,·) and (B,·) be multisemilattices

and h : A → B be a Benado-homomorphism. If ab 6=

∅, for all a,b ∈ A, then h(ab) = h(a)h(b) ∩ h(A), for

all a,b ∈ A.

Based on the previous result, we propose a new

deﬁnition of homomorphism on nd-grupoids, which

is stronger than Benado’s one and weaker than the al-

gebraic one. Furthermore, the underlying idea follows

the categorical meaning of morphism.

Deﬁnition 9. Let (A, ·) and (B, ·) be nd-groupoids. A

map h: A → B is said to be nd-homomorphism if, for

all a,b ∈ A, h(ab) = h(a)h(b) ∩ h(A).

Note that, h is nd-homomorphism if, for all

a,b,c ∈ A, the following conditions hold:

1. c ∈ ab implies h(c) ∈ h(a)h(b), that is, h is a

Benado-homomorphism.

2. h(c) ∈ h(a)h(b) implies that there exists c

∈ ab

such that h(c

) = h(c).

The following examples show that the notion of

nd-homomorphism is different from the two other

deﬁnitions given previously.

Example 10. Consider the hypergroupoid (A, ·) be-

ing A = [0, 1] and the hyperoperation · : A × A → 2

given by a · b = [0,max{a,b}]. Notice that this hy-

pergroupoid is a join space (it is commutative, as-

sociative, reproductive and veriﬁes the transposition

axiom). Let us consider a homomorphism h : A → A

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

deﬁned by

h(x) =

(

0 if x = 0

1 if x 6= 0

we have that

h(a · b) =

(

{0} if a = b = 0

{0,1} otherwise

and

h(a) · h(b) =

(

{0} if a = b = 0

[0,1] otherwise

and as a result, one obtains the equality h(a · b) =

h(a)·h(b)∩ h(A) but, in general, the proper inclusion

h(a · b) h(a) · h(b) holds.

Therefore, h is an nd-homomorphism but it is not

an algebraic-homomorphism. 

Example 11. Let us consider the set H = {a,b,c}

with the hyperoperation deﬁned by the following ta-

ble:

· a b c

a c c c

b c c c

c c c H

(H,·) is a commutative hypergrupoid. Let h : H →

H be the homomorphism

h(x) =

(

c if x = a

x otherwise

Im(h) = {b,c} and we have that:

h(a · a) = h(c) = c h(a) · h(a) ∩ Im(h) = Im(h)

h(a · b) = h(c) = c = h(a) · h(b) ∩ Im(h)

h(a · c) = h(c) = c h(a) · h(c) ∩ Im(h) = Im(h)

h(b · b) = h(c) = c = h(b) · h(b) ∩ Im(h)

h(b · c) = h(c) = c = h(b) · h(c) ∩ Im(h)

h(c · c) = h(H) = {b,c} = h(c) · h(c) ∩ Im(h)

Therefore, h is a Benado-homomorphism but it is

not an nd-homomorphism. 

We next discuss the different possible generaliza-

tions of the substructures of an nd-groupoid that con-

nect with the corresponding deﬁnitions of homomor-

phism.

The more general deﬁnition of subalgebra in

a non-deterministic setting was introduced by Be-

nado(Benado, 1954) for multilattices, and later used

by several authors (Davvaz, 2000; Gentile, 2006;

Corsini, 2003). A second deﬁnition follows the line of

the algebraic homomorphism and corresponds to the

embedding of the classical (deterministic) notion of

subgroupoid into the framework of nd-subgroupoids.

We introduce a third alternative deﬁnition below.

Deﬁnition 12. Let (A,·) be an nd-groupoid, X be a

non-empty subset of A and ∗: X × X → 2

an nd-

operation deﬁned on X.

• (X,∗) is a Benado-subgroupoid of (A,·) if a∗b ⊆

a · b, for all a, b ∈ X.

• (X,∗) is a Birkhoff-subgroupoid of (A,·) if a ∗

b = a · b, for all a,b ∈ X.

• (X,∗) is an nd-subgroupoid of (A,·) if a ∗ b = a ·

b ∩ X , for all a,b ∈ X.

Therefore, (X,∗) is a (Benado, Birkhoff, nd)-

subgroupoid if the inclusion map i : X → A is

a (Benado, algebraic, nd)-homomorphism. Like-

wise, the image space of a (Benado, algebraic,

nd)-homomorphism is a (Benado, Birkhoff, nd)-

subgroupoid of the codomain.

Lemma 13. Let (A,·) and (B,·) be nd-groupoids and

h: A → B be an nd-homomorphism. Let us consider

h(A) and the operation ∗ deﬁned as a

∗ b

= a

∩

h(A).

1. If (A,·) is commutative then (h(A),∗) is commu-

tative.

2. If (A,·) is associative then (h(A), ∗) is associative.

3. If (A,·) is idempotent then (h(A),∗) is idempotent.

4. If (A,·) is reproductive then (h(A),∗) is reproduc-

tive.

5. If (A,·) satisﬁes the transposition property then

(h(A),∗) satisﬁes the transposition property.

Proof. It is a matter of systematic calculations. As an

example we provide the proof of the case of associa-

tivity:



h(a) ∗ h(b)



∗ h(c) =



h(a)h(b) ∩ h(A)



∗ h(c) =

= h(ab) ∗ h(c) = h(ab)h(c) ∩ h(A) =

= h



(ab)c



= h



a(bc)



= h(a)h(bc) ∩ h(A) = h(a) ∗ h(bc) =

= h(a) ∗



h(b)h(c) ∩ h(A)



=h(a) ∗



h(b) ∗ h(c)





Proposition 14. Let (A,·) and (B, ·) be nd-groupoids

and h : A → B be an nd-homomorphism. Let us con-

sider h(A) and the operation ∗ deﬁned as a

∗ b

∩ h(A).

1. If (A,·) is a hypergroup then (h(A), ∗) is a hyper-

group.

2. If (A,·) is a join space then (h(A), ∗) is a join

space.

IJCCI 2009 - International Joint Conference on Computational Intelligence

Proof. It is a consequence of the previous lemma. 

In general, given a Benado-homomorphism between

nd-groupoids, the properties of the initial nd-groupoid

cannot be induced on the codomain, even with the

map being bijective. Furthermore, the inverse map

of a bijective Benado-homomorphism needs not be a

Benado-homomorphism.

Example 15. Let A = {a,b,c} and the two op-

erations deﬁned as x · y = {x,y} and x ? y = A.

Obviously, the identity mapping, I, is a Benado-

homomorphism from (A,·) to (A,?) which is not an

nd-homomorphism. However, I

−1

is not a Benado-

homomorphism: I

−1

(a ? b) = I

−1

(A) = A 6⊆ I

−1

(a) ·

−1

(b) = a · b = {a,b}. 

Theorem 16. Let h be a bijective mapping between

nd-groupoids. The following conditions are equiva-

lent:

(i) h and h

−1

are Benado-homomorphisms.

(ii) h is an algebraic-homomorphism.

(iii) h is an nd-homomorphism.

Proof. Obviously items (ii) and (iii) are equivalent

because the map is bijective.

(i) ⇒ (ii) Since h

−1

is a Benado-homomorphism,

one certainly has that h

−1



h(a) · h(b)



⊆ h

−1



h(a)



−1



h(b)



= ab. Thus, for an element x ∈ h(a) · h(b),

we have h

−1

(x) ⊆ a · b. As a consequence, x =



−1

(x)



∈ h(a · b).

(ii) ⇒ (i) Let x ∈ h

−1

). Then h(x) ∈ a



−1

)



· h



−1

)



and, since h is algebraic ho-

momorphism, h



−1

)



· h



−1

)



= h



−1

) ·

−1

)



. Thus, x ∈ h

−1





−1

) · h

−1

)





−1

) · h

−1

). 

As a consequence of the previous result, we will

call nd-isomorphism to any bijective algebraic homo-

morphism.

4 CONGRUENCES AND

HOMOMORPHISMS IN

ND-GROUPOIDS

In this section, we concentrate on the relation between

homomorphisms and congruence relations. To begin

with, we recall below the notion of congruence rela-

tion on an nd-groupoid that we are interested in.

Deﬁnition 17. Let (A,·) be an nd-groupoid. A con-

gruence on A is an equivalence relation ≡ which for

all a, b, c ∈ A satisﬁes that if a ≡ b, then ac ≡ bc and

ca ≡ cb, where X ≡ Y if and only if for all x ∈ X there

exists y ∈ Y such that x ≡ y and for all y ∈ Y there

exists x ∈ X such that x ≡ y.

The following result is a direct consequence of the

deﬁnition.

Proposition 18. Let (A,·) and (B, ·) be nd-groupoids

and h: A → B be an nd-homomorphism. The kernel

relation ≡

, deﬁned as

a ≡

b if and only if h(a) = h(b),

is a congruence relation.

It is obvious that an algebraic-homomorphism

provides a congruence relation as well. However, be-

ing an nd-homomorphism is not a necessary condition

to provide a congruence relation.

Example 19. Let (Z,·) with a · b = {a,b} and (N,?)

with a ? b = {0, a, b}. Let us deﬁne h : Z → N

as h(a) = |a|. Then, h is a surjective Benado-

homomorphism but is neither an algebraic or nd-

homomorphism. Straightforwardly, the kernel rela-

tion a ≡

b iff |a| = |b| is a congruence relation. 

Theorem 20. Let h: (A, ·) → (B,·) be a Benado-

homomorphism between nd-groupoids. The kernel re-

lation is a congruence relation if and only if h(ab) =



−1

(h(a)) · h

−1

(h(b))



, for all a,b ∈ A

Proof. Assume that the kernel relation is a congru-

ence. Since for all a ∈ A we have a ∈ h

−1

(h(a)),

and h is a Benado-homomorphism, then h(ab) ⊆



−1

(h(a)) · h

−1

(h(b))



For the other inclusion, consider h(x) ∈

h(h

−1

(h(a)) · h

−1

(h(b))), there exist y ∈ h

−1

(h(a))

and z ∈ h

−1

(h(b)) such that x ∈ yz. Since h(y) = h(a)

and h(z) = h(b), under the assumption that the kernel

relation is a congruence, yz ≡ ab. So, for x ∈ yz, there

exists x

∈ ab such that h(x) = h(x

) ∈ h(ab).

Conversely, assume the equality and let us prove

that the kernel is a congruence. It is sufﬁcient to note

the following chain of equalities for all a, b, c, d ∈ A

such that h(a) = h(b) and h(c) = h(d):

h(ac) =h(h

−1

(h(a)) · h

−1

(h(c)))

=h(h

−1

(h(b)) · h

−1

(h(d))) = h(bd)



Finally, it is worth to note that every homomorphism

h deﬁning a congruence relation can be canonically

decomposed as i ◦

h ◦ p, where

• p is the projection homomorphism p(x) = [x] and

[a] · [b] = {[x] | x ∈ ab}.

• h is the isomorphism deﬁned as h([x]) = h(x) and

a ·

b = h(h

−1

(a) · h

−1

(b)).

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

• i is the inclusion homomorphism correspond-

ing to the type of the homomorphism of h.

That is, if h is a Benado-homomorphism (re-

spectively, algebraic or nd-homomorphism) then

(h(A),·

) is a Benado-subgroupoid (respectively,

Birkhoff or nd-subgroupoid) and i is a Benado-

monomorphism (respectively, algebraic or nd-

monomorphism).

5 FUZZY HOMOMORPHISMS

ON ND-GROUPOIDS

A fuzzy relation is a mapping ϕ from A×B into [0,1],

that is to say, any fuzzy subset of A×B. The powerset

extension of a fuzzy relation is deﬁned as,

ϕ : 2

→ [0,1] with

ϕ(X,Y ) =



x∈X

y∈Y

ϕ(x,y)



∧



y∈Y

x∈X

ϕ(x,y)



The composition of fuzzy relations ϕ and ψ is deﬁned

as follows:

(ψ ◦ ϕ)(a, c) =

b∈B

ϕ(a,b) ∧ ψ(b, c)

A fuzzy relation ρ on A × A is said to be

1. reﬂexive if ρ(x,x) = 1, for every x ∈ A

2. symmetric if ρ(x, y) = ρ(y,x), for all x,y ∈ A

3. transitive if for all x,a,y ∈ A we have

ρ(x,a) ∧ ρ(a,y) ≤ ρ(x,y)

A reﬂexive, symmetric and transitive fuzzy rela-

tion on A is called a fuzzy equivalence. A fuzzy

equivalence ρ on A is called a fuzzy equality if for

any x,y ∈ A, ρ(x,y) = 1 implies x = y.

Deﬁnition 21 ((Cabrera, 2009)). A fuzzy equiva-

lence relation ρ on an nd-groupoid (A,·) is said to

be a fuzzy congruence relation if

ρ(a

) ≥

ρ(a

) ∧ ρ(a

) for all a

∈ A.

The fuzziﬁcation of the concept of function that

we adopt has been introduced in (Klawonn, 2000),

and also studied in (Demirci, 2000; Demirci, 2003;

Demirci, 2001), and more recently in (

Ciri

c et al.,

2009). We will introduce the extension of the notion

of perfect fuzzy function.

Deﬁnition 22 ((Demirci, 2000)). Let ρ and σ be fuzzy

equalities deﬁned on the sets A and B, respectively. A

partial fuzzy function ϕ from A to B is a mapping

ϕ : A × B → [0,1] satisfying the following conditions

for all a,a

∈ A and b,b

∈ B:

ext1 ϕ(a,b) ∧ ρ(a,a

) ≤ ϕ(a

,b)

ext2 ϕ(a,b) ∧ σ(b,b

) ≤ ϕ(a,b

)

part ϕ(a,b) ∧ ϕ(a,b

) ≤ σ(b,b

)

If, in addition, the following condition holds:

f1 For all a ∈ A there is b ∈ B such that ϕ(a,b) = 1

then we say that ϕ is a perfect fuzzy function.

It is not difﬁcult to show that the element b in con-

dition (f1) above is unique. As a result, every perfect

fuzzy function deﬁnes a crisp mapping from A to B

called crisp description of ϕ.

Deﬁnition 23. Let (A, ·) and (B, ·) be nd-groupoids

endowed with fuzzy equalities ρ and σ, respectively.

A perfect fuzzy function ϕ ∈ [0,1]

A×B

is said to

be a fuzzy homomorphism if for all a,a

∈ A and

b,b

∈ B, the following inequality holds:

compat ϕ(a,b) ∧ ϕ(a

) ≤

ϕ(aa

,bb

)

Moreover, ϕ is said to be complete if the two fol-

lowing conditions hold:

1. if

y∈Y

ϕ(a,y) = 1, then there exists y ∈ Y such

that ϕ(a,y) = 1.

2. if

x∈X

ϕ(x,b) = 1, then there exists x ∈ X such

that ϕ(x,b) = 1.

Remark: Hereafter, unless stated otherwise, we will

always consider that we are working with a complete

fuzzy homomorphism ϕ between nd-groupoids A and

B and fuzzy equalities ρ and σ, respectively.

Proposition 24. Given ϕ between A and B, the crisp

description h of ϕ is an algebraic homomorphism.

Proof. Let a

∈ A. As ϕ(a

,h(a

)) ∧

ϕ(a

,h(a

)) ≤

ϕ(a

,h(a

)h(a

)), we have that

a∈a

b∈h(a

)h(a

)

ϕ(a,b)∧

b∈h(a

)h(a

)

a∈a

ϕ(a,b) = 1

and so, for all a ∈ a

we have that

b∈h(a

)h(a

)

ϕ(a,b) = 1, and by completeness,

there exist b ∈ h(a

)h(a

) such that ϕ(a,b) = 1 and,

therefore, b = h(a). So h(a

) ⊆ h(a

)h(a

Conversely for all b ∈ h(a

)h(a

), we have that

a∈a

ϕ(a,b) = 1 and by completeness, there ex-

ist a ∈ a

such that h(a) = b, and so h(a

)h(a

) ⊆

h(a

). 

The notion of fuzzy homomorphism between nd-

groupoids behaves properly with respect to the com-

position of fuzzy relations, in that the composition

of fuzzy homomorphisms is a fuzzy homomorphism.

Furthermore, the composition is associative and there

exists an identity for this composition. As a result, the

class of nd-groupoids together with the fuzzy homo-

morphisms between them forms a category.

Let us concentrate now on the relationship be-

tween fuzzy homomorphism and congruences.

IJCCI 2009 - International Joint Conference on Computational Intelligence

Deﬁnition 25. The fuzzy kernel relation induced by

ϕ in A is deﬁned as ρ

(a,a

) = ϕ(a,h(a

)).

We adopt here the term kernel as an extension of

the crisp case because of the inequality

ϕ(a,b) ∧ ϕ(a

,b) ≤ ρ

(a,a

)

which can be checked by direct computation.

Theorem 26. Consider ϕ between A and B. The fuzzy

kernel relation ρ

is a congruence relation which in-

cludes the fuzzy equality ρ in A.

Proof. Let us see that ρ

is a congruence relation.

) =

a∈a

∈a

(a,a

) ∧

∈a

a∈a

(a,a

)

a∈a

∈a

ϕ(a,h(a

)) ∧

∈a

a∈a

ϕ(a,h(a

))

ϕ(a

,h(a

))

ϕ(a

,h(a

)h(a

)) by Prop. 24

≥ ϕ(a

,h(a

)) ∧ ϕ(a

,h(a

))

= ρ

) ∧ ρ

)

Now, let us show that ρ ≤ ρ

ρ(a,a

) = ρ(a,a

) ∧ ϕ(a

,h(a

))

≤ ϕ(a,h(a

)) = ρ

(a,a

) by (ext1)



In the rest of this section we will show the canon-

ical decomposition theorem for a complete fuzzy ho-

momorphism and a fuzzy congruence relation. For

suitable extensions on the notions of injectivity and

surjectivity we will rely on the deﬁnitions given

in (Demirci, 2000).

Deﬁnition 27. A perfect fuzzy function ϕ ∈ [0,1]

A×B

is said to be:

• surjective if, for all b ∈ B there exists a ∈ A such

that ϕ(a,b) = 1.

• injective if, for all a,a

∈ A and b ∈ B we have

ϕ(a,b) ∧ ϕ(a

,b) ≤ ρ(a,a

• bijective if it is injective and surjective.

The image set is Im ϕ = {b ∈ B | there exists a ∈

A with ϕ(a,b) = 1}.

In order to deﬁne the different homomorphisms

involved in the decomposition theorem, we have to

introduce the quotient set associated to a fuzzy equiv-

alence relation.

Deﬁnition 28. Let (A,·) be an nd-groupoid and ρ be a

fuzzy equivalence relation in A. An equivalence class

of an element a ∈ A is deﬁned as

ρ(a) ∈ [0,1]

with ρ(a)(a

) = ρ(a,a

)

The quotient set is deﬁned as A/

{

ρ(a) | a ∈ A

}

and a fuzzy equality ρ can be deﬁned in A/

ρ(ρ(a),ρ(a

)) = ρ(a,a

The fuzzy projection π from A to A/

is deﬁned as

π(a,ρ(a

)) = ρ(a,a

Proposition 29. Let (A,·) be an nd-groupoid, ρ a

fuzzy equality in A and ρ

be a fuzzy congruence rela-

tion in A that includes ρ. The fuzzy projection π from

A to A/

is a surjective fuzzy homomorphism where

the nd-operation in A/

is given by

) · ρ

) =

{

(d) | d ∈ a

}

and the fuzzy equality is ρ

Proof. We will only prove properties (ext1) and

(compat), as the rest of the required properties are

more or less straightforward computations.

ext1 We will use that ρ ≤ ρ

and ρ

is symmet-

ric and transitive: π(a

,ρ

)) ∧ ρ(a

) =

) ∧ ρ(a

) ≤ ρ

) ∧ ρ

) ≤

) = π(a

,ρ

)).

compat π(a

,ρ

)) ∧ π(a

,ρ

)) =

) ∧ ρ

) ≤

) =

π(a

,ρ

)ρ

)) 

Remark: In order to prove that the canonical inclu-

sion from the image of a homomorphism is an in-

jective fuzzy homomorphism, we recall the follow-

ing result from (Demirci, 2000): given ϕ between

A and B, there exists a unique crisp function f such

that ϕ(a,b) = σ( f (a), b). This f actually coincides

with the crisp description h of ϕ, which satisﬁes

ϕ(a,h(a)) = 1.

Lemma 30. Given ϕ between A and B, then the in-

clusion ι from Im ϕ to B deﬁned as ι(b,b

) = σ(b,b

)

is an injective fuzzy homomorphism.

Proof. We only prove property (compat):

Let b

∈ Im ϕ and a

∈ A such that h(a

) =

and h(a

) = b

. Then ι(b

) ∧ ι(b

) =

σ(b

) ∧ σ(b

) = σ(h(a

),b

) ∧ σ(h(a

),b

) =

ϕ(a

) ∧ ϕ(a

) ≤

ϕ(a

). Now, by

Proposition 24,

ϕ(a

) =

ι(h(a

),b

) =

ι(h(a

)h(a

),b

) =

ι(b

). 

Theorem 31. Any complete fuzzy homomorphism ϕ

from A to B can be canonically decomposed as ϕ =

ι ◦ϕ ◦π where π is the fuzzy projection from A to A/

ι is the inclusion from Im ϕ to B, and ϕ is the isomor-

phism from A/

to Im ϕ deﬁned as ϕ(ρ

(a),b) =

ϕ(a,b), and the nd-operation and the fuzzy equality

in Im ϕ being the corresponding restrictions of those

in B.

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS

Proof. Firstly, let us prove (ext1), (inj) and (surj), the

rest of properties are straightforward:

ext1 ϕ(ρ

(a),b) ∧ ρ

(ρ

(a),ρ

)) =

ϕ(a,b) ∧ ρ

(a,a

) = ϕ(a, b) ∧ ϕ(a,h(a

)) =

ϕ(a,b) ∧ ϕ(a,h(a

)) ∧ ϕ(a

,h(a

)) ≤

σ(b,h(a

))∧ϕ(a

,h(a

)) ≤ ϕ(a

,b) = ϕ(ρ

),b)

inj ϕ(ρ

(a),b) ∧ ϕ(ρ

),b) = ϕ(a,b) ∧ ϕ(a

,b) ≤

(a,a

) = ρ

(ρ

(a),ρ

)).

surj For all b ∈ Im ϕ there exists a ∈ A such that

ϕ(a,b) = 1 and then

ϕ(ρ

(a),b) = 1

Finally, let us check that ϕ = ι ◦

ϕ ◦ π:

(ι◦ϕ ◦ π)(a,b) =

)∈A/

∈Im ϕ



π(a,ρ

)) ∧ ϕ(ρ

),b

) ∧ ι(b

,b)



∈A

∈Imϕ



(a,a

) ∧ ϕ(a

) ∧ σ(b

,b)



(ext2)

≤

∈A



(a,a

) ∧ ϕ(a

,b)



(de f ρ

)

∈A



ϕ(a,h(a

)) ∧ ϕ(a

,b)



( f 1)

∈A



ϕ(a,h(a

)) ∧ ϕ(a

,h(a

)) ∧ ϕ(a

,b)



(part)

≤

∈A



ϕ(a,h(a

)) ∧ σ(h(a

),b)



(ext2)

≤ ϕ(a, b)

Conversely, ϕ(a, b) = σ(h(a),b) = π(a,ρ

(a)) ∧

ϕ(ρ

(a),h(a)) ∧ σ(h(a), b) ≤ (ι ◦ ϕ ◦ π)(a,b). 

6 CONCLUSIONS AND FUTURE

WORK

We have studied the relationship between the crisp

notions of congruence, homomorphism and substruc-

ture on some non-deterministic algebras which have

been used in the literature, i.e. hypergroups, and join

spaces, as a step towards developing a general theory

of L-fuzzy sets where L is a multilattice (a particu-

lar case of non-deterministic algebra). Moreover, we

have provided suitable extensions of these notions to

the fuzzy case.

As future work, we will study homomorphisms

and congruences in some algebraic nd-structures,

such as hyper-rings and hyper-near-rings, with the

aim of investigating the adequate notion of ideal for

them, together with the possible interactions with sev-

eral applications of these structures already published

in the literature. Then, we will analyze the behaviour

of the extension of a fuzzy homomorphism between

nd-groupoids to the powerset structures. At the same

time, we will investigate weaker deﬁnitions which ex-

tend that of nd-homomorphism, in order to to obtain

a more abstract and ﬂexible approach. The fuzzy no-

tions introduced in this paper will be studied in a more

general framework by substituting the unit interval

with a lattice-based structure.

Finally, it is worth to note that the notion of func-

tion compatible with certain structure is crucial in

the study of ﬁelds such as Functional Dependencies

in Databases, Attribute Implication in Formal Con-

cept Analysis, Association Rules in Datamining, etc.

Thus, we will analyze the possible applications de-

rived from our work on these research lines.

ACKNOWLEDGEMENTS

Partially supported by projects TIN2006-15455-C03-

01 and TIN2007-65819 (Science Ministry of Spain),

and P06-FQM-02049 (Junta de Andaluc

ıa).

REFERENCES

R. B

elohl

avek. Determinism and fuzzy automata. Informa-

tion Sciences, 143:205–209, 2002.

M. Benado. Les ensembles partiellement ordonn

es et le

eor

eme de rafﬁnement de Schreier. I.

Cehoslovack.

Mat.

Z., 4(79):105–129, 1954.

I. Cabrera, P. Cordero, G. Guti

errez, J. Mart

ınez, and

M. Ojeda-Aciego. Fuzzy congruence relations on nd-

groupoids. Intl J on Computer Mathematics, 2009. To

appear.

Ciri

c, J. Ignjatovi

c, and S. Bogdanovi

c. Uniform fuzzy

relations and fuzzy functions. Fuzzy Sets and Systems,

160(8):1054–1081, 2009.

P. Cordero, G. Guti

errez, J. Mart

ınez, M. Ojeda-Aciego,

and I. de las Pe

nas. Congruence relations on multilat-

tices. In Intl FLINS Conference on Computational In-

telligence in Decision and Control, FLINS’08, pages

139–144, 2008.

P. Corsini. A new connection between hypergroups and

fuzzy sets. Southeast Asian Bull. Math., 27(2):221–

229, 2003.

P. Corsini and V. Leoreanu. Applications of hyperstructure

theory. Kluwer, 2003.

B. Davvaz. Some results on congruences on semihyper-

groups. Bulletin of the Malaysian Mathematical Sci-

ences Society. Second Series, 23(1):53–58, 2000.

IJCCI 2009 - International Joint Conference on Computational Intelligence

M. Demirci. Fuzzy functions and their applications. J.

Math. Anal. Appl, 252:495–517, 2000.

M. Demirci. Gradation of being fuzzy function. Fuzzy Sets

and Systems, 119(3):383–392, 2001.

M. Demirci. Constructions of fuzzy functions based on

fuzzy equalities. Soft Computing, 7(3):199–207,

2003.

G. Gentile. Multiendomorphisms of hypergroupoids. In-

ternational Mathematical Forum, 1(9-12):563–576,

2006.

J. Goguen. L-fuzzy sets. J. Math. Anal. Appl., 18:145–174,

1967.

T. Janhunen, E. Oikarinen, H. Tompits, and S. Woltran.

Modularity aspects of disjunctive stable models. In

C. Baral, G. Brewka, and J. S. Schlipf, editors, LP-

NMR, volume 4483 of Lecture Notes in Computer Sci-

ence, pages 175–187. Springer, 2007.

F. Klawonn. Fuzzy points, fuzzy relations and fuzzy func-

tion. In V. Nov

ak and I. Perﬁlieva, editors, Discover-

ing World with Fuzzy Logic, pages 431–453. Physica-

Verlag, 2000.

J. Mart

ınez, G. Guti

errez, I. P. de Guzm

an, and P. Cordero.

Generalizations of lattices via non-deterministic oper-

ators. Discrete Math., 295(1-3):107–141, 2005.

J. Medina, M. Ojeda-Aciego, and J. Ruiz-Calvi

no. Fuzzy

logic programming via multilattices. Fuzzy Sets and

Systems, 158(6):674–688, 2007.

E. Oikarinen and T. Janhunen. A translation-based approach

to the veriﬁcation of modular equivalence. Journal

of Logic and Computation, 2008. Advance Access

published.

T. Petkovi

c. Congruences and homomorphisms of fuzzy au-

tomata. Fuzzy Sets and Systems, 157:444–458, 2006.

ON CONGRUENCES AND HOMOMORPHISMS ON SOME NON-DETERMINISTIC ALGEBRAS