=Paper=
{{Paper
|id=None
|storemode=property
|title=Concept Lattices in Fuzzy Relation Equations
|pdfUrl=https://ceur-ws.org/Vol-959/paper5.pdf
|volume=Vol-959
|dblpUrl=https://dblp.org/rec/conf/cla/DiazM11
}}
==Concept Lattices in Fuzzy Relation Equations==
https://ceur-ws.org/Vol-959/paper5.pdf
Concept lattices in fuzzy relation equations?
Juan Carlos Dı́az and Jesús Medina??
Department of Mathematics. University of Cádiz
Email: {juancarlos.diaz,jesus.medina}@uca.es
Abstract. Fuzzy relation equations are used to investigate theoretical
and applicational aspects of fuzzy set theory, e.g., approximate reasoning,
time series forecast, decision making and fuzzy control, etc.. This paper
relates these equations to a particular kind of concept lattices.
1 Introduction
Recently, multi-adjoint property-oriented concept lattices have been introduced
in [16] as a generalization of property-oriented concept lattices [10,11] to a fuzzy
environment. These concept lattices are a new point of view of rough set the-
ory [23] that considers two different sets: the set of objects and the set of at-
tributes.
On the other hand, fuzzy relation equations, introduced by E. Sanchez [28],
are associated to the composition of fuzzy relations and have been used to in-
vestigate theoretical and applicational aspects of fuzzy set theory [22], e.g., ap-
proximate reasoning, time series forecast, decision making, fuzzy control, as an
appropriate tool for handling and modeling of nonprobabilistic form of uncer-
tainty, etc. Many papers have investigated the capacity to solve (systems) of
fuzzy relation equations, e.g., in [1, 8, 9, 25, 26].
In this paper, the multi-adjoint relation equations are presented as a general-
ization of the fuzzy relation equations [24,28]. This general environment inherits
the properties of the multi-adjoint philosophy, consequently, e.g., several con-
junctors and residuated implications defined on general carriers as lattice struc-
tures can be used, which provide more flexibility in order to relate the variables
considered in the system.
Moreover, multi-adjoint property-oriented concept lattices and systems of
multi-adjoint relation equations have been related in order to obtain results that
ensure the existence of solutions in these systems. These definitions and results
are illustrated by a toy example to improve the readability and comprehension
of the paper.
Among all concept lattice frameworks, we have related the multi-adjoint
property-oriented concept lattices to the systems of multi-adjoint relation equa-
tions, e.g., the extension and intension operators of this concept lattice can be
?
Partially supported by the Spanish Science Ministry TIN2009-14562-C05-03 and by
Junta de Andalucı́a project P09-FQM-5233.
??
Corresponding author.
used to represent multi-adjoint relation equations, and, as a result, the solu-
tions of these systems of relation equations can be related to the concepts of the
corresponding concept lattice.
The more important consequence is that this relation provides that the prop-
erties given, e.g., in [2–4,12,14,17,18,27] can be applied to obtain many properties
of these systems. Indeed, it can be considered that the algorithms presented, e.g.,
in [5, 6, 15] obtain solutions for these systems.
The plan of this paper is the following: in Section 2 we will recall the multi-
adjoint property-oriented concept lattices as well as the basic operators used and
some properties; later, in Section 3, an example will be introduced to motivate
the multi-adjoint relation equations. Once these equations have been presented,
in Section 4 the multi-adjoint property-oriented concept lattices and the systems
of multi-adjoint relation equations will be related in order to obtain results which
ensure the existence of solutions in these systems; the paper ends with some
conclusions and prospects for future work.
2 Multi-adjoint property-oriented concept lattices
The basic operators in this environment are the adjoint triples, which are formed
by three mappings: a non-commutativity conjunctor and two residuated impli-
cations [13], which satisfy the well-known adjoint property.
Definition 1. Let (P1 , ≤1 ), (P2 , ≤2 ), (P3 , ≤3 ) be posets and & : P1 × P2 → P3 ,
. : P3 × P2 → P1 , - : P3 × P1 → P2 be mappings, then (&, ., -) is an adjoint
triple with respect to P1 , P2 , P3 if:
1. & is order-preserving in both arguments.
2. . and - are order-preserving on the first argument1 and order-reversing
on the second argument.
3. x ≤1 z . y iff x & y ≤3 z iff y ≤2 z - x, where x ∈ P1 , y ∈ P2
and z ∈ P3 .
Example of adjoint triples are the Gödel, product and Lukasiewicz t-norms
together with their residuated implications.
Example 1. Since the Gödel, product and Lukasiewicz t-norms are commuta-
tive, the residuated implications satisfy that .G =-G , .P =-P and .L =-L .
Therefore, the Gödel, product and Lukasiewicz adjoint triples are defined on
[0, 1] as:
&P (x, y) = x · y z -P x = min(1, z/x)
(
1 if x ≤ z
&G (x, y) = min(x, y) z -G x =
z otherwise
&L (x, y) = max(0, x + y − 1) z -G x = min(1, 1 − x + z)
1
Note that the antecedent will be evaluated on the right side, while the consequent
will be evaluated on the left side, as in logic programming framework.
2
In [19] more general examples of adjoint triples are given.
The basic structure, which allows the existence of several adjoint triples for
a given triplet of lattices, is the multi-adjoint property-oriented frame.
Definition 2. Given two complete lattices (L1 , 1 ) and (L2 , 2 ), a poset (P, ≤)
and adjoint triples with respect to P, L2 , L1 , (&i , .i , -i ), for all i = 1, . . . , l, a
multi-adjoint property-oriented frame is the tuple
(L1 , L2 , P, 1 , 2 , ≤, &1 , .1 , -1 , . . . , &l , .l , -l )
Multi-adjoint property-oriented frames are denoted as (L1 , L2 , P, &1 , . . . , &l ).
Note that the notation is similar to a multi-adjoint frame [18], although the
adjoint triples are defined on different carriers.
The definition of context is analogous to the one given in [18].
Definition 3. Let (L1 , L2 , P, &1 , . . . , &l ) be a multi-adjoint property-oriented
frame. A context is a tuple (A, B, R, σ) such that A and B are non-empty sets
(usually interpreted as attributes and objects, respectively), R is a P -fuzzy rela-
tion R : A × B → P and σ : B → {1, . . . , l} is a mapping which associates any
element in B with some particular adjoint triple in the frame.2
From now on, we will fix a multi-adjoint property-oriented frame and context,
(L1 , L2 , P, &1 , . . . , &l ), (A, B, R, σ).
↓N
Now we define the following mappings ↑π : LB A
2 → L1 and : LA B
1 → L2 as
g ↑π (a) = sup{R(a, b) &σ(b) g(b) | b ∈ B} (1)
N
f ↓ (b) = inf{f (a) -σ(b) R(a, b) | a ∈ A} (2)
Clearly, these definitions3 generalize the classical possibility and necessity
operators [11] and they form an isotone Galois connection [16]. There are two
dual versions of the notion of Galois connetion. The most famous Galois connec-
tion, where the maps are order-reversing, is properly called Galois connection,
and the other in which the maps are order-preserving, will be called isotone Ga-
lois connection. In order to make this contribution self-contained, we recall their
formal definitions:
Let (P1 , ≤1 ) and (P2 , ≤2 ) be posets, and ↓ : P1 → P2 , ↑ : P2 → P1 mappings,
the pair (↑ , ↓ ) forms a Galois connection between P1 and P2 if and only if: ↑ and
↓
are order-reversing; x ≤1 x↓↑ , for all x ∈ P1 , and y ≤2 y ↑↓ , for all y ∈ P2 .
The one we adopt here is the dual definition: Let (P1 , ≤1 ) and (P2 , ≤2 ) be
posets, and ↓ : P1 → P2 , ↑ : P2 → P1 mappings, the pair (↑ , ↓ ) forms an isotone
Galois connection between P1 and P2 if and only if: ↑ and ↓ are order-preserving;
x ≤1 x↓↑ , for all x ∈ P1 , and y ↑↓ ≤2 y, for all y ∈ P2 .
2
A similar theory could be developed by considering a mapping τ : A → {1, . . . , l}
which associates any element in A with some particular adjoint triple in the frame.
3
From now on, to improve readability, we will write &b , -b instead of &σ(b) , -σ(b) .
3
A concept, in this environment, is a pair of mappings hg, f i, with g ∈ LB , f ∈
N
L , such that g ↑π = f and f ↓ = g, which will be called multi-adjoint property-
A
oriented formal concept. In that case, g is called the extension and f , the inten-
sion of the concept. The set of all these concepts will be denoted as MπN [16].
Definition 4. A multi-adjoint property-oriented concept lattice is the set
N
↑π
MπN = {hg, f i | g ∈ LB A
2 , f ∈ L1 and g = f, f ↓ = g}
in which the ordering is defined by hg1 , f1 i hg2 , f2 i iff g1 2 g2 (or equivalently
f1 1 f2 ).
The pair (MπN , ) is a complete lattice [16], which generalize the concept
lattice introduced in [7] to a fuzzy environment.
3 Multi-adjoint relation equations
This section begins with an example that motivates the definition of multi-
adjoint relation equations, which will be introduced later.
3.1 Multi-adjoint logic programming
A short summary of the main features of multi-adjoint languages will be pre-
sented. The reader is referred to [20, 21] for a complete formulation.
A language L contains propositional variables, constants, and a set of logical
connectives. In this fuzzy setting, the usual connectives are adjoint triples and
a number of aggregators.
The language L is interpreted on a (biresiduated) multi-adjoint lattice,4
hL, , .1 , -1 , &1 , . . . , .n , -n , &n i, which is a complete lattice L equipped with
a collection of adjoint triples h&i , .i , -i i, where each &i is a conjunctor in-
tended to provide a modus ponens-rule with respect to .i and -i .
A rule is a formula A .i B or A -i B, where A is a propositional symbol
(usually called the head) and B (which is called the body) is a formula built from
propositional symbols B1 , . . . , Bn (n ≥ 0), truth values of L and conjunctions,
disjunctions and aggregations. Rules with an empty body are called facts.
A multi-adjoint logic program is a set of pairs hR, αi, where R is a rule and
α is a value of L, which may express the confidence which the user of the system
has in the truth of the rule R. Note that the truth degrees in a given program
are expected to be assigned by an expert.
Example 2. Let us to consider a multi-adjoint lattice
h[0, 1], ≤, ←G , &G , ←P , &P , ∧L i
4
Note that a multi-adjoint lattice is a particular case of a multi-adjoint property-
oriented frame.
4
where &G and &P are the Gödel and product conjunctors, respectively, and
←G , ←P their corresponding residuated implications. Moreover, the Lukasie-
wicz conjunctor ∧L will be used in the program [13].
Given the set of variables (propositional symbols)
Π = {low oil, low water, rich mixture, overheating, noisy behaviour,
high fuel consumption}
the following set of multi-adjoint rules form a multi-adjoint program, which may
represent the behaviour of a motor.
hhigh fuel consumption ←G rich mixture ∧L low oil, 0.8i
hoverheating ←G low oil, 0.5i
hnoisy behaviour ←P rich mixture, 0.8i
hoverheating ←P low water, 0.9i
hnoisy behaviour ←G low oil, 1i
The usual procedural is to measure the levels of “oil”, “water” and “mix-
ture” of a specific motor, after that the values for low oil, low water and
rich mixture are obtained, which are represented in the program as facts, for
instance, the next ones can be added to the program:
hlow oil ←P >, 0.2i
hlow water ←P >, 0.2i
hrich mixture ←P >, 0.5i
Finally, the values for the rest of variables (propositional symbols) are com-
puted [20]. For instance, in order to attain the value for overheating(o, w), for
a level of oil, o, and water, w, the rules hoverheating ←G low oil, ϑ1 i and
hoverheating ←P low water, ϑ2 i are considered and its value is obtained as:
overheating(o, w) = (low oil(o) &G ϑ1 ) ∨ (low water(w) &P ϑ2 ) (3)
Now, the problem could be to recompute the weights of the rules from
experimental instances of the variables, that is, the values of overheating,
noisy behaviour and high fuel consumption are known for particular mea-
sures of low oil, low water and rich mixture.
Specifically, given the levels of oil, o1 , . . . , on , the levels of water, w1 , . . . , wn ,
and the measures of mixture, t1 , . . . , tn , we may experimentally know the values
of the variables: noisy behaviour(ti , oi ), high fuel consumption(ti , oi ) and
overheating(oi , wi ), for all i ∈ {1, . . . , n}.
Considering Equation (3), the unknown elements could be ϑ1 and ϑ2 instead
of overheating(o, w). Therefore, the problem now is to look for the values of
ϑ1 and ϑ2 , which solve the following system obtained after assuming the exper-
imental data for the propositional symbols, ov1 , o1 , w1 , . . . , ovn , on , wn .
overheating(ov1 ) = (low oil(o1 ) &G ϑ1 ) ∨ (low water(w1 ) &P ϑ2 )
.. .. .. ..
. . . .
overheating(ovn ) = (low oil(on ) &G ϑ1 ) ∨ (low water(wn ) &P ϑ2 )
5
This system can be interpreted as a system of fuzzy relation equations in
which several conjunctors, &G and &P , are assumed. Moreover, these conjunctors
could be neither non-commutative nor associative and defined in general lattices,
as permit the multi-adjoint framework.
Next sections introduce when these systems have solutions and a novel method
to obtain them using concept lattice theory.
3.2 Systems of multi-adjoint relation equations
The operators used in order to obtain the systems will be the generalization of the
sup-∗-composition, introduced in [29], and inf-→-composition, introduced in [1].
From now on, a multi-adjoint property-oriented frame, (L1 , L2 , P, &1 , . . . , &l )
will be fixed.
In the definition of a multi-adjoint relation equation an interesting mapping
σ : U → {1, . . . , l} will be considered, which relates each element in U to an
adjoint triple. This mapping will play a similar role as the one given in a multi-
adjoint context, defined in the previous section, for instance, this map provides
a partition of U in preference sets. A similar theory may be developed for V
instead of U .
Let U = {u1 , . . . , um } and V = {v1 , . . . , vn } be two universes, R ∈ L2 U ×V
an unknown fuzzy relation, σ : U → {1, . . . , l} a map that relates each element
in U to an adjoint triple, and K1 , . . . , Kn ∈ P U , D1 , . . . , Dn ∈ L1 V arbitrarily
chosen fuzzy subsets of the respective universes.
A system of multi-adjoint relation equations with sup-&-composition, is the
following system of equations
_
(Ki (u) &u R(u, v)) = Di (v), i ∈ {1, . . . , n} (4)
u∈U
where &u represents the adjoint conjunctor associated to u by σ, that is, if
σ(u) = (&s , .s , -s ), for s ∈ {1, . . . , l}, then &u is exactly &s .
If an element v of V is fixed and the elements Ki (uj ), R(uj , v) and Di (v) are
written as kij , xj and di , respectively, for each i ∈ {1, . . . , n}, j ∈ {1, . . . , m},
then System (4) can particularly be written as
k11 &u1 x1 ∨ · · · ∨ k1m &um xm = d1
.. .. .. .. (5)
. . . .
kn1 &u1 x1 ∨ · · · ∨ knm &um xm = dn
where kij and di are known and xj must be obtained.
Hence, for each v ∈ V , if we solve System (5), then we obtain a “column”
of R (i.e. the elements R(uj , v), with j ∈ {1, . . . , m}), thus, solving n similar
systems, one for each v ∈ V , the unknown relation R is obtained.
Example 3. Assuming Example 2, in this case, we will try to solve the problem
about to obtain the weights associated to the rules from particular observed data
for the propositional symbols.
6
The propositional symbols (variables) will be written in short as: hfc, nb,
oh, rm, lo and lw, and the measures of particular cases of the behaviour of the
motor will be: hi , ni , ovi , ri , oi , wi , for hfc, nb, oh, rm, lo and lw, respectively,
in each case i, with i ∈ {1, 2, 3}.
For instance, the next system associated to overheating is obtained from
the computation provided in Example 2.
oh(ov1 ) = (lo(o1 ) &G ϑoh oh
lo ) ∨ (lw(w1 ) &P ϑlw )
oh(ov2 ) = (lo(o2 ) &G ϑoh oh
lo ) ∨ (lw(w2 ) &P ϑlw )
oh(ov3 ) = (lo(o3 ) &G ϑoh oh
lo ) ∨ (lw(w3 ) &P ϑlw )
where ϑoh oh
lo and ϑlw are the weights associated to the rules with head oh. Similar
systems can be obtained to high fuel consumption and noisy behaviour.
Assuming the multi-adjoint frame with carrier L = [0, 1] and the Gödel and
product triples, these systems are particular systems of multi-adjoint relational
equations. The corresponding context is formed by the sets U = {rm, lo, lw, rm∧L
lo}, V = {hfc, nb, oh}; the mapping σ that relates the elements lo, rm ∧L lo to
the Gödel triple, and rm, lw to the product triple; the mappings K1 , . . . , Kn ∈
P U , defined as the values given by the propositional symbols in U on the ex-
perimental data, for instance, if u = lo, then K1 (lo) = lo(o1 ), . . . , Kn (lo) =
lo(on ); and the mappings D1 , . . . , Dn ∈ L1 V , defined analogously, for instance,
if v = rm, then D1 (rm) = rm(r1 ), . . . , Dn (rm) = rm(rn ).
Finally, the unknown fuzzy relation R ∈ L2 U ×V is formed by the weights of
the rules in the program.
In the system above, oh has been the element v ∈ V fixed. Moreover, as
there do not exist rules with body rm and rm ∧L lo, that is, the weights for
that hypothetical rules are 0, then the terms (rm(ri ) &G 0 = 0 and (rm(ri ) ∧L
lo(oi ) &P 0 = 0 do not appear.
Its counterpart is a system of multi-adjoint relation equations with inf---
composition, that is,
^
(R(u, v) -uj Kj∗ (v)) = Ej (u), j ∈ {1, . . . , m} (6)
v∈V
considered with respect to unknown fuzzy relation R ∈ L1 U ×V , and where
K1∗ , . . . , Km
∗
∈ P V and E1 , . . . , Em ∈ L2 U . Note that -uj represents the corre-
sponding adjoint implication associated to uj by σ, that is, if σ(uj ) = (&s , .s
, -s ), for s ∈ {1, . . . , l}, then -uj is exactly -s . Remark that in System 6,
the implication -uj does not depend of the element u, but of j. Hence, the
implications used in each equation of the system are the same.
If an element u of U is fixed, fuzzy subsets K1∗ , . . . , Km∗
∈ P V , E1 , . . . , Em ∈
U ∗
L2 are assumed, such that Kj (vi ) = kij , R(u, vi ) = yi and Ej (u) = ej , for each
i ∈ {1, . . . , n}, j ∈ {1, . . . , m}, then System (6) can particularly be written as
y1 -u1 k11 ∧ · · · ∧ yn -u1 kn1 = e1
.. .. .. .. (7)
. . . .
y1 -um k1m ∧ · · · ∧ yn -um knm = em
7
Therefore, for each u ∈ U , we obtain a “row” of R (i.e. the elements R(u, vi ), with
i ∈ {1, . . . , n}), consequently, solving m similar systems, the unknown relation
R is obtained.
Systems (5) and (7) have the same goal, searching for the unknown relation
R although the mechanism is different.
Analyzing these systems, we have that the left side of these systems can be
represented by the mappings CK : Lm n n m
2 → L1 , IK ∗ : L1 → L2 , defined as:
CK (x̄)i = ki1 &u1 x1 ∨ · · · ∨ kim &um xm , for all i ∈ {1, . . . , n} (8)
IK ∗ (ȳ)j = y1 -uj k1j ∧ · · · ∧ yn -uj knj , for all j ∈ {1, . . . , m} (9)
where x̄ = (x1 , . . . , xm ) ∈ Lm n
2 , ȳ = (y1 , . . . , yn ) ∈ L1 , and CK (x̄)i , IK ∗ (ȳ)j
are the components of CK (x̄), IK ∗ (ȳ), respectively, for each i ∈ {1, . . . , n} and
j ∈ {1, . . . , m}.
Hence, Systems (5) and (7) can be written as:
CK (x1 , . . . , xm ) = (d1 , . . . , dn ) (10)
IK ∗ (y1 , . . . , yn ) = (e1 , . . . , em ) (11)
respectively.
4 Relation between multi-adjoint property-oriented
concept lattices and multi-adjoint relation equation
This section shows that Systems (5) and (7) can be interpreted in a multi-
adjoint property-oriented concept lattice. And so, the properties given to the
N
isotone Galois connection ↑π and ↓ , as well as to the complete lattice MπN can
be used in the resolution of these systems.
First of all, the environment must be fixed. Hence, a multi-adjoint context
(A, B, S, σ) will be considered, such that A = V 0 , B = U , where V 0 has the same
cardinality as V , σ will be the mapping given by the systems and S : A × B → P
is defined as S(vi0 , uj ) = kij . Note that A = V 0 is related to the mappings Ki ,
since S(vi0 , uj ) = kij = Ki (uj );
Now, we will prove that the mappings defined at the end of the previous
section are related to the isotone Galois connection. Given µ ∈ LB 2 , such that
µ(uj ) = xj , for all j ∈ {1, . . . , m}, the following equalities are obtained, for each
i ∈ {1, . . . , n}:
CK (x̄)i = ki1 &u1 x1 ∨ · · · ∨ kim &um xm
= S(vi0 , u1 ) &u1 µ(u1 ) ∨ · · · ∨ S(vi0 , um ) &um µ(um )
= sup{S(vi0 , uj ) &uj µ(uj ) | j ∈ {1, . . . , m}}
= µ↑π (vi0 )
↑π
Therefore, the mapping CK : Lm n
2 → L1 is equivalent to the mapping : LB
2 →
A m B
L1 , where an element x̄ in L2 can be interpreted as a map µ in L2 , such that
8
µ(uj ) = xj , for all j ∈ {1, . . . , m}, and the element CK (x̄) as the mapping µ↑π ,
such that µ↑π (vi0 ) = CK (x̄)i , for all i ∈ {1, . . . , n}.
An analogy can be developed applying the above procedure to mappings IK ∗
N
↓N
and ↓ , obtaining that the mappings IK ∗ : Ln1 → Lm 2 and : LA B
1 → L2 are
equivalent.
As a consequence, the following result holds:
Theorem 1. The mappings CK : Lm n n m
2 → L1 , IK ∗ : L1 → L2 , establish an iso-
m m
tone Galois connection. Therefore, IK ∗ ◦ CK : L2 → L2 is a closure operator
and CK ◦ IK ∗ : Ln1 → Ln1 is an interior operator.
As (CA , IK ∗ ) is an isotone Galois connection, any result about the solvability
of one system has its dual counterpart.
The following result explains when these systems can be solved and how a
solution can be obtained.
N
Theorem 2. System (5) can be solved if and only if hλ↓d¯ , λd¯i is a concept
of MπN , where λd¯ : A = {v1 , . . . , vn } → L1 , defined as λd¯(vi ) = di , for all
N
i ∈ {1, . . . , n}. Moreover, if System (5) has a solution, then λ↓d¯ is the greatest
solution of the system.
Similarly, System (7) can be solved if and only if hµē , µē↑π i is a concept of
MπN , where µē : B = {u1 , . . . , um } → L2 , defined as µē (uj ) = ej , for all j ∈
{1, . . . , m}. Furthermore, if System (7) has a solution, then µ↑ē π is the smallest
solution of the system.
The main contribution of the relation introduced in this paper is not only the
above consequences, but a lot of other properties for Systems (5) and (7) that
can be stabilized from the results proved, for example, in [2–4, 12, 14, 17, 18, 27].
Next example studies the system of multi-adjoint relation equations presented
in Example 3.
Example 4. The aim will be to solve a small system in order to improve the
understanding of the method. In the environment of Example 3, the following
system will be solved assuming the experimental data: oh(ov1 ) = 0.5, lo(o1 ) =
0.3, lw(w1 ) = 0.3, oh(ov2 ) = 0.7, lo(o2 ) = 0.6, lw(w2 ) = 0.8, oh(ov3 ) = 0.4,
lo(o3 ) = 0.5, lw(w3 ) = 0.2.
oh(ov1 ) = (lo(o1 ) &G ϑoh oh
lo ) ∨ (lw(w1 ) &P ϑlw )
oh(ov2 ) = (lo(o2 ) &G ϑoh oh
lo ) ∨ (lw(w2 ) &P ϑlw )
oh(ov3 ) = (lo(o3 ) &G ϑoh oh
lo ) ∨ (lw(w3 ) &P ϑlw )
where ϑoh oh
lo and ϑlw are the variables.
The context is: A = V 0 = {1, 2, 3}, the set of observations, B = U = {lo, lw},
σ associates the propositional symbol lo to the Gödel triple and lw to the
product triple. The relation S : A × B → [0, 1] is defined in Table 1.
Therefore, considering the mapping λoh : A → [0, 1] associated to the values
of overheating in each experimental case, that is λoh (1) = 0.5, λoh (2) = 0.7,
9
Table 1. Relation S.
low oil low water
1 0.3 0.3
2 0.6 0.8
3 0.5 0.2
and λoh (3) = 0.4; and the mapping CK : [0, 1]2 → [0, 1]3 , defined in Equation (8),
the system above can be written as
CK (ϑoh oh
lo , ϑlw ) = λoh
Since, by the comment above, there exists µ ∈ [0, 1]B , such that CK (ϑoh oh
lo , ϑlw ) =
↑π B ↑π
µ , the goal will be to attain the mapping µ ∈ [0, 1] , such that µ = λoh ,
N
which can be found if and only if ((λoh )↓ , λoh ) is a multi-adjoint property-
oriented concept in the considered context, by Theorem 2.
N
First of all, we compute (λoh )↓ .
N
(λoh )↓ (lo) = inf{λoh (1) -G S(1, lo), λoh (2) -G S(2, lo), λoh (3) -G S(3, lo)}
= inf{0.5 -G 0.3, 0.7 -G 0.6, 0.4 -G 0.5}
= inf{1, 1, 0.4} = 0.4
↓N
(λoh ) (lw) = inf{0.5 -P 0.3, 0.7 -P 0.8, 0.4 -P 0.2}
= inf{1, 0.875, 1} = 0.875
N
Now, the mapping (λoh )↓ ↑π is obtained.
N N N
(λoh )↓ ↑π (1) = sup{S(1, lo) &G (λoh )↓ (lo), S(1, lw) &P (λoh )↓ (lw)}
= sup{0.3 &G 0.4, 0.3 &P 0.875}
= sup{0.3, 0.2625} = 0.3
↓N ↑π
(λoh ) (2) = sup{0.6 &G 0.4, 0.8 &P 0.875} = 0.7
↓N ↑π
(λoh ) (3) = sup{0.5 &G 0.4, 0.2 &P 0.875} = 0.4
N
Therefore, ((λoh )↓ , λoh ) is not a multi-adjoint property-oriented concept and
thus, the considered system has no solution, although if the experimental value
for oh had been 0.3 instead of 0.5, the system would have had a solution.
These changes could be considered in several applications where noisy vari-
ables exist and their values can be conveniently changed to obtain approximate
solutions for the systems. Thus, if the experimental data for overheating are
oh(ov1 ) = 0.3, oh(ov2 ) = 0.7 and oh(ov2 ) = 0.4, then the original system will
have at least one solution and the values ϑoh oh
lo , ϑlw will be 0.4, 0.875, respectively
for a solution. Consequently, the truth for the first rule is lower than for the
second or it might be thought that it is more determinant in obtaining higher
10
values for lw than for lo. Another possibility is to consider that this conclusion
about the certainty of the rules is not correct, in which case another adjoint
triple might be associate to lo.
As a result, the properties introduced in several fuzzy formal concept anal-
ysis frameworks can be applied in order to obtain solutions of fuzzy relation
equations, as well as in the multi-adjoint general framework.
Furthermore, in order to obtain the solutions of Systems (5) and (7), the
algorithms developed, e.g., in [5, 6, 15], can be used.
5 Conclusions and future work
Multi-adjoint relation equations have been presented that generalize the existing
definitions presented at this time. In this general environment, different conjunc-
tors and residuated implications can be used, which provide more flexibility in
order to relate the variables considered in the system.
A toy example has been introduced in the paper in order to improve its
readability and reduce the complexity of the definitions and results.
As a consequence of the results presented in this paper, several of the prop-
erties provided, e.g., in [2–4, 12, 14, 17, 18, 27], can be used to obtain additional
characteristics of these systems.
In the future, we will apply the results provided in the fuzzy formal con-
cept analysis environments to the general systems of fuzzy relational equations
presented here.
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