=Paper=
{{Paper
|id=None
|storemode=property
|title=Using Intensifying Hedges to Reduce Size of Multi-adjoint Concept Lattices with Heterogeneous Conjunctors
|pdfUrl=https://ceur-ws.org/Vol-972/paper21.pdf
|volume=Vol-972
|dblpUrl=https://dblp.org/rec/conf/cla/KonecnyMO12
}}
==Using Intensifying Hedges to Reduce Size of Multi-adjoint Concept Lattices with Heterogeneous Conjunctors==
https://ceur-ws.org/Vol-972/paper21.pdf
Using intensifying hedges to reduce
size of multi-adjoint concept lattices
with heterogeneous conjunctors
Jan Konecny1 , Jesús Medina2 , and Manuel Ojeda-Aciego3
1
Dept. Computer Science, Palacky University, Olomouc, Czech Republic⋆
2
Dept. Mathematics, University of Cádiz, Spain⋆⋆
3
Dept. Applied Mathematics, University of Málaga, Spain⋆ ⋆ ⋆
Abstract. In this work we focus on the use of intensifying hedges as a
tool to reduce the size of the recently introduced multi-adjoint concept
lattices with heterogeneous conjunctors.
1 Introduction and preliminaries
Formal concept analysis (FCA) is a very active topic for several research groups
throughout the world [6, 1, 3, 5, 7, 8, 10, 11]. In this work, the authors aim to
merge recent advances obtained in this area: on the one hand, the use of hedges
as structures which allow to modulate the size of fuzzy concept lattices [4] and,
on the other hand, the consideration of heterogeneous conjunctors in the general
approach to fuzzy FCA so-called multi-adjoint framework [9].
One of the key features of the latter approach is that some quasi-closure
operators arise which, although do not directly allow to prove the complete
lattice structure of the resulting set of concepts as usual, i.e. in terms of a
Galois connection, actually do provide means to manually build the operators
for suprema and infima of a set of concepts. The core notion in [9] is that of P -
connected pair of posets which, in some sense, turns out to be a more abstract
notion than a truth-stressing hedge. As a consequence of this observation, due to
Radim Belohlavek, we now focus on the use of the specific properties of hedges
in order to import some results related to the size of fuzzy concept lattices to
the more general framework of [9].
The structure of the paper is the following: in Section 2 the preliminary
definitions are introduced, interested readers will obtain further comment on
the intuitions underlying the definitions in the original papers [4, 9]; the main
results are presented in Section 3.
⋆
Partially supported by grant No. P202/10/0262 of the Czech Science Foundation.
⋆⋆
Partially supported by Spanish Ministry of Science and FEDER funds through
project TIN09-14562-C05-03 and Junta de Andalucı́a project P09-FQM-5233.
⋆⋆⋆
Partially supported by Spanish Ministry of Science and FEDER funds through
project TIN09-14562-C05-01 and Junta de Andalucı́a project P09-FQM-5233.
c 2012 by the paper authors. CLA 2012, pp. 245–256. Copying permitted only for
private and academic purposes. Volume published and copyrighted by its editors.
Local Proceedings in ISBN 978–84–695–5252–0,
Universidad de Málaga (Dept. Matemática Aplicada), Spain.
246 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
2 Preliminaries
In this section, we introduce the basic definitions and preliminary results which
will be used later in the core of this work.
Definition 1. Let (L, , ⊤, ⊥) be a complete lattice, a truth-stressing hedge in
L is a mapping ∗ : L → L satisfying, for each x, y ∈ L,
∗(⊤) = ⊤, (1)
∗(x) x, (2)
x y implies ∗ (x) ∗(y), (3)
∗(∗(x)) = ∗(x) (4)
fix(∗) denotes set of fixed points of ∗ in L, i.e. fix(∗) = {a ∈ L | ∗(a) = a}.
In [3, 4] truth-stressing hedges were used to decrease size of a concept lattice
(in fact, the truth-stressing hedges were defined on a residuated lattice).
Later in this work, we will need the following lemmas.
Lemma 1. Let (L, ) be a complete lattice, for any mapping ∗ : L → L satisfy-
ing (2), (3), and (4) we have, for each xi ∈ L,
_ _ ^ ^
∗(xi ) = ∗( ∗(xi )) and ∗ ( ∗(xi )) = ∗( xi ). (5)
i∈I i∈I i∈I i∈I
W
In addition, if we have xj = i∈I xi for some j ∈ I then
_ _
∗( xi ) = ∗(xi ). (6)
i∈I i∈I
V
Similarly, if we have xj = i∈I xi for some j ∈ I then
^ ^
∗( xi ) = ∗(xi ). (7)
i∈I i∈I
Lemma 2. (a) Let ∗ : L → L be a mapping satisfying (2), (3), and (4). Then
fix(∗) is a ∨-subsemilattice of L.
(b) Let K be a ∨-subsemilattice of L then the mapping ∗K : L → L defined by
W
∗K (x) = {y ∈ K | y ≤ x}
satisfies (2), (3), and (4).
(c) ∗fix(∗) = ∗ and fix(∗K ) = K.
By Lemma 2 the set fix(∗) of truth-stressing hedge ∗ is a ∨-subsemilattice.
Now we will introduce the basic notions of multi-adjoint concept lattices with
heterogeneous conjunctors, in order to show how both frameworks, hedges and
heterogeneous conjunctors, can be merged.
Using intensifying hedges to reduce size of multi-adjoint concept lattices 247
Firstly, let us introduced a bit of terminology: in the rest of this work we will
call a mapping ∗ : L → L satisfying (2), (3), and (4) an intensifying hedge,
following the terminology introduced in [2]. In terms of interior structures (L, ),
a mapping satisfying (2)–(4) is an interior operator on the lattice of truth degrees.
The two main notions on which multi-adjoint concept lattices with heteroge-
neous conjunctors is defined are given below: the P -connection between posets,
and the adjoint triples.
Definition 2. Given the posets (P1 , ≤1 ), (P2 , ≤2 ) and (P, ≤), we say that P1
and P2 are P -connected if there exist non-decreasing mappings ψ1 : P1 → P ,
φ1 : P → P1 , ψ2 : P2 → P and φ2 : P → P2 verifying that φ1 (ψ1 (x)) = x, and
φ2 (ψ2 (y)) = y, for all x ∈ P1 , y ∈ P2 .
Definition 3. Let (P1 , ≤1 ), (P2 , ≤2 ), (P3 , ≤3 ) be posets, and consider mappings
& : P1 × P2 → P3 , ւ : P3 × P2 → P1 , տ : P3 × P1 → P2 , then (&, ւ, տ) is an
adjoint triple with respect to P1 , P2 , P3 if: x ≤1 z ւ y iff x & y ≤3 z iff
y ≤2 z տ x, where x ∈ P1 , y ∈ P2 and z ∈ P3 .
From Lemma 2 we immediately obtain the following proposition:
Corollary 1. Consider the posets (P1 , ≤1 ), (P2 , ≤2 ) and (P, ≤), and assume
that L1 and L2 are P -connected, then:
(a) If ψ1 ◦ φ1 is contractive (i.e. satisfies (2)) then P1 is isomorphic to a ∨-
subsemilattice of P .
(b) If ∗ : P1 → P1 is an intensifying hedge (i.e. satisfies properties (2), (3), and
(4)) then the composition ψ1 ◦ ∗ ◦ φ1 : P → P is an intensifying hedge in
fix(ψ1 ◦ φ1 ).
Lemma 3. Let (L, ), (L1 , 1 ), (L2 , 2 ) be lattices and let (&, ւ, տ) be an ad-
joint triple. For a, ai ∈ L1 , b, bi ∈ L2 , we have
W W W W
i∈I
(ai & b) = ( 1i∈I a) & b and i∈I
(a & bi ) = a &( 2i∈I bi ) (8)
Definition 4. A multi-adjoint frame is a tuple
(L1 , L2 , P, &1 , ւ1 , տ1 , . . . , &n , ւn , տn )
where Li are complete lattices and P i a poset, such that (&i , ւi , տi ) is an
adjoint triple with respect to L1 , L2 , P for all i = 1, . . . , n.
Definition 5. Let (L1 , L2 , P, &1 , . . . , &n ) be a multi-adjoint frame, a multi-
adjoint 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, . . . , n} is a mapping which associates any
element in B with some particular adjoint triple in the frame.
248 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
Given a complete lattice (L, ) such that L1 and L2 are L-connected, a
multi-adjoint frame (L1 , L2 , P, &1 , . . . , &n ), and a context (A, B, R, σ), we can
cσ
define the mappings ↑cσ : LB → LA and ↓ : LA → LB defined for all g ∈ LB
and f ∈ LA as follows:
g ↑cσ (a) = ψ1 (inf{R(a, b) ւσ(b) φ2 (g(b)) | b ∈ B}) (9)
↓cσ
f (b) = ψ2 (inf{R(a, b) տσ(b) φ1 (f (a)) | a ∈ A}) (10)
The notion of concept is defined as usual. A concept is a pair hg, f i satisfying
cσ
g ∈ LB , f ∈ LA and that g ↑cσ = f and f ↓ = g.
Definition 6. Given the complete lattices (L1 , 1 ), (L2 , 2 ) and (L, ), where
L1 and L2 are L-connected, the set of multi-adjoint L-connected concepts asso-
ciated to a multi-adjoint frame (L1 , L2 , P, &1 , . . . , &n ) and context (A, B, R, σ)
is given by ML = {hg, f i | hg, f i is a concept}.
The main theorem of concept lattices in [9], proves that ML has the structure
of a complete lattice:
Theorem 1 ([9]). Given complete lattices (L1 , 1 ), (L2 , 2 ) and (L, ), where
L1 and L2 are L-connected, a context (A, B, R, σ), and a multi-adjoint frame
(L1 , L2 , L, &1 , . . . , &n ), the multi-adjoint L-connected concept lattice ML is ac-
tually a complete lattice with the meet and join operators f, g : ML ×ML → ML
defined below, for all hg1 , f1 i, hg2 , f2 i ∈ ML ,
c
hg1 , f1 i f hg2 , f2 i = hψ2 ◦ φ2 (g1 ∧ g2 ), (f1 ∨ f2 )↓ ↑c i
c
hg1 , f1 i g hg2 , f2 i = h(g1 ∨ g2 )↑c ↓ , ψ1 ◦ φ1 (f1 ∧ f2 )i
The order which corresponds to f and g is defined as
hg1 , f1 i hg2 , f2 i iff φ2 (g1 ) ≤ φ2 (g2 ) (iff φ1 (f2 ) ≤ φ1 (f1 ))
In what follows M denotes multi-adjoint L-connected concept lattice of given
context (A, B, R, σ). We will also omit subscript σ(b) and write just ւ instead
of ւσ(b) .
3 Reducing the size of multi-adjoint concept lattices
The size of the concept lattice M can be reduced either by a suitable selection
of a ∨-subsemilattice of L1 (and/or L2 ) and the use of a restriction of &. The
following proposition says that the selection of ∨-subsemilattices of L1 (resp. L2 )
yields a reduction of size of concept lattice and, moreover, preserves intents (or
extents) of the original concept lattice, meaning that each intent of the reduced
concept lattice is an intent of the original concept lattice.
Using intensifying hedges to reduce size of multi-adjoint concept lattices 249
Proposition 1. Let A = (L1 , L2 , P, &1 , . . . , &n ), A′ = (K1 , L2 , P, &′1 , . . . , &′n )
be multi-adjoint frames, s.t. K1 is a ∨-subsemilattice of L1 , and &′1 , . . . , &′n
are restrictions of &1 , . . . , &n to K1 × L2 and ψ1′ = ψ1 , ψ2′ = ψ2 , φ′2 = φ2 ,
φ′1 = ∗K1 ◦φ1 , where ∗K1 is the hedge associated to K1 as introduced in Lemma 2.
Then, Int(MA′ ) ⊆ Int(MA ) where Int(M) denotes the set of intents in M.
Proof (sketch). We have z տ′ x = z տ x, for each x ∈ K1 , z ∈ P , whence
′
f ↓ = f ↓ , for each f : A → ψ1 (K1 ) where ψ1 (K1 ) ∈ L is image of ψ1 (note that
ւ′ is well-defined since K1 is ∨-subsemilattice) and thus by Proposition 16 in
[9] Ext(MA′ ) ⊆ Ext(MA ). ⊔
⊓
Remark 1. One can state a dual proposition to Proposition 1 for intents. Let A =
(L1 , L2 , P, &1 , . . . , &n ), A′ = (L1 , K2 , P, &′1 , . . . , &′n ) be multi-adjoint frames,
s.t. K2 is a ∨-subsemilattice of L2 , and &′1 , . . . , &′n are restrictions of &1 , . . . , &n
to L1 × K2 and φ′2 = ∗K2 ◦ φ2 .
The following proposition says that by selection of ∨-subsemilattices of both
L1 and L2 we obtain a reduction of the size as well. However, the preservation
of intents (or extents) is lost.
Proposition 2. Let A = (L1 , L2 , L, &1 , . . . , &n ), A′ = (K1 , K2 , L, &′1 , . . . , &′n )
be multi-adjoint frames, s.t. K1 is a ∨-subsemilattice of L1 , K2 is a ∨-subsemi-
lattice of L2 , and &′1 , . . . , &′n are restrictions of &1 , . . . , &n to K1 × K2 , and
φ′1 = ∗K1 ◦ φ1 , φ′2 = ∗K2 ◦ φ2 . Then we have |MA′ | ≤ |MA |.
Proof (sketch). By applying Proposition 1 and Remark 1 we obtain the result.
⊔
⊓
In the next result we show how to generate new adjoint triples using hedges.
Lemma 4. Assume (&, ւ, տ) is an adjoint triple with respect to L1 , L2 , P ,
and ∗1 : L1 → L1 , ∗2 : L2 → L2 are hedges, then x &∗ y = ∗1 (x) & ∗2 (y) has two
residuated implications ւ∗ , տ∗ which form a new adjoint triple with respect to
L1 , L2 , P , if and only if the following equalities hold:
_
∗1 (z ւ ∗2 (y)) = ∗1 ( {x | x &∗ y ≤ z}) (11)
_
∗2 (z տ ∗1 (x)) = ∗2 ( {y | x &∗ y ≤ z}) (12)
Proof. “⇒”: Let (&∗ , ւ∗ , տ∗ ) be an adjoint triple. We have
x &∗ y ≤ z iff y 2 z տ∗ x
by definition. In particular, we obtain
∗1 (x) &∗ ∗2 (y) ≤ z iff ∗2 (y) 2 z տ∗ ∗1 (x)
and ∗1 (x) &∗ ∗2 (y) = ∗1 (∗1 (x)) & ∗2 (∗2 (y)) = ∗1 (x) & ∗2 (y) = x &∗ y. Hence, we
have
x &∗ y ≤ z iff ∗2 (y) 2 z տ∗ ∗1 (x)
250 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
From (3) and (4) we obtain that
∗2 (y) 2 z տ∗ ∗1 (x) implies ∗2 (y) 2 ∗2 (z տ∗ ∗1 (x))
and due to (2) we have
∗2 (y) 2 ∗2 (z տ∗ ∗1 (x)) implies ∗2 (y) 2 z տ∗ ∗1 (x)
Therefore, we have
x &∗ y ≤ z iff ∗2 (y) 2 ∗2 (z տ∗ ∗1 (x)). (13)
Analogously, we obtain
∗1 (x) & ∗2 (y) ≤ z iff ∗2 (y) 2 ∗2 (z տ ∗1 (x)) (14)
By setting y = (z տ ∗1 (x)), in Equation (13), and y = (z տ∗ ∗1 (x)), in Equa-
tion (15), we obtain equivalent inequalities ∗2 (z տ ∗1 (x)) ∗2 (z տ∗ ∗1 (x)),
∗2 (z տ ∗1 (x)) ∗2 (z տ∗ ∗1 (x)) respectively. Thus we have
∗2 (z տ ∗1 (x)) = ∗2 (z տ∗ ∗1 (x)).
Which is equal to (12). The first equation (11) can be obtained dually.
“⇐”: Assume (12) holds true. By properties of adjointness, to show that &∗
generates an adjoint triple we need to show that
R = {y | ∗1 (x) & ∗2 (y) ≤ z}
has a greatest element.
In the previous part, we proven that
∗1 (x) & ∗2 (y) ≤ z iff ∗2 (y) 2 ∗2 (z տ ∗1 (x)) (15)
hence R = {y | ∗2 (y) ∗2 (zWտ ∗1 (x))}. Now, if R has no greatest element, i.e.
W
R∈ / R, then we have ∗2 ( R) 6 ∗2 (z տ ∗1 (x)) which is a contradiction with
the assumption. By the contradiction we proved that R has a greatest element.
⊔
⊓
Proposition 3. Let A = (L1 , L2 , P, &1 , . . . , &n ) be a multi-adjoint frame ∗1 , ∗2
be hedges on L1 and L2 , respectively. Let A′ = (fix(∗1 ), fix(∗2 ), P, &′1 , . . . , &′n )
s.t. &′1 , . . . , &′n are restrictions of &1 , . . . , &n to fix(∗1 ) × fix(∗2 ), and φ′1 = ∗1 ◦
φ1 , φ′2 = ∗2 ◦φ2 . Let A∗ = (L1 , L2 , P, &∗1 , . . . , &∗n ) be a multi-adjoint frame where
&∗i is defined by a &∗i b = ∗1 (a) &i ∗2 (b), for all i ∈ {1, . . . , n}, and the conditions
in Lemma 4 are satisfied. Then (MA′ , ′ ) and (MA∗ , ∗ ) are isomorphic.
Proof. Let K = (A, B, R, σ) be a formal context, denote by ↑ , ↓ concept-forming
operators induced by K and A′ and denote by ⇑ , ⇓ concept-forming operators
induced by K and A∗ . Furthermore, denote compositions ψ1 ◦ ∗1 ◦ φ1 and ψ2 ◦
∗2 ◦ φ2 by •1 and •2 respectively.
Using intensifying hedges to reduce size of multi-adjoint concept lattices 251
For each mapping g : B → L we have
^
•1 (g ⇑ (a)) = •1 (ψ1 (R(a, b) ւ∗ φ2 (g(b))))
1
^
= ψ1 ∗1 (φ1 ψ1 (R(a, b) ւ∗ (φ2 (g(b)))))
^ _ 1
= ψ1 ∗1 ( {x | ∗1 (x) & ∗2 (φ2 (g(b))) ≤ R(a, b))})
1 1
(∆) ^ _
= ψ1 ( {∗1 (x) | ∗1 (x) & ∗2 (φ2 (g(b))) ≤ R(a, b))})
1 1
^ _
= ψ1 ( {x ∈ fix(∗1 ) | x & ∗2 (φ2 (g(b))) ≤ R(a, b))})
^1 1
= ψ1 (R(a, b) ւ′ ∗2 (φ2 (g(b))))
1
^
= ψ1 (R(a, b) ւ′ φ2 ψ2 ∗2 (φ2 g(b)))
^1
= ψ1 (R(a, b) ւ′ φ2 •2 (g(b)))
1
= (•2 ◦ g)↑ (a)
whereW(∆) is due to Lemma 1 (6) and the fact that & generates adjoint triple and
thus 1 {x | ∗1 (x) & ∗2 (φ2 (g(b))) ≤ R(a, b))}) has a greatest elements. Dually, we
have •2 ◦ (f ⇓ ) = (•1 ◦ f )↓ for each mapping f : A → L. From that we have
g ↑ = •1 ◦ (g ⇑ ) and f ↓ = •2 ◦ (f ⇓ )
for each g : B → fix(•2 ), f : A → fix(•1 ). As a result of the previous equalities,
we have that •2 is a surjective mapping Ext(MA∗ ) → Ext(MA′ ) and •1 is a
surjective mapping Int(MA∗ ) → Int(MA′ ). In addition, for g ∈ Ext(MA∗ ) we
have
^
•2 (g)⇑ (a) = ψ1 R(a, b) ւ∗ φ2 ψ2 ∗2 φ2 (g(b))
^1 _
= ψ1 {x | ∗1 (x) & ∗2 ∗2 (φ2 (g(b))) ≤ R(a, b)}
^1 _2
= ψ1 {x | ∗1 (x) & ∗2 (φ2 (g(b))) ≤ R(a, b)}
^1 2
= ψ1 R(a, b) ւ∗ φ2 (g(b)))
1
= g ⇑ (a)
and dually •1 (f )⇓ = f ⇓ . Putting it together, we have g = g ⇑⇓ = •1 (g ⇑ )⇓ =
•2 (g)↑⇓ showing that ↑⇓ is injective; whence •1 , •2 are bijections.
To show that •1 , •2 are order-preserving let hg1 , f1 i , hg2 , f2 i ∈ MA∗ . An
extent of hg1 , f1 i ∧ hg2 , f2 i is equal to ψ2 φ2 (g1 ∧ g2 ) by the main Theorem in [9].
For g1 , g2 ∈ Ext(MA∗ ) we have
•2 ψ2 φ2 (g1 ∧ g2 ) = ψ2 ∗2 φ2 ψ2 φ2 (g1 ∧ g2 )
= ψ2 ∗2 φ2 (g1 ∧ g2 )
= ψ2 ∗2 φ2 •2 (g1 ∧ g2 )
(∆)
= ψ2 φ′2 (•2 (g1 ) ∧ •2 (g2 ))
252 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
Equality (∆) is due to Corollary 1(b) since note that g1 , g2 are fixpoints of ψ2 ◦φ2 .
Now, let g1 , g2 ∈ Ext(MA′ ). We have
(ψ2 φ′2 (g1 ∧ g2 ))↑⇓ = (ψ2 ∗2 φ2 (g1 ∧ g2 ))↑⇓
= (•1 (g1 ∧ g2 ))↑⇓
= (•1 (g1↑↓ ∧ g2↑↓ ))↑⇓
= (•1 (•1 (g1↑⇓ ) ∧ •1 (g2↑⇓ ))↑⇓
(∆)
= (•1 •1 (g1↑⇓ ∧ g2↑⇓ ))↑⇓
= (•1 (g1↑⇓ ∧ g2↑⇓ ))↑⇓
= (g1↑⇓ ∧ g2↑⇓ )⇑⇓
(∇)
= ψ2 φ2 (g1↑⇓ ∧ g2↑⇓ )
Equality (∆) is due to Corollary 1(b) since g1 , g2 are fixpoints of ψ2 ◦φ2 ; equality
(∇) is due to [9, Lemma 21].
This proves that •1 , •2 , ↑⇓, and ↓⇑ are order-preserving. ⊔
⊓
Example 1. Consider the multi-adjoint frame depicted in Fig. 1 (structures are
the same as in [9, Example 3 (Fig. 2)] (where all &i ’s coincide). Figure 2 de-
picts a formal context with two objects and two attributes, together with their
associated multi-adjoint concept lattice.
Concept lattices with truth-stressing hedges
In this part, we follow the way in which the hedges are used in [4], i.e. we
generalize concept-forming operators using intensifying hedges. Then we show
how this is related to the theory described above.
We define the concept-forming operators as follows
^
g △ (a) = ψ1 R(a, b) ւ ∗2 (φ2 (g(b))),
1b∈B
^
f ▽ (b) = ψ2 R(a, b) տ ∗1 (φ1 (f (a))).
2a∈A
Note that this is not strictly the same approach as used in Proposition 3
since ւ and տ are residua of the original adjoint operators &, not the altered
operators &∗ . In fact, generally there is no base operation & such that (·) ւ ∗2 (·)
and (·) տ ∗1 (·) are its residua, since we do not generally have
x ≤ z ւ ∗2 (y) iff y ≤ z տ ∗1 (x)
for each x ∈ L1 , y ∈ L2 , z ∈ L.
Lemma 5. Assume (&, ւ, տ) is an adjoint triple, ∗1 , ∗2 are intensifying hedges,
and ւ⋄ , տ⋄ being defined as z ւ⋄ y = z ւ ∗2 y, and z տ⋄ x = z տ ∗1 x; then
Using intensifying hedges to reduce size of multi-adjoint concept lattices 253
v
δ
d t
u γ
b c z β
y
α
a x
abcd αβγδ xyz tuv xy z t uv
ψ1 x t u v ψ2 x y t v φ1 a b a b c d φ2 α β γ γ δ δ
ւαβγδ տa b c d
&αβγ δ t d d c c t δβ δ β
a xxxx u d ddd u δ δ δ δ
b xy v v v d ddb v δ δ γ γ
c xy y t x d aaa x δααα
d xy vu y d d ca y δβ γ β
z d aaa z δααα
Fig. 1. L1 (top left), L (top middle), L2 (top right), connection operators φ1 , φ2 , ψ1 , ψ2
(middle), adjoint triple (h&, տ, ւi) (bottom).
(v, v)(u, x)
12 (v, u)(u, t)
1uv (u, v)(v, x)
2vy
(v, y)(u, v)
(u, u)(v, t)
(u, y)(v, v)
Fig. 2. Multi-adjoint formal context with two objects and two attributes (left) and the
multi-adjoint concept lattice associated to the context (right).
254 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
δ
d
γ
b c β
α
a
(v, v)(x, x) (v, v)(x, x)
(x, t)(y, x) (v, x)(x, t) (x, v)(y, x) (v, x)(x, y)
(x, x)(t, t) (x, x)(v, v)
Fig. 3. Intensifying hedge on L1 (top left) and L2 (top right); concept lattices MA′
(bottom left), MA∗ (bottom right) of the formal context in Fig. 2; labels of nodes of
MA′ and MA∗ represent characteristic vectors of corresponding extents and intents.
Using intensifying hedges to reduce size of multi-adjoint concept lattices 255
ւ⋄ , տ⋄ are part of an adjoint triple with conjunctor &⋄ if and only if for all
x, y the following equality holds
x & ∗2 (y) = ∗1 (x) & y
and, in this case the previous value is the definition of &⋄ .
Proof. For all x, y, z, on the one hand, we have
∗1 (x) & y ≤ z iff y 2 z տ ∗1 (x) iff y 2 z տ⋄ x.
On the other hand, we have
x & ∗2 (y) ≤ z iff x 1 z ւ ∗2 (y) iff x 1 z ւ⋄ y.
Thus we have
y 2 z տ⋄ x iff x 1 z ւ⋄ y
is equivalent to
x & ∗2 (y) ≤ z iff ∗1 (x) & y ≤ z,
which is equivalent to x & ∗2 (y) = ∗1 (x) & y. ⊔
⊓
However, the concept-forming operators △, ▽ are in one-to-one correspon-
dence with concept-forming operators ↑, ↓ with restrictions of L1 and L2 to
subsemilattices fix(∗1 ) and fix(∗2 ):
^
•1 (g △ (a)) = •1 (ψ1 R(a, b) ւ ∗2 φ2 (g(b)))
b∈B
^
= ψ1 ∗1 ( R(a, b) ւ ∗2 (φ2 (g(b))))
1b∈B
^
= ψ1 ∗1 (R(a, b) ւ ∗2 (φ2 (g(b))))
1b∈B
^ _
= ψ1 ∗1 {x | x & ∗2 (φ2 (g(b))) ≤ R(a, b)}
1b∈B 1
^ _
= ψ1 {∗1 (x) | x & ∗2 (φ2 (g(b))) ≤ R(a, b)}
1b∈B 1
(∆) ^ _
= ψ1 {∗1 (x) | ∗1 (x) & ∗2 (φ2 (g(b))) ≤ R(a, b)}
1b∈B 1
^ _
= ψ1 {x ∈ fix(∗1 ) | x & ∗2 (φ2 (g(b))) ≤ R(a, b)}
1b∈B 1
^ _
= ψ1 {x ∈ fix(∗1 ) | x & φ′2 (g(b)) ≤ R(a, b)}
1b∈B 1
^
= ψ1 R(a, b) ւ φ′2 (g(b))
1b∈B
↑
= g (a)
where equality (∆) holds because each x satisfying x & y ≤ z satisfies ∗2 (x) & y ≤
z as well; and because each ∗2 (x) such that ∗2 (x) & y ≤ z there is x′ (explicitly,
∗2 (x)) with ∗2 (x′ ) = ∗2 (x) such that x′ & y ≤ z. Dually, one can show •2 (f ▽ ) =
f ↓.
256 Jan Konecny, Jesús Medina and Manuel Ojeda-Aciego
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