7. Generalized extremal lattices with maximum breadth
7.4. Mixed generators
We will frequently deal with at least two incidence relations. Therefore, from now on we denote derivation by writing the incidence relation in a superscript. Generally, the letterI will denote the incidence relation of some “original” contextK, whereasJwill mean the incidence relation of the wideningKgØm.
The main mathematical object which we use to prove Claim 7.2.3 is a notion which has two aspects having opposite characteristics. Informally speaking: on the one hand, the notion which we introduce resembles an extent because it will be maximal with respect to some subset. On the other hand, it will be like a minimal generator too, in that it will be minimal regarding some other subset. Now, we progressively make these thoughts more concrete. Consider some arbitrary set of objectsSof some formal context. The definition of minimal generators, together with monotonicity of the closure operator yields:
Sis a minimal generator ô
(
Sztgu)
I I ‰SI I for each objectginS. (7.1) In an antipodal manner, the following equivalence dictates whetherSis an extent:Sis an extent ô
(
S Y tgu)
I I ‰SI I for each objectgnot inS. (7.2) One can see the equivalence in 7.2 as stating that an extent is a “maximal generator” (actually, it is the unique) of a closure.If we previously fix some arbitrary subsetR Ď Gand require condition in (7.1) to be valid only for objectsg P R and the second condition (in (7.2)) to hold only for objects
7.4. Mixed generators 87
7.4.1 Definition (mixed generator) LetR Ď Gbe fixed. A setS Ď Gis called aR-mixed generator (of the extentSI I) if, for everyg P G, both implications below hold:
i
)
g P(
S X R)
ñ(
Sztgu)
I I ‰SI Iii
)
g R(
S Y R)
ñ(
S Y tgu)
I I ‰SI I.♦
One valid question is whether every extent has a mixed generator. Under the condition that the object set is finite, Proposition 7.4.2 will answer this in the affirmative.
Note thatG-mixed generators are minimal generators and thatH-mixed generators are extents. We ocasionally refer to aR-mixed generator simply by mixed generator or by mixgen if there is no possibility for ambiguity.
We are particularly interested in the case whenRis the set of objects not having some fixed attribute, that is, whenR
=
GzmI. For this reason, we set the notationmI=
GzmI. Note that, in this case, the setRis precisely the set of objects whose derivations are changed by the widening operation.To visualize examples of mixed generators, consider Figure 7.6, which represents the context of Figure 7.3 using the same principle as Figure 7.4 (i. e. representing a clarified context through the hypergraph of its co-object-intents). Further, setR
=
mI=
tg, hu. In Figure 7.6, the two objects belonging toRare (represented by) ellipses whereas the other three objects (the ones not inR) are closed polygonal curves with rounded corners. The setS
=
th, i, j, kumay be verified as being a mixed generator, since the removal of any element belonging toS X Rcauses its closure (equivalently, its derivation) to change and there is no element inGzRwhich can be added toSwithout changing its closure. Note thatSis not an extent. Similarly,tg, h, i, j, kuis an extent but not a mixed generator. Lastly,ti, ju is both an extent and a mixed generator, whereas the setth, juis neither an extent nor a mixed generator. g h i j k m n o p qFigure 7.6.: Hypergraph of co-object-intents of the context in Figure 7.3.
In the next subsection, we prove elementary but necessary properties of mixed generators which will allow us to effectively work with them.
88 7. Generalized extremal lattices with maximum breadth
7.4.1. Elementary properties of mixed generators
We begin this subsection by showing the general existence of mixed generators, provided that the object set is finite. This is a particular case of the statement below, because an extentSalways satisfies conditionii
)
of mixgens. In other words, it satisfies the hypothesis of the proposition below.7.4.2 Proposition IfSis a finite set satisfying conditionii
)
in the definition of mixed generators,thenScontains a mixed generator ofSI I.
Proof Conditionii
)
forSis equivalent togI ĞSIfor eachg P Gz(
S Y R)
. IfSalso satisfies condition i)
, the claim follows trivially. Otherwise, pick some object g P S X Rsuch that(
Sztgu)
I=
SIand defineT :=
Sztgu. Note thatT Y RandS Y Rare the same set, implyingGz(
T Y R) =
Gz(
S Y R)
. SinceTI=
SI, we have that conditionii)
for mixed generators still holds forT. We repeat the removal of such an object until conditioni)
isfulfilled. ˝
Any mixed generatorShas a dual aspect by definition, in thatShas an extensional part and a minimal part. We mean the first to beS X R, whereas the latter isSzR. The next proposition shows an intuitive fact, namely, that the extensional parts of two mixed generators of the same extent must coincide.
7.4.3 Proposition LetSandTbe mixed generators withSI I
=
TI I. Then,SzR=
TzR.Proof Letg P SzR. Then,gI ĚSI
=
TI, implying(
T Y tgu)
I I=
TI I. Using the contrapo- sition of conditionii)
in the definition of mixgens gives thatg P T Y R. Because ofg R R, it follows thatg P T. The dual containment, i. e.TzR Ď SzR, is established in the same way.˝Proposition 7.4.4 describes which mixed generators are extents as well.
7.4.4 Proposition Let
(
G, M, I)
be a context andSbe aR-mixed generator. Then,Sis an extentif and only if
(
SI IzS)
XR=
H.Proof The direct implication is clear since SI IzS
=
HwheneverSis an extent. For the converse, we prove the contraposition. Suppose thatSis not an extent and takeg P SI IzS. Note that g P SI I implies
(
S Y tgu)
I I=
SI I. Conditionii)
of the definition of mixgens forces, therefore, thatg P S Y R. Because ofg R S, it holds thatg P R. ˝The following proposition is technical, but its usefulness shall become more clear later in the context of Proposition 7.5.3.
7.4.5 Proposition Let
(
G, M, I)
be a context,R=
m I for some attributemandS Ď Gbe a7.5. Describing intents of the wideningKgØm 89