VENTAS POR LINEAS DE COEXITO
2. MARCOS DE REFERENCIA Y SISTEMA DE HIPÓTESIS
2.1 Marco contextual
In this subsection we provide an alternative description of Dep.
We introduce the category Cover which is essentially the arrow category of Relf. Its hom-sets admit a natural closure structure, so thatDepis the restriction ofCoverto closed morphisms.
This closure structure is really just a more detailed explanation of the maximumR-witnesses(R−,R+), revealing
that their construction is functorial in nature. Given what we already know, it is not hard to prove. However it is useful because it allows us to work with morphisms ‘modulo closure’ in a precise sense.
Motivation 4.4.1. Of particular importance is the following basic fact. Given a finite set of Dep-endomorphisms
{(Ra
−,Ra+)∶G → G∶a∈Σ}then:
(Ra1
− ,Ra+1)#⋯#(R−an,Ra+n) is the closure of (R−a1;⋯;Ra−n, R+an;⋯;Ra+1)
That is, we may use the usual relational composition in each component and then close once. The restriction to endomorphisms is unnecessary. The reason we emphasise it stems from our interest in nondeterministic acceptance of regular languages. Later on, the endomorphisms (Ra
−,Ra+) ∶ G → G will be viewed as the a-transitions of two
classical nondeterministic automata, one with states Gs (the ‘lower one’) and the other with states Gt (the ‘upper one’). These paired nondeterministic automata naturally accept a single regular language, using only the definition of Dep-composition. Then using the above fact, this language is precisely the language accepted by the lower nfa, or equivalently the reverse of the language accepted by the upper nfa. That is, our ‘categorical’ notion of language acceptance corresponds to the classical notion of nondeterministic acceptance. ∎
Definition 4.4.2 (The categoryCover). The objects ofCoverare the relations between finite sets i.e. the objects of
Dep. A morphism(Rl,Rr)∶G → His a pair of relations Rl⊆Gs×HsandRr⊆Ht×Gtsuch that:
Rl;H=G;R˘r
ThenidG ∶=(∆Gs,∆Gt)and composition is defined(Rl,Rr);(Sl,Sr)∶=(Rl;Sl,Sr;Rr). ∎
Definition 4.4.3 (Dep-morphism associated to a Cover-morphism). A Cover-morphism (Rl,Rr) ∶ G → H has an
associated Dep-morphism R∶G → H, namelyRl;H=∶R∶=G;R˘r. ∎
Note 4.4.4 (Coveris isomorphic to the arrow category ofRelf). Arguably the most natural category whose objects are relations between finite sets is thearrow category Arr(Relf)of the category of finite sets and relationsRelf. It is isomorphic toCoverby (i) reversing the type ofR+, and (ii) changing composition appropriately (use pairwise relational
composition). In fact, Arr(Relf)corresponds to the comma category IdRelf ↓IdRelf, whereas Cover corresponds to
IdRelf ↓(−)˘ ∶Rel
op
f → Relf. Small comma categories always arise as natural pullbacks in Cat, the category of small
Cover is a well-defined category by Note 4.4.4 above. Given (Rl,Rr) ∶ G → H s.t. Rl;H = G;R˘r, taking the relational converse yieldsRr; ˘G=H˘;R˘l i.e. aCover-morphism(Rr,Rl)∶H →˘ G˘. This defines a self-duality and acts in the same way as Dep’s self-duality (Definition4.1.12); we denote it by the same symbol.
Lemma 4.4.5(Self-duality ofCover). Coveris well-defined and self-dual via (−)∨∶Coverop→Cover,
G∨∶
=G˘ (Rl,Rr)∶G → H (Rl,Rr)∨∶=(Rr,Rl)∶H →˘ G˘
with witnessing natural isomorphism α∶IdCover ⇒(−)∨○((−)∨)op with action αG ∶=idG=(∆Gs,∆Gt).
Cover’s hom-sets admit a natural ordering i.e. pairwise inclusion. We now define a natural closure operator uniformly on each such poset.
Definition 4.4.6 (The posetCover(G,H)). For each pair of relations(G,H)we define the finite poset:
Cover(G,H)∶=(Cover(G,H),≤(G,H)) where (R1,R2)≤(G,H)(S1,S2)∶⇐⇒ R1⊆S1 andR2⊆S2.
This poset of morphisms admits a natural closure operatorclG,H∶Cover(G,H)→Cover(G,H)defined:
clG,H(Rl,Rr)∶=(R●l,R●r)where:
R●
l ∶={(gs, hs)∈Gs×Hs∶hs∈clH(Rl[gs])} R
●
r∶={(ht, gt)∈Ht×Gt∶gt∈clG˘(Rr[ht])} using the closure operatorsclH=H↓○H↑ andcl
˘ G=G˘
↓○G˘↑from Definition4.1.4. ∎
Each finite poset Cover(G,H) is actually a finite lattice: the bottom is (∅,∅)∶G → H and the join is pairwise
binary union (the meet structure is induced). We now prove that these closure operators are well-defined and construct the associated components (R−,R+). Furthermore they naturally interact with the self-duality and compositional
structure. Lemma 4.4.7.
1. For anyCover-morphism (Rl,Rr)∶G → H we have:
(Rl,Rr)≤(G,H)(R●l,R●r) Rl●;H=Rl;H G;R˘r=G;(R●r)˘
so thatclG,H(Rl,Rr)∶G → H is also aCover-morphism.
2. clG,H is a well-defined closure operator on the finite posetCover(G,H).
3. The closure of aCover-morphism (Rl,Rr)∶G → H can be described in the following three ways.
i. The components (R−,R+)of its associated Dep-morphism R.
ii. The pairwise union of allCover-morphisms(Sl,Sr)∶G → Hsuch that Sl;H=Rl;H.
iii. The pairwise union of all Cover-morphisms(Sl,Sr)∶G → Hsuch that G;S˘r=G;R˘l.
4. The closure operatorsclG,H commute withCover’s self-duality i.e.
clH∨,G∨((Rl,Rr)∨)=(clG,H(Rl,Rr))∨
5. The closure operatorsclG,H are well-behaved w.r.t.Cover-composition i.e.
clG,H(Rl,Rr)=clG,H(Sl,Sr)
clG,I((Rl,Rr);(Tl,Tr))=clG,I((Sl,Sr);(Tl,Tr)) clH,I(Sl,Sr)=clH,I(Tl,Tr)
clG,I((Rl,Rr);(Sl,Sr))=clG,I((Rl,Rr);(Tl,Tr))
Proof.
1. The left statement follows becauseclH and clG˘ are extensive. The central and right statement follow because for allgs∈Gs andht∈Ht,
R● l;H[gs] =H[R●l[gs]] =H[clH(Rl[gs])] =H↑○H↓○H↑(R l[gs]) =H↑(R l[gs]) by(↑↓↑) =Rl;H[gs] R● r; ˘G[ht] =G[R˘ ●r[ht]] =G˘↑(cl ˘ G(Rr[ht])) =G˘↑○G˘↓○G˘↑(R r[ht]) =G˘↑(R r[ht]) by(↑↓↑) =Rr; ˘G[ht]
ThenclG,H(Rl,Rr)is a well-definedCover-morphism using the fact that(Rl,Rr)is.
2. ThatclG,H is a well-defined function follows from the previous statement. That it is monotonic, extensive and
idempotent follows becauseclH andclG˘possess these properties.
3. Given aCover-morphism (Rl,Rr) letR be its associated Dep-morphism and(R−,R+)the latter’s associated
component relations. Then:
R−[gs] =H↓(R[gs]) by definition =H↓○(Rl;H)↑({g s}) by assumption =H↓○H↑○R↑ l({gs}) by(↑○) =clH(Rl[gs]) by definition =R● l[gs] by definition R+[ht] =G˘↓(R[h˘ t]) by definition =G˘↓○(Rr; ˘G)↑({h t}) by assumption =G˘↓○G˘↑○R↑ r({ht}) by(↑○) =clG˘(Rr[ht]) by definition =R● r[ht] by definition for allgs∈Gs andht∈Ht. This proves the first statement. By Lemma 4.1.10.2 we know that (R−,R+)is the
union of allCover-morphisms(Sl,Sr)∶G → Hsuch thatSl;H=R=G;S˘r. Then the second and third statement follow byRl;H=RandR=G;R˘l respectively.
4. Follows from the definitions:
clH∨,G∨((Rl,Rr)∨)=cl˘
H,G˘(Rr,Rl)=(R●r,R●l)=(R●l,R●r)∨=(clG,H(Rl,Rr))∨
5. To prove the first rule, assume we have Cover-morphisms (Rl,Rr),(Sl,Sr) ∶ G → H with the same closure, and also a Cover-morphism (Tl,Tr)∶ H → I. We need only show that (Rl;Tl)● = (Sl;Tl)● because the other component relation is uniquely determined. Then we calculate:
(Rl;Tl)●[gs] =clI(Rl;Tl[gs]) by definition =I↓○I↑○T↑ l ○R ↑ l({gs}) by definition =I↓○T↑(R l[gs]) sinceT =Tl;I =I↓○T↑○cl
H(Rl[gs]) sinceT a Dep-morphim =I↓○T↑○cl
H(Sl[gs]) sinceR●l =Sl● =(Sl;Tl)●[gs] reasoning in reverse
for allgs∈Gs. The second rule follows by dualising (using (4)), applying the first rule, and dualising again. Each closure operatorclG,H induces an equivalence relation on its respective hom-set i.e. the kernel:
kerclG,H⊆Cover(G,H)×Cover(G,H)
which relates those morphisms with the same closure. Then by Lemma4.4.7.5 these relations are collectively compatible with Cover-composition and thus induce a ‘quotient category’. We denote the composition of morphisms in this category by ‘#’ i.e. the same symbol we use to denoteDep-composition. This is warranted because these two categories are isomorphic.
Definition 4.4.8 (The categoryCover/cl). It has the same objects asCover, whereas its hom-sets are:
Cover/cl(G,H)∶=Cover(G,H)/kerclG,H
i.e. the equivalence classes ofCover-morphisms relative to kerclG,H. Let us denote the associated surjective canonical
maps byJ⋅KG,H∶Cover(G,H)↠Cover/cl(G,H). Then identity morphisms and composition are defined: idG∶=JidGKG,G=J(∆G
s,∆Gt)KG,G J(Rl,Rr)KG,H# J(Sl,Sr)KH,I∶=J(R1,Rl);(Sl,Sr)KG,I
We also define two identity-on-objects functors:
J⋅K∶Cover→Cover/cl JGK∶=G (Rl,Rr)∶G → H
J(Rl,Rr)K∶=J(Rl,Rr)KG,H∶G → H I∶Cover/cl→Dep IG∶=G J(Rl,Rr)KG,H∶G → H
R∶=Rl;H=G;R˘r∶G → H
and also the composite functor cl ∶=I○J⋅K∶Cover→Dep. Recalling thatDep-morphisms Rmay be identified with their associated components(R−,R+), we may abuse notation by equivalently defining:
cl(Rl,Rr)∶=clG,H(Rl,Rr)=(R−,R+)
∎
Theorem 4.4.9 (Depas a quotient category ofCover).
1. Cover/cl is a well-defined category andJ⋅K∶Cover→Cover/clis a well-defined functor. 2. I∶Cover/cl→Dep is a well-defined isomorphism of categories.
3. cl∶Cover→Dep is a well-defined functor and preserves the ordering on morphisms i.e.
(Rl,Rr)≤(G,H)(Sl,Sr) Ô⇒ R⊆S (or equivalently (R−,R+)≤G,H(S−,S+))
Proof.
1. Follows via Lemma4.4.7.5, also see section II.8 on ‘Quotient functors’ in MacClane’s book.
2. I’s action on objects and morphisms is well-defined, noting that elements of the same equivalence class induce the sameDep-morphism by definition. Concerning preservation of identity morphisms:
IidG =IJ(∆Gs,∆Gt)KG,G
=IJ(G−,G+)KG,G (∆Gs,∆Gt)aG-witness, Lemma4.4.7.3
=G−;G
=G
and regarding preservation of composition:
I(J(Rl,Rr)KG,H# J(Sl,Sr)KH,I) =IJ(Rl,Rr);(Sl,Sr)KG,I by definition
=R#S by Corollary??
Next,I is faithful because distinct equivalence classes induce distinctDep-morphisms. It is full by passing from
R∶G → HtoJ(R−,R+)KG,H. Finally it acts like the identity on objects, so we have an isomorphism of categories.
3. Consequently the composite functor cl ∶= I ○J⋅K is well-defined. Then it preserves the natural ordering on
morphisms: givenRl⊆Sl thenR=Rl;H⊆Sl;H=S because relational composition is monotonic separately in each argument.
We now deduce an important property, viewingDep-morphisms as components(R−,R+).
Corollary 4.4.10. For any n≥0 and any chain of Dep-morphisms((Ri
−,Ri+)∶Gi→ Gi+1)1≤i≤n,
(R1
−,R1+)#⋯#(Rn−,Rn+)=cl(G1,Gn+1)((R
l
−,R1+);⋯;(Rn−,Rn+))
By the usual convention, the case n=0 is the fact that (G−,G+)=cl(G,G)(∆Gs,∆Gt).
4.5
Dedekind-MacNeille completions
Definition 4.5.1 (Dedekind-MacNeille completion of finite posets). Given any finite poset Pthen:
1. itsDedekind-MacNeille completion is the finite join-semilattice DM(P)∶=Open≰P=(O(≰P,∪,∅).
2. its associated canonical order-embedding is defined:
eP∶P→(O(≰P),⊆) where eP(p)∶= ≰P[p]=↑Pp.
noting that≰P[p]= ≤P[p]= ≤P[p]=↑Pp. ∎
Theorem 4.5.2 ([Dedekind-MacNeille embedding for finite posets).
eP∶P→UDM(P)is a well-defined order-embedding, and preserves all meets and joins which exist inP.
Proof. ePis a well-defined function becauseDM(P)has carrierO(≰P)={≰P[X]∶X ⊆P}. Then:
p1≤Pp2 ⇐⇒ ↑Pp2⊆↑Pp1 ⇐⇒ ↑Pp1⊆↑Pp2 ⇐⇒ eP(p1)≤Open≰PeP(p2). so thatePis an order-embedding. Next, given that ⋁PX exists we’ll show thatePpreserves this join:
⋁ Open≰P e[X]= ⋃ p∈P ↑Pp= ⋂ p∈P ↑Pp=↑P⋁ P X=eP(⋁ P X).
Finally suppose that⋀PX exists. Recalling Definition4.2.4.3, the join-semilattice of open setsOpen≰P is isomorphic
to the join-semilattice of closed sets(C(≰P),∨, P)whose meet is intersection. This isomorphism acts on the embedding
image as follows:
≰P[p] ↦ ≰↓P(≰P[p])={p′∈P∶≰P[p′]⊆ ≰P[p]}
!
={p′∈P∶p′≤Pp}=↓Pp
where in the marked equality we recall thatePis an order-embedding. Then since:
⋀ (C(≰P),∨,P) {↓Pp∶p∈X}= ⋂ p∈X ↓Pp=↓P⋀ P X
applying the inverse join-semilattice isomorphism we deduce thatePpreserves the meet⋀PX.