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COMETIC FUNCTORS AND REPRESENTING
ORDER-PRESERVING MAPS BY PRINCIPAL LATTICE
CONGRUENCES
G´
ABOR CZ´
EDLI
Dedicated to the memory of E. Tam´as Schmidt
Abstract. Let Latsd
5and Pos+
01 denote the category of selfdual bounded
lattices of length 5 with {0,1}-preserving lattice homomorphisms and
that of bounded ordered sets with {0,1}-preserving isotone maps, re-
spectively. For an ob ject Lin Latsd
5, the ordered set of principal con-
gruences of the lattice Lis denoted by Princ(L). By means of congru-
ence generation, Princ : Latsd
5→Pos+
01 is a functor. We prove that
if Ais a small subcategory of Pos+
01 such that every morphism of A
is a monomorphism, understood in A, then Ais the Princ-image of
an appropriate subcategory of Latsd
5. This result extends G. Gr¨atzer’s
earlier theorems where Aconsisted of one or two objects and at most
one non-identity morphism, and the author’s earlier result where all
morphisms of Awere 0-separating and no hom-set had more the two
morphisms. Furthermore, as an auxiliary tool, we derive some families
of maps, also known as functions, from injective maps and surjective
maps; this can be useful in various fields of mathematics, not only in
lattice theory. Namely, for every small concrete category A, we define
a functor F
com, called cometic functor, from Ato the category Set of
sets and a natural transformation πcom, called cometic projection, from
F
com to the forgetful functor of Ainto Set such that the F
com-image of
every monomorphism of Ais an injective map and the components of
πcom are surjective maps.
1. Prerequisites and outline
This paper consists of an easy category theoretical part followed by a
more involved lattice theoretical part.
The category theoretical first part, which consists of Sections 2 and 3, is
devoted to certain families of maps, also known as functions. Only some easy
concepts are needed from category theory; their definitions will be recalled
in the paper. Hence, there is no prerequisite for this part. Our purpose is to
2000 Mathematics Subject Classification. 06B10, 18B05. July 20, 2018.
Key words and phrases. Cometic functor, cometic projection, natural transformation,
injective map, monomorphism, principal congruence, lattice congruence, lifting diagrams,
ordered set, poset, quasi-colored lattice, preordering, quasiordering, isotone map, mono-
tone map.
This research was supported by NFSR of Hungary (OTKA), grant number K 115518.
1
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 2
derive some families of maps from injective maps and surjective maps. This
part can be interesting in various fields of algebra and even outside algebra.
The lattice theoretical second part is built on the first part. The readers
of the second part are not assumed to have deep knowledge of lattice theory;
a little part of any book on lattices, including Gr¨atzer [8] and Nation [15],
is sufficient.
Outline. The paper contains two theorems and it is structured as follows.
Section 2 recalls some basic concepts from category theory. In Section 3, we
introduce cometic functors and cometic projections, and prove Theorem 3.6
on them. In Section 4, we formulate Theorem 4.7 on the representation of
families of isotone maps by principal lattice congruences. The rest of the
sections are devoted to the proof of this theorem. First, Section 5 gives a
heuristic overview of the proof. Section 6 tailors the toolkit developed for
quasi-colored lattices in Cz´edli [4] to the present environment; when reading
this section, [4] should be nearby. In Section 7, we prove a lemma that
allows us to work with certain homomorphisms efficiently. Finally, with the
help of cometic functors and cometic projections, Section 8 completes the
proof of Theorem 4.7.
2. Introduction to the category theory part
2.1. Notation, terminology, and the rudiments. Recall that a category
Ais a system hOb(A),Mor(A),◦i formed from a class Ob(A) of objects,
a class Mor(A) of morphisms, and a partially defined binary operation ◦
on Mor(A) such that Asatisfies certain axioms. Each f∈Mor(A) has
asource object X∈Ob(A) and a target object Y∈Ob(A); the collec-
tion of morphisms with source object Xand target object Yis denoted by
Mor(X, Y ) or MorA(X, Y ). The axioms require that Mor(X, Y ) is a set
for all X, Y ∈Ob(A), every Mor(X, X) contains a unique identity mor-
phism 1
1
1X,f◦gis defined and belongs to Mor(X, Z ) iff f∈Mor(Y, Z)
and g∈Mor(X, Y ), this multiplication is associative, and the identity mor-
phisms are left and right units with respect to the multiplication. Note that
Mor(X, Y ) is often called a hom-set of Aand Mor(A) is the disjoint union
of the hom-sets of A. If Aand Bare categories such that Ob(A)⊆Ob(B)
and Mor(A)⊆Mor(B), then Ais a subcategory of B. If Ais a category
and Ob(A) is a set, then Ais said to be a small category.
Definition 2.1. If Ais a category such that
(i) every object of Ais a set, possibly with a structure on it,
(ii) for all X, Y ∈Ob(A) and f∈Mor(X, Y ), fis a map from Xto Y,
and
(iii) the operation is the usual composition of maps,
then Ais a concrete category. Note the rule (f◦g)(x) = fg(x), that is, we
compose maps from right to left. Note also that Mor(X, Y ) does not have
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 3
to contain all possible maps from Xto Y. The category of all sets with all
maps between sets will be denoted by Set.
Remark 2.2. In category theory, the concept of concrete categories is usu-
ally based on forgetful functors and it has a more general meaning. Since
this paper is not only for category theorists, we adopt Definition 2.1, which
is conceptually simpler but, apart from mathematically insignificant techni-
calities, will not reduce the generality of our result, Theorem 3.6.
For an arbitrary category Aand f∈Mor(A), if f◦g1=f◦g2implies
g1=g2for all g1, g2∈Mor(A) such that both f◦g1and f◦g2are defined,
then fis a monomorphism in A. Note that if Ais a subcategory of B, then
a monomorphism of Aneed not be a monomorphism of B. In a concrete cat-
egory, an injective morphism is always a monomorphism but not conversely.
The opposite (that is, left-right dual) of the concept of monomorphisms is
that of epimorphisms. We say that f∈Mor(A) is an isomorphism in A
if there is a g∈Mor(A) such that both f◦gand g◦fare identity mor-
phisms. Every isomorphism is both a monomorphism and epimorphism.
Next, let Aand Bbe categories. An assignment F:A→Bis a functor if
F(X)∈Ob(B) for every X∈Ob(A), F(f)∈MorB(F(X), F (Y)) for every
f∈MorA(X, Y ), Fcommutes with ◦, and Fmaps the identity morphisms
to identity morphisms. If F(f) = F(g) implies f=gfor all X, Y ∈Ob(A)
and all f, g ∈MorA(X, Y ), then Fis called a faithful functor. Although
category theory seems to avoid talking about equality of objects, to make
our theorems stronger, we introduce the following concept.
Definition 2.3. For categories Aand Band a functor F:A→B,Fis a
totally faithful functor if, for all f, g ∈Mor(A), F(f) = F(g) implies that
f=g.
Remark 2.4. Let F:A→Bbe a functor. Then Fis totally faithful iff it
is faithful and, for all X, Y ∈Ob(A), F(X) = F(Y) implies that X=Y.
Proof. First, assume that Fis totally faithful. Clearly, Fis faithful. Let
X, Y ∈Ob(A) such that F(X) = F(Y). Then F(1
1
1X) = 1
1
1F(X)=1
1
1F(Y)=
F(1
1
1Y). Using that Fis totally faithful, we obtain that 1
1
1X=1
1
1Y, whereby
X=Y. Second, to see the converse implication, assume that Fis faithful
and, in addition, it satisfies the implication from Remark 2.4. Let f1∈
MorA(X1, Y1) and f2∈MorA(X2, Y2) such that F(f1) = F(f2). Then
F(f1) = F(f2) belongs to MorB(F(X1), F (Y1)) ∩MorB(F(X2), F (Y2)), so
this intersection is not empty. Since Mor(B) is the disjoint union of the
hom-sets of B, we obtain that hF(X1), F (Y1)i=hF(X2), F (Y2)i. Hence,
by our additional assumption on F,hX1, Y1i=hX2, Y2i. This allows us to
apply that Fis faithful, and we conclude that f1=f2, showing that Fis
totally faithful.
For a concrete category A, the well-known
forgetful functor GA
forg :A→Set will often be denoted by G
forg (2.1)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 4
if the superscript Ais understood from the context. (The mnemonic in
the subscript comes from “forgetful”.) This functor sends objects, which
are structures, to their underlying sets and acts identically on morphisms,
which are maps. For a functor F:A→B, the F-image of Ais the category
F(A) = h{F(X) : X∈Ob(A)},{F(f) : f∈Mor(A)},◦i. (2.2)
Next, let Fand Gbe functors from a category Ato a category B. A natu-
ral transformation κ:F→Gis a system hκX:X∈Ob(A)iof morphisms of
Bsuch that the component κXof κat Xbelongs to MorB(F(X), G(X)) for
every X∈Ob(A), and for every X, Y ∈Ob(A) and every f∈MorA(X, Y ),
the diagram
F(X)F(f)
−−−−→ F(Y)
κX
y
κY
y
G(X)G(f)
−−−−→ G(Y)
commutes, that is, κY◦F(f) = G(f)◦κX. If all the components κX
of κare isomorphisms in B, then κis a natural isomorphism. If there is
a natural isomorphism κ:F→G, then Fand Gare naturally isomorphic
functors. Naturally isomorphic functors are, sometimes, also called naturally
equivalent.
3. Cometic functors and projections
Our purpose is to derive some families of maps from injective and sur-
jective maps. In order to do so, we introduce some concepts. The third
component of an arbitrary triplet hx, y, ziis obtained by the third projection
pr(3), in notation, pr(3) (hx, y, zi) = z.
Definition 3.1. Given a small concrete category A, a triplet c=hf, x, yi
is an eligible triplet of Aif there exist X, Y ∈Ob(A) such that f∈
MorA(X, Y ), x∈X,y∈Y, and f(x) = y. The third component of
c=hf, x, yiwill also be denoted by
πcom
Y(hf, x, yi) := pr(3)(hf , x, yi) = y=f(x),provided that y∈Y.
For x∈X∈Ob(A),
~v triv(x) = ~v triv
X(x) denotes h1
1
1X, x, xi,
the trivial triplet at x. Note the obvious rule
πcom
X(~v triv
X(x)) = x, for x∈X. (3.1)
Definition 3.2. Given a small concrete category A(see Definition 2.1), we
define the cometic functor
F
com =FA
com :A→Set
associated with Aas follows. For each Y∈Ob(A), we let
F
com(Y) := {hf , x, yi:hf, x, yiis an eligible triplet of Aand y∈Y}.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 5
For Y , Z ∈Ob(A) and g∈MorA(Y, Z), we define F
com(g) as the map
F
com(g) : F
com(Y)→F
com(Z),defined by
hf, x, yi 7→ hg◦f, x, g (y)i.
The map X→F
com(X), defined by x7→ ~v triv (x), will be denoted by ~v triv
X.
We could also denote an eligible triplet hf, x, yiby xf
7→ y, but techni-
cally the triplet is a more convenient notation than the f-labeled “\mapsto”
arrow. However, in this paragraph, let us think of eligible triplets as ar-
rows. The trivial arrows ~v triv
X(x) with x∈Xcorrespond to the elements
of X. Besides these arrows, F
com(X) can contain many other arrows, which
are of different lengths and of different directions in space but with third
components in X. This geometric interpretation of F
com(X) resembles a real
comet; the trivial arrows form the nucleus while the rest of arrows the coma
and the tail. This explains the adjective “cometic”.
Lemma 3.3. F
com =FA
com from Definition 3.2 is a totally faithful functor.
Proof. First, we prove that F
com := FA
com is a functor. Obviously, the F
com-
image of an identity morphism is an identity morphism. Assume that
X, Y, Z ∈Ob(A), f∈MorA(X, Y ), g∈MorA(Y , Z), c=hh, x, yi ∈
F
com(X), and let us compute:
F
com(g)◦F
com(f)(c) = F
com(g)F
com(f)(c)
=F
com(g)hf◦h, x, f (y)i=hg◦(f◦h), x, g(f(y))i
=h(g◦f)◦h, x, (g◦f)(y)i=F
com(g◦f)(c).
Hence, F
com(g)◦F
com(f) = F
com(g◦f) and F
com is a functor. In order to prove
that F
com is faithful, assume that X, Y ∈Ob(A), f, g ∈MorA(X, Y ), and
F
com(f) = F
com(g); we have to show that f=g. This is clear if X=∅.
Otherwise, for x∈X,
hf◦1
1
1X, x, f (x)i=F
com(f)(~v triv(x)) = F
com(g)(~v triv(x)) = hg◦1
1
1X, x, g(x)i.
Comparing either the third components (for all x∈X), or the first com-
ponents, we conclude that f=g. Thus, F
com is faithful. Finally, if X, Y ∈
Ob(A) and X*Y, then there is an x∈X\Y. Since ~v triv(x)∈F
com(X)\
F
com(Y), we conclude that F
com is totally faithful.
Definition 3.4. Let Abe a small concrete category, let GA
forg :A→Set be
the forgetful functor, see (2.1), and keep Definition 3.2 in mind. Then the
transformation
πcom =πcom,A:F
com →GA
forg
whose components are defined by
πcom
X:F
com(X)→Xand πcom
X(c) := pr(3)(c),
for X∈Ob(A) and c∈F
com(X), is the cometic projection associated with
A. (Note that πcom
Xis simply the restriction of the third projection pr(3) to
F
com(X).)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 6
Lemma 3.5. The cometic projection defined above is a natural transforma-
tion and its components are surjective maps.
Proof. Let X, Y ∈Aand f∈Mor(X, Y ). We have to prove that the
diagram
F
com(X)F
com(f)
−−−−−→ F
com(Y)
πcom
X
y
πcom
Y
y
Xf
−−−−→ Y
(3.2)
commutes. For an arbitrary triplet c=hh, x, yi ∈ F
com(X), we have that
πcom
Y◦F
com(f)(c) = πcom
YF
com(f)(c)=πcom
Yhf◦h, x, f (y)i
=f(y) = fπcom
X(c)= (f◦πcom
X)(c),
which proves the commutativity of (3.2). Finally, for X∈Ob(A) and
x∈X,x=πcom
X(~v triv(x)). Thus, the components of πcom are surjective.
Now, we are in the position to state the main result of this section; it also
summarizes Lemmas 3.3 and 3.5.
Theorem 3.6. Let Abe a small concrete category.
(A) For the cometic functor F
com =FA
com and the cometic projection πcom =
πcom,Aassociated with A, the following hold.
(i) F
com :A→Set is a totally faithful functor and πcom :F
com →
GA
forg is a natural transformation whose components are surjective
maps.
(ii) For every f∈Mor(A),fis a monomorphism in Aif and only
if F
com(f)is an injective map.
(B) Whenever F:A→Set is a functor and κ:F→GA
forg is a natural
transformation whose components are surjective maps, then for ev-
ery morphism f∈Mor(A), if F(f)is an injective map, then fis a
monomorphism in A.
By part (B), we cannot “translate” more morphisms to injective maps
than those translated by F
com. In this sense, part (B) is the converse of part
(A) (with less assumptions on the functor). A category Ais finite if both
Ob(A) and Mor(A) are finite sets. The following remark will automatically
follow from the proof of Theorem 3.6.
Remark 3.7. If Ain Theorem 3.6 is a finite concrete category, then so is
its F
com-image, F
com(A); see (2.2).
Proof of Theorem 3.6. (Ai) is the conjunction of Lemmas 3.3 and 3.5.
In order to prove part (B), let Abe a small concrete category, let F:A→
Set be a functor, and let κ:F→GA
forg be a natural transformation with
surjective components. Assume that Y, Z ∈Ob(A) and f∈MorA(Y, Z)
such that F(f) is injective. In order to prove that fis a monomorphism in
A, let X∈Ob(A) and g1, g2∈MorA(X, Y ) such that f◦g1=f◦g2; we
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 7
have to show that g1=g2. That is, we have to show that, for an arbitrary
x∈X,gi(x) does not depend on i∈ {1,2}. By the surjectivity of κX, we
can pick an element a∈F
com(X) such that x=κX(a). Since f◦g1=f◦g2,
F(f)F(gi)(a)=F(f)◦F(gi)(a) = F(f◦gi)(a)
does not depend on i∈ {1,2}. Hence, the injectivity of F(f) yields that
F(gi)(a) does not depend on i∈ {1,2}. Since κis a natural transformation,
F(X)F(gi)
−−−−→ F(Y)
κX
y
κY
y
Xgi
−−−−→ Y
is a commutative diagram, and we obtain that
gi(x) = giκX(a)= (gi◦κX)(a) = κY◦F(gi)(a) = κYF(gi)(a).
Hence, gi(x) does not depend on i∈ {1,2}, because neither does F(gi)(a).
Consequently, g1=g2. Thus, fis a monomorphism, proving part (B).
In order to prove the “only if” direction of (Aii), assume that X, Y ∈
Ob(Y) and f∈MorA(X, Y ) is a monomorphism in the category A. We
have to show that F
com(f) is injective. In order to do so, let ci=hhi, zi, xii ∈
F
com(X) such that F
com(f)(c1) = F
com(f)(c2). Since the middle components
in
hf◦h1, z1, f (x1)i=F
com(f)(c1) = F
com(f)(c2) = hf◦h2, z2, f (x2)i
are equal, we have that z1=z2. Since fis a monomorphism, the equality
of the first components yields that h1=h2. Since c1and c2are eligible
triplets, the first two components determine the third. Hence, c1=c2and
F
com(f) is injective, as required. This proves the “only if” direction of part
(Aii).
Finally, the “if” direction of (Aii) follows from (Ai) and (B).
Remark 3.8. There are many examples of monomorphisms in small con-
crete categories that are not injective. For example, let f:X→Ybe a
non-injective map between two distinct sets. Consider the category Awith
Ob(A) = {X, Y }and Mor(A) = {1
1
1X,1
1
1Y, f }; then fis a monomorphism in
A. For a bit more general example, see Example 4.10.
Remark 3.9. Let Abe as in Theorem 3.6, X, Y ∈Ob(A), and let fbelong
to Mor(X, Y ). Since ~v triv
Xfrom Definition 3.2 is a right inverse of πcom
X, the
commutativity of (3.2) yields easily that f=πcom
Y◦F
com(f)◦~v triv
X. Note,
however, that ~v triv is not a natural transformation in general.
Remark 3.10. Let Abe as in Theorem 3.6. As an easy consequence of
the theorem, every monomorphism of F
com(A) is an injective map. In this
sense, F
com(A) is “better” than A. Since F
com(A) is obtained by the cometic
functor, one might, perhaps, call it the celestial category associated with A.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 8
4. Introduction to the lattice theory part
From now on, the paper is mainly for lattice theorists. Motivated by
the history of the congruence lattice representation problem, which culmi-
nated in Wehrung [17] and R˚uˇziˇcka [16], Gr¨atzer in [9] has recently started
an analogous new topic of lattice theory. Namely, for a lattice L, let
Princ(L) = hPrinc(L),⊆i denote the ordered set of principal congruences
of L. A congruence is principal if it is generated by a pair ha, biof elements.
Ordered sets (also called partially ordered sets or posets) and lattices with
0 and 1 are called bounded. If Lis a bounded lattice, then Princ(L) is a
bounded ordered set. Conversely, Gr¨atzer [9] proved that every bounded
ordered set Pis isomorphic to Princ(L) for an appropriate bounded lattice
Lof length 5. The ordered sets Princ(L) of countable but not necessarily
bounded lattices Lwere characterized in Cz´edli [3]. There are also results
that represent two or more bounded ordered sets together with some iso-
tone maps simultaneously by means of principal congruences of lattices; the
present paper extends these results. In order to review these earlier results in
an economic way and to formulate our theorem later, we need the following
definition.
Definition 4.1. We define the following four categories.
(i) Lat+
01 is the category of at least 2-element bounded lattices with {0,1}-
preserving lattice homomorphisms.
(ii) Lat5is the category of lattices of length 5 with {0,1}-preserving lattice
homomorphisms.
(iii) Latsd
5is the category of selfdual bounded lattices of length 5 with
{0,1}-preserving lattice homomorphisms.
(iv) Pos+
01 is the category of at least 2-element bounded ordered sets with
{0,1}-preserving isotone (that is, order-preserving) maps.
The superscript + above is to remind us that the least structures, the
singleton ones, are excluded. Note that Latsd
5is a subcategory of Lat5,
which is a subcategory of Lat+
01. Note also that if Xand Yare ordered sets
and |Y|= 1, then Mor(X, Y ) consists of the trivial map and Mor(Y, X)6=∅
iff |X|= 1. Hence, we do not loose anything interesting by excluding the
singleton ordered sets from Pos+
01. A similar comment applies for singleton
lattices, which are excluded from Lat+
01.
For an algebra Aand x, y ∈A, the principal congruence generated by
hx, yiis denoted by con(x, y) or conA(x, y). For lattices, the following ob-
servation is due to Gr¨atzer [10]; see also Cz´edli [2] for the injective case.
Note that Princ(A) is meaningful for every algebra A.
Lemma 4.2. If Aand Bare algebras of the same type and f:A→Bis a
homomorphism, then
Princ(f) = ζf,A,B : Princ(A)→Princ(B), defined by
conA(x, y)7→ conB(f(x), f (y)),(4.1)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 9
is a 0-preserving isotone map. Thus, for every concrete category Aof similar
algebras with all homomorphisms as morphisms, Princ is a functor from A
to the category of ordered sets having 0with 0-preserving isotone maps.
Proof. We only have to prove that ζf,A,B is a well-defined map, since the rest
of the statement is obvious. That is, we have to prove that if conA(a, b) =
conA(c, d), then conB(f(a), f (b)) = conB(f(c), f(d)). Clearly, it suffices to
prove that if a, b, c, d ∈Asuch that ha, bi ∈ conA(c, d), then hf(a), f (b)i ∈
conB(f(c), f (d)). According to a classical lemma of Mal’cev [14], see also
Fried, Gr¨atzer and Quackenbush [5, Lemma 2.1], the containment ha, bi ∈
conA(c, d) is witnessed by a system of certain equalities among terms applied
for certain elements of A. Since fpreserves these equalities, hf(a), f (b)i ∈
conB(f(c), f (d)), as required.
It follows from Lemma 4.2 that
Princ: Latsd
5→Pos+
01,defined by
X7→ Princ(X) for X∈Ob(Latsd
5) and
f7→ ζf,X,Y for f∈Mor(X, Y ),
(4.2)
is a functor. Note that Princ could similarly be defined with Lat+
01 or Lat5
as its domain category. Prior to Definition 4.4, we observe the following.
Lemma 4.3. In the category Pos+
01, the monomorphisms, the epimorphisms,
and the isomorphisms are exactly the injective {0,1}-preserving isotone maps,
the surjective {0,1}-preserving isotone maps, and the order isomorphisms,
respectively.
Proof. All maps in the proof are assumed to be {0,1}-preserving and isotone.
It is well-known that an injective map is a monomorphism and a surjective
map is an epimorphism. In order to prove the converse, assume that f:X→
Yis a non-injective morphism in Pos+
01. Pick x16=x2∈Xsuch that
f(x1) = f(x2), and let Z={0≺z≺1}be a three-element chain. Define
the {0,1}-preserving isotone map gi:Z→Xby the rule gi(z) = xi. Since
g16=g2but f◦g1=f◦g2,fis not injective. Next, assume that f:X→Y
is a non-surjective morphism of Pos+
01, pick a y∈Y\f(X), and pick two
elements, y1and y2, outside Y. On the set Y0:= (Y\ {y})∪ {y1, y2}, define
the ordering relation by the rule u < v iff either {u, v} ∩ {y1, y2}=∅and
u <Yv, or u=yiand y <Yv, or v=yiand u <Yyfor some i∈ {1,2}.
Note that y1and y2are incomparable. Let gi:Y→Y0be defined by u7→ u
if u6=yand y7→ yi. Then g1, g2∈Mor(Pos+
01), g1◦f=g2◦fbut
g16=g2, showing that fis not an epimorphism. Finally, if h:X→Y
is an order isomorphism, then it is an isomorphism in category theoretical
sense. Conversely, if h∈MorPos+
01 (X, Y ) is an isomorphism in category
theoretical sense, then it has an inverse in MorPos+
01 (Y, X ), whereby his an
order isomorphism.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 10
Definition 4.4. Let Abe a small category and let F
pos :A→Pos+
01 be a
functor. Following Gillibert and Wehrung [6], we say that a functor
ELift :A→Latsd
5or ELift :A→Lat5
lifts the functor F
pos with respect to the functor Princ, if F
pos is naturally
isomorphic to the composite functor Princ ◦ELift.
Note that the existence of ELift :A→Latsd
5above is a stronger require-
ment than the existence of ELift :A→Lat5. Every ordered set hP;≤i can
be viewed as a small category whose objects are the elements of Pand, for
X, Y ∈P,|Mor(X, Y )|= 1 for X≤Yand |Mor(X, Y )|= 0 for XY.
Small categories obtained in this way are called categorified posets. Based
on Lemma 4.3, the known results on representations of isotone maps by
principal congruences can be stated in the following two propositions. A
map is 0-separating if the only preimage of 0 with respect to this map is 0.
Proposition 4.5 (Cz´edli [4]).Assume that Ais a categorified poset. If
F
pos :A→Pos+
01 is a functor such that F
pos(f)is 0-separating for all f∈
Mor(A), then there exists a functor ELift :A→Latsd
5that lifts F
pos with
respect to Princ.
Note that [4] extends the result of Cz´edli [2], in which Ais the categorified
two-element chain but F(f) is still 0-separating. As another extension of
[2], Gr¨atzer dropped the injectivity in the following statement, which we
translate to our terminology as follows.
Proposition 4.6 (Gr¨atzer [10]).If Ais the categorified two-element chain,
then for every functor F
pos :A→Pos+
01, there exists a functor ELift :A→
Lat5that lifts F
pos with respect to Princ.
Equivalently, in a simpler language and using the notation given in (4.1),
Proposition 4.6 asserts that if X1and X2are nontrivial bounded ordered
sets and f:X1→X2is a {0,1}-preserving isotone map, then there exist
lattices L1and L2of length 5, order isomorphisms κi: Princ(Li)→Xifor
i∈ {1,2}, and a {0,1}-preserving lattice homomorphism g:L1→L2such
that the diagram
Princ(L1)ζg,L1,L2
−−−−−→ Princ(L2)
κ1
y
κ2
y
X1
f
−−−−→ X2
is commutative, that is, f=κ2◦ζg,L1,L2◦κ−1
1.
Now we are in the position to formulate the second theorem of the paper.
Theorem 4.7. Let Abe a small category such that
every f∈Mor(A)is a monomorphism in A.(4.3)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 11
Then for every faithful functor F
pos :A→Pos+
01, there exists a faithful
functor
ELift :A→Latsd
5
that lifts F
pos with respect to Princ. Furthermore, if F
pos is totally faithful,
then there exists a totally faithful ELift that lifts F
pos with respect to Princ.
Apart from some remarks and examples at the end of the present section,
the rest of the paper is devoted to the proof of this theorem. After the
necessary constructions and preparatory statements given in Sections 6 and
7, the proof is completed in Section 8 right after Corollary 8.2.
Remark 4.8. Based on Wehrung [18], an anonymous referee of Cz´edli [4]
has pointed out that a faithful functor from an arbitrary small category to
Pos+
01 cannot be lifted with respect to Princ in general; see [4, Observa-
tion 6.5] for details. Therefore, assumption (4.3) cannot be omitted from
Theorem 4.7
Remark 4.9. Subsection 2.3 of [4], which is due to the above-mentioned
referee, can be adopted to the present paper. That is, if PLat5denotes
the category of polarity lattices of length 5 with polarity-preserving lattice
homomorphisms, then Theorem 4.7 remains valid if we replace by Latsd
5by
PLat5. (Keeping the size limited, we do not elaborate the straightforward
details.)
Observe that Propositions 4.5 and 4.6 are particular cases of Theorem 4.7,
since every morphism of a categorified poset is a monomorphism and the
functors in these statements are automatically faithful. In order to avoid
the feeling that Proposition 4.6 or similar situations are the only cases where
Theorem 4.7 takes care of non-injective isotone maps, we give an example.
Example 4.10. Let D1, D2⊆Ob(Pos+
01) such that D1and D2are disjoint
sets and D1is nonempty. We define a small category A=A(Pos+
01, D1, D2)
by the equalities Ob(A) = D1∪D2and
Mor(A) = {f∈MorPos+
01 (X, Y ) : either X, Y ∈D1and fis a
monomorphism in Pos+
01,or X∈D2and Y∈D1,
or X=Y∈D2and f=1
1
1X}.
(4.4)
Then all morphisms in Aare monomorphisms in Abut, clearly, many of
them are not injective in general. (The same is true for all subcategories of
A. Also, the same holds even if we start from a variety of general algebras
rather than from Pos+
01. By Lemma 4.3, we can replace “monomorphism”
by “injective” in the second line of (4.4).) Now if F
pos :A→Pos+
01 is the
inclusion functor defined by X7→ Xfor objects and f7→ ffor morphisms,
then Theorem 4.7 yields a totally faithful functor ELift :A→Latsd
5that lifts
F
pos with respect to Princ.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 12
Proof. We prove that all morphisms in Aabove are monomorphisms in
A. Let f∈MorA(X, Y ), and assume that g1, g2∈MorA(Z, X) such that
f◦g1=f◦g2. If X∈D2, then Z=Xand g1=1
1
1Z=g2. Otherwise
X, Y ∈D1and fis a monomorphism in Pos+
01, whence we conclude the
equality g1=g2again. Thus, fis a monomorphism in A.
Example 4.11. In a self-explanatory (simpler but less precise) language,
we mention two particular cases of Example 4.10. First, we can represent
all automorphisms of a bounded ordered set simultaneously by principal
congruences. Second, if we are given two distinct bounded ordered sets X
and Y, then we can simultaneously represent all {0,1}-preserving isotone
X→Ymaps by principal congruences.
5. The main ideas for the proof of Theorem 4.7
5.1. Outlining the role of gadgets and quasi-colored lattices. In or-
der to construct a lattice Lwith a given Princ(L) (up to isomorphism), we
will use uniform building blocks, which are called gadgets; see Gr¨atzer [7, 9]
for this terminology, and see Cz´edli [2, 3, 4] and Gr¨atzer [9, 10] based on
similar gadgets. These gadgets and those in the present paper serve the
following purpose. Assume that we want to construct a lattice L=Sι<κ Lι
as a directed union of a well-ordered system of sublattices Lιto represent
an ordered set hX;≤i as Princ(L). Let conLι(ax, bx) and conLι(ay, by) be
incomparable congruences of Lιcorresponding to xand y, respectively, such
that x<yin X. Then we merge Lιand a copy Gof our gadget to obtain
Lι+1 such that {ax, bx, ay, by} ⊆ Lι∩Gand conG(ax, bx)≤conG(ay, by)
forces that conLι+1 (ax, bx)≤conLι+1 (ay, by). In order to avoid that unde-
sired inequalities among principal congruences of Lι+1 enter, we need some
insight into the transition from Princ(Lι) to Princ(Lι+1). This insight will
be provided by quasi-colorings, which were introduced in Cz´edli [1] and were
successfully used for principal lattice congruences in Cz´edli [2, 3, 4]. Besides
that quasi-colorings conveniently determine the principal congruences (this
will be precisely formulated in Lemma 8.1 and Corollary 8.2), there is a nat-
ural way to merge them when the corresponding lattices are merged. See
Subsection 1.5 in Cz´edli [3] for an alternative introduction to these ideas.
5.2. On the rest of the ideas. This subsection is not necessary for the
rest of the paper, but it gives information for those who want to understand
the rest of ideas without reading the rigorous and long proofs and definitions
that we present in the remaining part of the paper.
Examples 2.2 and 3.1 of Cz´edli [4] (with Figures 1–4 there) show most of
the ideas needed in the particular case where Ais a categorified poset and
our isotone maps are 0-separating; see Proposition 4.5 here. Since we do not
assume 0-separation, we also need the quotients (see Figures 2 and 3 here)
of our gadgets, see Figure 1. (Of course, these quotients will be merged
with their duals to turn them selfdual quasi-colored lattices.) The isotone
map ψ31 in [4, Figure 1] is not injective since ψ31(q3) = ψ31(r3) = q1. It is
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 13
described below [4, Figure 3] how the lattices Liare obtained from the aux-
iliary lattices Wiin [4, Figure 3]. Let ψ∗
31 denote the lattice homomorphism
L3→L1that corresponds to ψ31 in the sense that Princ(ψ∗
31) will represent
ψ31. Observe that ψ∗
31 is injective on the set of (thin) basic edges; otherwise
the method of [4] would collapse.
In order to prove Theorem 4.7, we have a lot of isotone maps ψand
we have to make them injective maps ψ∗on the sets of basic edges. We
apply the cometic functor to obtain injective maps that can used to define
these maps ψ∗. Armed with these ψ∗, Figures 1, 2, and 3 (here), and the
above-mentioned ideas taken from [4], we have a rough idea how to prove
Theorem 4.7.
6. Gadgedts, quasi-colored lattices and a toolkit for them
6.1. Gadgets and basic facts. We follow the terminology of Cz´edli [4].
If νis a quasiorder, that is, a reflexive transitive relation, then hx, yi ∈ ν
will occasionally be abbreviated as x≤νy. For a lattice or ordered set L=
hL;≤i and x, y ∈L,hx, yiis called an ordered pair of Lif x≤y. If x=y,
then hx, yiis a trivial ordered pair. The set of ordered pairs of Lis denoted
by Pairs≤(L). If X⊆L, then Pairs≤(X) will stand for X2∩Pairs≤(L). We
also need the notation Pairs≺(L) := {hx, yi ∈ Pairs≤(X) : x≺y}for the set
of covering pairs. By a quasi-colored lattice we mean a structure
L=hL, ≤;γ;H, νi
where hL;≤i is a lattice, hH;νiis a quasiordered set, γ: Pairs≤(L)→His
asurjective map, and for all hu1, v1i,hu2, v2i ∈ Pairs≤(L),
(C1) if γ(hu1, v1i)≤νγ(hu2, v2i), then con(u1, v1)≤con(u2, v2) and
(C2) if con(u1, v1)≤con(u2, v2), then γ(hu1, v1i)≤νγ(hu2, v2i).
This concept is taken from Cz´edli [4]; see Gr¨atzer, Lakser, and Schmidt [13],
Gr¨atzer [7, page 39], and Cz´edli [1, 3] for the evolution of this concept. It
follows easily from (C1), (C2), and the surjectivity of γthat if hL, ≤;γ;H, νi
is a quasi-colored bounded lattice, then hH;νiis a quasiordered set with a
least element and a greatest element; possibly with many least elements and
many greatest elements. Let U(H) stand for the set of greatest elements.
For hx, yi ∈ Pairs≤(L), γ(hx, yi) is called the color (rather than the quasi-
color) of hx, yi, and we say that hx, yiis colored (rather than quasi-colored)
by γ(hx, yi). For T⊆H, we say that hx, yiis T-colored if γ(hx, yi)∈
T. Usually, the following convention applies to our figures of quasi-colored
lattices that contain thick edges and, possibly, also thin edges: if γis a
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 14
quasi-coloring, then for an ordered pair hx, yi,
γ(hx, yi) =
0,iff x=y,
w, if x≺yis a thin edge labeled by w,
u∈U(H),if the interval [x, y] contains is a thick edge,
γ(hx0, y0i),if [x, y] and [x0, y0] are transposed intervals.
(6.1)
If Hhas exactly one largest element 1 = 1Hand so U(H) = {1}, then
our figures determine the corresponding quasi-colorings by convention (6.1).
Note, however, that this convention only partially applies to Figure 6, which
is not a quasi-colored lattice. The quasi-colored lattice
Gup
2(p, q) := hGup
2(p, q), λup
2pq;γup
2pq;H2(p, q), ν2pq i
in Figure 1, taken from Cz´edli [4] where it was denoted by Gup (p, q), is our
upward gadget of type 2. Its quasi-coloring is defined by (6.1); note that
γup
2pq(hcpq
4, dpq
4i) = q. Using the quotient lattices
Figure 1. The upward gadget of rank 2
Gup
0(p, q) := Gup
2(p, q)/con(aq, bq) and
Gup
1(p, q) := Gup
2(p, q))/con(ap, bp),(6.2)
we also define the gadgets
Gup
0(p, q) := hGup
0(p, q), λup
0pq;γup
0pq;H0(p, q), ν0pq iand
Gup
1(p, q) := hGup
1(p, q), λup
1pq;γup
1pq;H1(p, q), ν1pq i
of rank 0 and rank 1, respectively; see Figures 2 and 3. Note that the rank is
length([ap, bp]) + length([aq, bq]). We obtain the downward gadgets Gdn
2(p, q),
Gdn
1(p, q), and Gdn
0(p, q) of ranks 2, 1, and 0 from the corresponding upward
gadgets by dualizing; see Cz´edli [4, (4.3)]. Instead of dpq
ij and, if applicable,
cpq
ij and epq, their elements are denoted by dij
pq,cij
pq, and epq; see [4]. By a
single gadget we mean an upper or lower gadget. The adjective “upper” or
“lower” is the orientation of the gadget. A single gadget of rank jwithout
specifying its orientation is denoted by G∀
j(p, q).
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 15
Figure 2. The upward gadget of rank 1
Figure 3. The upward gadget of rank 0
In case of all our gadgets G∀
j(p, q), we automatically assume that p6=q.
Also, we always assume that for i, j ∈ {0,1,2}, the ordered pairs hp, qi,hu, vi,
and the strings s,t∈ {up,dn}are such that hp, q, i, si 6=hu, v, j, ti,
the intersection of Gt
i(p, q) and Gs
j(u, v) is
as small as it follows from the notation. (6.3)
This convention allows us to form the union Gdb
i(p, q) of Gup
i(p, q) and Gdn
i(p, q),
for i∈ {0,1,2}, which we call a double gadget of rank i. While Gdb
1(p, q) and
Gdb
0(p, q) are given in Figures 4 and 5, the double gadget Gdb
2(p, q) of rank
2 is depicted in Cz´edli [4, Figure 4]. Observe that all the thin edges are
q-colored in Gdb
1(p, q) and, in lack of thin edges, all the edges are 1-colored in
Gdb
0(p, q). For i∈ {0,1,2},Gdb
i(p, q) is a selfdual lattice; we will soon point
out that Gdb
i(p, q) is a quasi-colored lattice. Note that
In each of G∀
j(p, q), con(ap, bp)≤con(aq, bq); we will use
our gadgets to force this inequality in larger lattices. (6.4)
Of course, the inequality in (6.4) is important only for j= 2, since it trivially
holds for j∈ {0,1}.
For S⊆X×X, the least quasiorder including Sis denoted by quo(S) =
quoX(S); we write quo(a, b) rather than quo({ha, bi}).
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 16
Figure 4. The double gadget of rank 1
Figure 5. The double gadget of rank 0
Lemma 6.1. Assume that L=hL;≤Li=hL;λLiis a lattice of length
5, and let 0< apbp<1and 0< aqbq<1in Lsuch that, with
j:= length([ap, bp]) + length([aq, bq]),
ap∨Laq= 1, bp∧Lbq= 0, L ∩Gup
j(p, q) = {0, ap, bp, aq, bq,1},
0≤length([ap, bp]) ≤length([aq, bq]) ≤1,
length([0, bp]) ≤2 + length([ap, bp]),length([ap,1]) ≤2 + length([ap, bp]),
length([0, bq]) ≤2 + length([aq, bq]),length([aq,1]) ≤2 + length([aq, bq]).
Let
LM
M
M:= L∪Gup
j(p, q)and λM
M
M:= quo(λL∪λup
jpq );
see [4, Figure 8] for j= 2. Then LM
M
M=hLM
M
M;λM
M
Mi, also denoted by LM
M
M
jpq or
hLM
M
M
jpq ;≤M
M
Mi, is a lattice of length 5. Also, both Land Gup
j(p, q)are {0,1}-
sublattices of LM
M
M.
Since the lattices required by Theorem 4.7 are selfdual, we will use selfdual
gadgets, which are defined under the name “double gadgets” as follows.
Definition 6.2. Within LM
M
M, the (sublattice) Gup
j(p, q) is the upper gadget
from hap, bpito haq, bqi. By duality, we can analogously glue the lower gadget
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 17
Gdn
j(p, q) into Lfrom hap, bpito haq, bqi. Applying Lemma 6.1, its dual, and
(6.3), we can glue the double gadget Gdb
j(p, q) into Lfrom hap, bpito haq, bqi.
Proof of Lemma 6.1. For j= 2, the lemma coincides with [4, Lemma 4.5]
while the case j < 2 is analogous but simpler. Hence, it would suffice to
say that the proof in [4] works without any essential modification. However,
since we will need some formulas from the proof later, we give some details
for j∈ {0,1,2}. In order to simplify our equalities below, we denote Gup
j(p, q)
by Gup
jand, in subscript position, by G. As in [4], we can still use the
sublattice
B=B(p, q) := {0, ap, bp, aq, bq,1}=L∩Gup
j(p, q),
the closure operators
∗:Gup
j→B, where x∗is the smallest element of B∩ ↑
Gx,
•:L→B, where x•is the smallest element of B∩ ↑
Lx,
and, dually, the interior operators
∗:Gup
j→B, where x∗is the largest element of B∩ ↓
Gx,
•:L→B, where x•is the largest element of B∩ ↓
Lx;
which were introduced in [4, (4.9) and (4.10)]. Since our gadgets are ”wide
enough” in some geometric sense, the operators above are well-defined. As
in [4, (4.11)],
λM
M
Mis an ordering, λM
M
MeL=λL, λM
M
MeG=λup
pq,
for x∈Land y∈Gup
j, x ≤M
M
My⇐⇒ x•≤Gy⇐⇒ x≤Ly∗,
for x∈Gup
jand y∈L, x ≤M
M
My⇐⇒ x∗≤Ly⇐⇒ x≤Gy•.
(6.5)
Denote the lattice operations in Land Gup
jby ∨L,∧L, and ∨G,∧G, respec-
tively. For x, y ∈LM
M
M, we have that
if x∈L\Gup
jand y∈Gup
j\L, then x∧M
M
My=x∧Ly∗,(6.6)
if x∈L\Gup
jand y∈Gup
j\L, then x∨M
M
My=x•∨Gy, (6.7)
if x, y ∈L, then x∧M
M
My=x∧Ly, and x∨M
M
My=x∨Ly, (6.8)
if x, y ∈Gup
j,then x∧M
M
My=x∧Gy, and x∨M
M
My=x∨Gy. (6.9)
Based on (6.5), these equations are proved by exactly the same argument as
their particular cases, [4, (4.12)–(4.15)] for j= 2. It follows from (6.6)–(6.9)
that LM
M
Mis a lattice.
6.2. Large lattices. In this subsection and the next one, we use our double
gadgets to build a “large” quasi-colored lattice for a given quasiordered set
of colors; this immediate plan will be verified by (the proof of) Lemma 6.4.
It will turn out later from Lemma 8.1 and Corollary 8.2 that Lemma 6.4
implies the representability of a given ordered set by principal congruences.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 18
Figure 6. M4×3and L−(H, Z, U ;∅,∅), which is not quasi-colored
Moreover, Lemma 6.4 gives us even more; it gives sufficient flexibility, which
is needed to simultaneously represent many ordered sets and isotone maps.
Let Hbe a set and Z, U ⊂Hsuch that
0∈Z, 1∈U, and Z∩U=∅. (6.10)
This notation is explained by our intention: Zand Uwill be the set of
“zeros” (least elements) and that of “units” (largest elements) of Hsome-
what later. The selfdual simple lattice on the left of Figure 6 is denoted by
M4×3; see also [4, Figure 9] for another diagram. (The two square-shaped
gray-filled elements will play a special role in Lemma 7.2.) Also, we denote
by
L−(H, Z, U ;∅,∅) = hL−(H, Z, U ;∅,∅); λL−(H,Z,U;∅,∅)i(6.11)
the lattice on the right, where Z={0, x, y . . . }and H\Z={1, u, v, w, . . . }.
Of course, 1 ∈U⊆H\Z. The lattice given in (6.11) is almost the same as
that on the right of [4, Figure 9]. Note, however, that |Z|and |U|can be ar-
bitrarily large cardinals. Note also that for z∈Z,az=bz. The role of M4×3
in the construction is two-fold. First, it is a simple lattice and it guarantees
that all the thick edges are 1-colored, that is, they generate the largest con-
gruence, even if |H|= 2. Second, M4×3guarantees that L−(H, Z, U ;∅,∅)
is of length 5. Since ha1, b1iis 1-colored according to labeling but this edge
does not generate the largest congruence, L−(H, Z, U ;∅,∅) is not a quasi-
colored lattice (at least, not if 1 is intended to be a largest elements in H).
So we cannot be satisfied yet. In order to make this edge and all the har, bri,
for r∈U, generate the largest congruence, Definition 6.2 allows us
to glue, for each r∈U, a distinct copy of Gdb
2(p, q)
into L−(H, Z, U ;∅,∅) from ha0
1, b0
1ito har, bri.(6.12)
(No matter if we glue the gadgets one by one by a transfinite induction or
glue them simultaneously, we obtain the same.) It follows from Lemma 6.1
that we obtain a lattice in this way; we denote this lattice by
L(H, Z, U ;∅,∅) = hL(H, Z, U ;∅,∅); λH,Z,U;∅,∅i.
Note that after adding the above-mentioned gadgets to L−(H, Z, U ;∅,∅),
all edges of the gadgets in (6.12) become thick; (6.13)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 19
this follows from (6.1) and (6.4). Let
νH,Z,U ;∅,∅= quo((Z×H)∪(H×U)),
and define γH,Z,U ;∅,∅by convention (6.1). It is straightforward to see that
L(H, Z, U ;∅,∅) =
hL(H, Z, U ;∅,∅), λH,Z,U ;∅,∅;γH,Z,U ;∅,∅;H, νH,Z,U;∅,∅i(6.14)
is a quasi-colored lattice.
Next, to obtain larger lattices, we are going to insert gadgets into the
lattice L(H, Z, U ;∅,∅) in a certain way. It will prompt follow Lemma 6.1
that we obtain lattices; in particular, λH,Z,U ;I ,J in (6.18) will be a lattice
order. Assume that
Iand Jare subsets of H×Hsuch that p6=qand the
implications (q∈Z⇒p∈Z) and (p∈U⇒q∈U)
hold for every hp, qi ∈ I∪J.
(6.15)
With this assumption, we define the rank of a pair hp, qi ∈ I∪Jas follows:
r(hp, qi) :=
0,if p, q ∈Z,
1,if p∈Zand q∈H\Z,
2,if p, q ∈H\Z.
(6.16)
Let us agree that, for every hp, qi ∈ I∪Jand j:= r(hp, qi),
Gup
j(p, q)∩L(H, Z, U ;∅,∅) = {0, ap, bp, aq, bq,1}and
Gdn
j(p, q)∩L(H, Z, U ;∅,∅) = {0, ap, bp, aq, bq,1}.(6.17)
Taking Conventions (6.3) and (6.17) into account, we define
L(H, Z, U ;I, J ) := L(H, Z, U ;∅,∅)∪[
hp,qi∈I
Gup
r(hp,qi)(p, q)
∪[
hp,qi∈J
Gdn
r(hp,qi)(p, q),and
λH,Z,U ;I,J := quoλH,Z,U ;∅,∅∪[
hp,qi∈I
λup
r(hp,qi)pq
∪[
hp,qi∈J
λdn
r(hp,qi)pq .
(6.18)
Based on Lemma 6.1 and its dual, a trivial transfinite induction yields that
L(H, Z, U ;I, J ) = hL(H, Z, U ;I, J ); λH,Z,U ;I,J i
is a lattice of length 5. Clearly, if I=J, then this lattice is selfdual. Let
us emphasize that whenever we use the notation L(H, Z, U;I, J ), (6.15) is
assumed.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 20
Remark 6.3. For later reference, we note that for lattices of the form
(6.18), we treat ap,bp,cpq
ij ,dpq
ij ,cij
pq, etc. as if they were tuples ha, pi,hb, pi,
hc, p, q, i, ji,hd, p, q, i, j i,hcdual, p, q, i, ji, etc.. Therefore,
L(H1, Z1, U1;I1, J1) = L(H2, Z2, U2;I2, J2) iff
hH1, Z1, U1, I1, J1i=hH2, Z2, U2, I2, J2i.
6.3. Large quasi-colored lattices. Assuming (6.10), let H−ZU := H\
(Z∪U). Also, let νH,Z,U ;∅,∅= quo((Z×H)∪(H×U)). Note that each
z∈Zis a least element of hH;νH,Z,U;∅,∅iand each u∈Uis a largest
element. Also, for any two distinct p, q ∈H−Z U ,pand qare incomparable,
that is, none of hp, qiand hq, pibelongs to νH,Z,U;∅,∅. With convention
(6.15), let
νH,Z,U ;I,J := quoH(νH,Z,U ;∅,∅∪I∪J)
= quo((Z×H)∪(H×U)∪I∪J).
Based on (6.17), it is easy to see that
γH,Z,U ;I,J := γH,Z,U ;∅,∅∪[
hp,qi∈I
γup
r(hp,qi)pq ∪[
hp,qi∈J
γdn
r(hp,qi)pq (6.19)
is a well-defined map from Pairs≤(L(H, Z, U;I, J )) to H.
Lemma 6.4. Assume (6.15). Then
L(H, Z, U ;I, J )
:= hL(H, Z, U ;I, J ), λH,Z,U ;I ,J ;γH,Z,U;I ,J ;H, νH,Z,U ;I ,J i(6.20)
is a quasi-colored lattice of length 5. If I=J, then it is a selfdual lattice.
Proof. If Z={0}and r(hp, qi) = 2 for all hp, q i ∈ I∪J, then the statement
is practically the same as [4, Lemma 4.6]. (Although 1 /∈U=∅in [4,
Lemma 4.6], this does not make any difference.) As in [4], the only nontrivial
task is to show (C2). This argument in [4] has two ingredients, and these
ingredients also work in the present situation.
First, let αbe the equivalence on L(H, Z, U ;I, J ) whose non-singleton
equivalence classes are the [ap, bp] for p∈H−ZU , the [cpq
i, dpq
i] for hp, qi ∈ I
and i∈ {1,...,5}, and the [di
pq, ci
pq] for hp, qi ∈ Jand i∈ {1,...,5}.
Using the Technical Lemma from Gr¨atzer [11], cited in [4, Lemma 4.1], it
is straightforward to see that αis a congruence. Clearly, αis distinct from
∇
L(H,Z,U ;I,J), the largest congruence of L(H, Z, U ;I, J ). Like in [4, (4.28)],
this implies easily that, for any hx, yi ∈ Pairs≤(L(H, Z, U ;I, J )),
γH,Z,U ;I ,J (hx, yi) = 1 ⇐⇒ con(x, y) = ∇
L(H,Z,U ;I,J).
The second ingredient of the proof is to show that
if p, q ∈H−Z U , con(ap, bp)≤con(aq, bq)6=∇
L(H,Z,U ;I,J), and
p6=q, then hp, qi=hγH,Z,U ;I ,J (hap, bpi), γH,Z,U;I ,J (haq, bqi)i
belongs to νH,Z,U ;I ,J ;
(6.21)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 21
compare this with [4, (4.29)]. The inequality con(ap, bp)≤con(aq, bq) is
equivalent to the containment hap, bpi ∈ con(aq, bq). This containment is
witnessed by a shortest sequence of consecutive prime intervals in the sense
of the Prime-projectivity Lemma of Gr¨atzer [12]; note that this lemma is
cited in [4, Lemma 4.2]. If one of the prime intervals in the sequence gener-
ates ∇
L(H,Z,U ;I,J), then the easy direction of the Prime-projectivity Lemma
yields that con(aq, bq) = ∇
L(H,Z,U ;I,J), a contradiction. Hence, none of these
prime intervals generates ∇
L(H,Z,U ;I,J). Thus, since (C1) is easily verified
in the same way as in [4], none of these prime intervals is 1-colored. In
other words, all prime intervals of the sequence are thin edges. Gadgets of
rank 0 contain no thin edges, so the sequence avoids them. The same holds
for the gadgets mentioned in (6.12) and (6.13). Gadgets of rank 1 contain
too few thin edges, so the sequence can only make a loop in them; this is
impossible since we consider the shortest sequence. Thus, the sequence goes
in the sublattice that we obtain by omitting all gadgets of rank less than 2,
all gadgets occurring in (6.13), and all elements az=bzfor z∈Z. So we
can work in this sublattice, which is the same as the lattice considered in
[4, (4.29)]. Consequently, the proof of [4, (4.29)] yields (6.21). Thus, (C2)
holds.
7. From quasiorders to homomorphisms
For a quasiordered set hH;νi, we define
Z(H) = Z(H, ν) := {x∈H: (∀y∈H) (hx, yi ∈ ν)}and
U(H) = U(H, ν) := {x∈H: (∀y∈H) (hy, xi ∈ ν)}.(7.1)
These are the set of smallest elements (the notation comes from “zeros”)
and that of largest elements (“units”). If νis clear from the context, we
prefer the notations Z(H) and U(H) to Z(H, ν) and U(H, ν ), respectively.
In this section, we are only interested in the following particular case of the
quasi-colored lattices L(H, Z, U ;I, J ).
Definition 7.1. For a quasiordered set H=hH;νi, assume that
0∈Z(H),1∈U(H),and 0 6= 1. (7.2)
With this assumption, we define
L(H, ν) = hL(H, ν ), λH,ν;γH,ν ;H, νias L(H, Z(H), U (H); ν, ν)(7.3)
according to (6.20); this is possible since (6.15) clearly holds. Let us note
that ν=νH,Z(H),U (H);ν,ν and, clearly, L(H, ν ) is a selfdual lattice of length
5.
For quasiordered sets hH1;ν1iand hH2;ν2i, a map f:H1→H2is isotone
if hx, yi ∈ ν1implies hf(x), f (y)i ∈ ν2for all x, y ∈H1. Now, we are in
the position to state the main lemma of this subsection. By (6.18) and
(7.3), our lattices are extensions of lattices of the form given in Figure 6.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 22
So the parenthetical sentence above (6.11) explains what the distinguished
elements are in the following lemma.
Lemma 7.2. Let hH1;ν1iand hH2;ν2ibe quasiordered sets, both with 0
and 1such that 06= 1. If f:H1→H2is an injective isotone map such
that f(Z(H1)) ⊆Z(H2)and f(U(H1)) ⊆U(H2), then there exists a unique
{0,1}-preserving lattice homomorphism g:L(H1, ν1)→L(H2, ν2)such that
g(ap) = af(p)and g(bp) = bf(p),for all p∈H1,(7.4)
and the g-image of the square-shaped gray-filled atom and coatom, see Fig-
ure 6, is the square-shaped gray-filled atom and coatom, respectively.
By (7.1), 0 ∈Z(Hi), 1 ∈U(Hi), and Z(Hi)∩U(Hi) = ∅hold for
i∈ {1,2}. The assumption of injectivity cannot be omitted from this lemma,
because if fis a non-injective {0,1}-preserving homomorphism, then (7.4)
yields that the kernel of gcollapses some ap6=aq, so this kernel is the largest
congruence, contradicting g(0) = 0 6=1=g(1).
Proof of Lemma 7.2. First, we deal with the uniqueness of g. Since g(0) =
06=1=g(1), the kernel congruence ker(g) of gcannot collapse a thick (that
is, a U(H1)-colored) edge. Since all edges of M4×3are thick, the restric-
tion geM4×3of gto M4×3is injective. Since no other sublattice of L2than
M4×3itself is isomorphic to M4×3, it follows that g(M4×3) is the unique
M4×3sublattice of L(H2;ν2). Observe that except for the two doubly ir-
reducible atoms and the two doubly irreducible coatoms, each element of
M4×3is a fixed point of all automorphisms of M4×3. Therefore, since gpre-
serves the “square-shaped gray-filled” property, we conclude that geM4×3is
uniquely determined. The g-images of the apand bp,p∈H1, are determined
by the assumption on g. Observe that an upper gadget Gup
2(p, q) has ex-
actly two non-trivial congruences, con(ap, bp) and con(aq, bq); Gup
1(p, q) has
only con(aq, bq), and Gup
0(p, q) has none. The same holds for lower gadgets.
Therefore, since ker(g) cannot collapse a thick edge, it follows easily that
the restriction of gto any gadget is uniquely determined. Therefore, gis
unique.
In the rest of the proof, we intend to show the existence of g. We will define
an appropriate gas the union of some partial maps. Let gM4×3denote the
unique isomorphism from the M4×3sublattice of L(H1, ν1) onto the M4×3
sublattice of L(H2, ν2) such that gM4×3preserves the “square-shaped gray-
filled” property. For i∈ {1,2}, we denote νi\ {hx, xi:x∈Hi}by ν+
i. Next,
let hp, qi ∈ ν+
1and j:= r(hp, qi); according to (6.16) with Z:= Z(H1, ν1).
By the construction of L(H1, ν1), see (6.18), (7.3), and Definition 6.2, the
gadget Gup
j(p, q) is a {0,1}-sublattice of L(H1, ν1) from hap, bpito haq, bqi.
Let p0=f(p), q0=f(q), and j0=r(hp0, q0i). Besides that fis isotone, we
frequently need the assumption that it is injective; at present, we conclude
hp0, q0i ∈ ν+
2from these assumptions. (Later, we will not always emphasize
similar consequences of these assumptions.) It follows from hp0, q0i ∈ ν+
2and
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 23
the construction of L(H2, ν2) that Gup
j0(p0, q0) is a gadget in L(H2, ν2) from
hap0, bp0ito haq0, bq0i. We obtain from f(Z(H1)) ⊆Z(H2) that
j0≤j. (7.5)
According to (6.2), we can take the unique surjective {0,1}-preserving lat-
tice homomorphism gup
pq :Gup
j(p, q)→Gup
j0(p0, q0) such that gup
pq (ap) = ap0,
gup
pq (bp) = bp0,gup
pq (aq) = aq0, and gup
pq (bq) = bq0. We take the {0,1}-preserving
lattice homomorphism gdn
pq :Gdn
j(p, q)→Gdn
j0(p0, q0) analogously. Note that
gM4×3maps a0
1∈L(H1, ν1) onto a0
1∈L(H2, ν2), and the same is true for b0
1.
For u∈U(H1), we know that f(u)∈U(H2). By construction, there is an
upper gadget of rank 2 from ha0
1, b0
1ito hau, buiin L(H1, ν1), and we have
an upper gadget of rank 2 from ha0
1, b0
1ito haf(u), bf(u)iin L(H2, ν2). The
unique isomorphism from the first gadget to the second such that a0
17→ a0
1,
b0
17→ b0
1,au7→ af(u), and bu7→ bf(u)is denoted by gup
10u. Here 10in the
subscript is only a symbol, which does not belong to H1∪H2. We define
the isomorphism gdn
10ubetween the corresponding lower gadgets similarly.
For hp1, q1i,hp2, q2i ∈ ν+
1and u∈U(H1), any two of the homomorphisms
gM4×3,gup
p1q1,gdn
p1q1,gup
p2q2,gdn
p2q2,gup
10u, and gdn
10uagree on the intersection of
their domains. Therefore,
g:= gM4×3∪[
hp,qi∈ν+
1
gup
pq ∪[
hp,qi∈ν+
1
gdn
pq ∪[
u∈U(H1)
gup
10u∪[
u∈U(H1)
gdn
10u
is a well-defined {0,1}-preserving map from L(H1, ν1) to L(H2, ν2).
Figure 7. hup,dni,q=r, and hj, j0, k, k0i=h2,1,2,2i
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 24
Next, we are going to show that, for all x, y ∈L(H1;ν1),
g(x∨y) = g(x)∨g(y) and g(x∧y) = g(x)∧g(y). (7.6)
Clearly, we can assume that {x, y}∩M4×3=∅and no single gadget contains
both xand y. Therefore, {0,1} ∩ {x, y}=∅and there are single gadgets
G∀
j(p, q) and G∀
k(r, s) containing xand y, respectively. Of course, p6=qand
r6=s; however, we do not know more than |{p, q, r, s}| ∈ {2,3,4}. (It may
even happen that hr, si=hq, pi.) We can work in the union S:= G∀
j(p, q)∪
G∀
k(r, s), which is a sublattice by (6.6)–(6.9); see also the upper parts of
Figures 7, 8, and 9. Alternatively, Sis a sublattice by Lemma 6.1. Let
p0:= f(p), q0:= f(q), r0:= r(p), s0:= f(s), and S0:= G∀
j0(p0, q0)∪G∀
k0(r0, s0);
see the lower parts of Figures 7, 8, and 9, where g(x) is geometrically below
xfor every x∈S. Again, S0is a sublattice by (6.6)–(6.9). (Note that if
hp, qior hr, siis of the form h10, uiwith u∈U(H1), then we have to extend f
by 107→ 10, since 10/∈H1.) Let j:= r(hp, qi), k:= r(hr, si), j0:= r(hp0, q0i),
and k0:= r(hr0, s0i).
We know from (7.5) that j0≤jand, similarly, k0≤k. Hence, by the
definition of our gadgets of rank less than 2, there are congruences α1and
α2of G∀
j(p, q) and G∀
k(r, s) and surjective homomorphisms (namely, the
natural projections) g1:G∀
j(p, q)→G∀
j0(p0, q0) and g2:G∀
k(r, s)→G∀
k0(r0, s0)
such that α1is the kernel of g1and α2is the kernel of g2. In Figures 7,
8, and 9, the nontrivial α1-blocks and nontrivial α2-blocks are indicated by
dotted lines.
By the definition of g,g1∪g2is the restriction geSof gto S. Thus, to
verify (7.6), we need to show that g1∪g2:S→S0is a homomorphism.
It suffices to show that α1∪α2is a congruence of S, (7.7)
because then S0is the quotient lattice of Smodulo α1∪α2and g1∪g2is the
natural projection homomorphism of Sto this quotient lattice. There are
several cases but all of them can be settled similarly. We only discuss those
given by Figures 7, 8, and 9. By Gr¨atzer [11], each of these cases would
be quite easy, although a bit tedious. However, to indicate that the rest of
cases are similar, we give slightly more sophisticated arguments for them.
Note that these figures also use the injectivity of f; for example, this is why
p06=s0and q06=r0in Figure 8.
In case of Figure 7, let H={0, p, q, r, s, 1}and
ν= quo{hp, qi,hq, ri,hr, qi,hr, si} ∪ ({0} × H)∪(H× {1}).
(In general, the quasiordered set hH;νiis quite different from hH1;ν1iand
hH2;ν2i.) Using that Sis a sublattice of the quasi-colored lattice L(H, ν), see
Lemma 6.4 and Definition 7.1, it is easy to see that α1∪α2is a congruence
of S. Namely, we can quite easily show that α1∪α2= conS(ap, bp). Clearly,
conS(ap, bp) collapses the p-colored edges. If it collapsed a t-colored edge for
some t∈ {q, r, s, 1}in S, then it would collapse the same edge (with the
same color) in L(H, ν), but then (C2) would give t≤νp, a contradiction.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 25
Figure 8. hup,upi,{p, q, r, s}| = 4, and hj, j0, k, k0i=
h2,1,2,1i
In case of Figure 8, let hH;νibe the six element lattice in which there
are exactly two maximal chains, {0≺p≺q≺1}and {0≺r≺s≺1}.
The same argument as above shows that conS(ap, bp) collapses the p-colored
edges and only those, while conS(ar, br) collapses exactly the r-colored edges.
In order to see that α1∪α2is a congruence, it suffices to show that α1∪α2=
conS(ap, bp)∨conS(ar, br). Clearly, α1∪α2⊆conS(ap, bp)∨conS(ar, br). As-
sume that hx, yi ∈ Pairs≺(S) such that hx, yi ∈ conS(ap, bp)∨conS(ar, br). In
other words, conS(x, y)≤conS(ap, bp)∨conS(ar, br). Since a covering pair of
a lattice always generates a join-irreducible congruence and the congruence
lattice of a lattice is distributive, it follows that conS(x, y)≤conS(ap, bp) or
conS(x, y)≤conS(ar, br). Hence, hx, yi ∈ α1or hx, yi ∈ α2, and we obtain
the required inclusion, α1∪α2⊇conS(ap, bp)∨conS(ar, br).
For Figure 9, we use the same hH;νias for Figure 7 and, practically, the
same argument as for Figure 8 to show that α1∪α2= conS(ar, br). By
(7.7), this completes the proof of Lemma 7.2.
8. Completing the lattice theoretical part
For a quasiordered set hH, νi, we let Θν=ν∩ν−1. It is known that Θν
is an equivalence relation, and the definition
hx/Θν, y/Θνi ∈ ν/Θν
def
⇐⇒ hx, yi ∈ ν(8.1)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 26
Figure 9. hup,dni,q=r, and hj, j0, k, k0i=h2,0,2,1i
turns the quotient set H/Θνinto an ordered set hH;νi/Θν, which is also
denoted by hH/Θν;ν/Θνi. The following lemma is a straightforward conse-
quence of (C1) and (C2), see [2, Lemma 3.1], [3, Lemma 2.1], or [4, Lemma
4.7], where the inverse isomorphism is considered. Although the lemma was
only formulated for the particular quasi-colored lattices constructed in these
papers, its easy proof makes it valid for every quasi-colored lattice, so it is
time to formulate it more generally.
Lemma 8.1. For every quasi-colored lattice hL, ≤;γ;H, νi,Princ(L)is iso-
morphic to hH;νi/Θνand the map hPrinc(L); ⊆i → hH;νi/Θν, defined by
con(x, y)7→ γ(hx, yi)/Θν, is an order isomorphism.
As a consequence of this lemma and our construction, or (the proof of)
[4, Lemma 4.7], we obtain the following corollary.
Corollary 8.2. If hH;νiis a quasiordered set satisfying (7.2), then the map
ζH,ν :hH;νi/Θν→ hPrinc(L(H, ν )); ⊆i
defined by p/Θν7→ con(ap, bp)is an order isomorphism.
Proof of Theorem 4.7. Let F
pos :A→Pos+
01 be a faithful functor as in the
theorem, and let
B:= F
pos(A).
For X∈Ob(A) and f∈Mor(A), F
pos(X) is an ordered set and F
pos(f) is
an isotone map; we will use the notation
hX;≤Xi:= F
pos(X) and f:= F
pos(f).
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 27
In B, two ordered sets with the same underlying set but different order-
ings are two distinct objects. Since we do not want to identify distinct
objects when we forget their orderings, we index the underlying sets as
follows. For hY;νi ∈ Ob(B), we let G0
forg(hY;νi) := Y× {ν}. For g∈
Mor(hY1;ν1i,hY2;ν2i), we let
g0=G0
forg(g) : Y1× {ν1} → Y2× {ν2},defined by hu, ν1i 7→ hg(u), ν2i.
In this way, we have defined a totally faithful functor G0
forg :B→Set; the
subscript comes from “forgetful” and the prime reminds us that G0
forg is
slightly different from the forgetful functor G
forg. For hX;νi ∈ Ob(B) and
u∈X, if νis understood, we often write X0and u0instead of X× {ν}
and hu, νi. With this abbreviation, g0=G0
forg(g) : Y0
1→Y0
2is defined by
u07→ (g(u))0. Hence, for X, Y ∈A,f∈MorA(X, Y ), and u∈X,
X0=X× {≤X}=G0
forg(F
pos(X)), u0=hu, ≤Xi ∈ X0,
f0=G0
forg(F
pos(f)) : X0→Y0,and f0(u0) = (f(u))0=hf(u),≤Yi.(8.2)
The image
C:= G0
forg(B)=(G0
forg ◦F
pos)(A)
is a small concrete category, a subcategory of Set; its objects and morphisms
are the X0for X∈Ob(A) and the f0for f∈Mor(A), respectively, as
described in (8.2). We claim that
all morphisms of Care monomorphisms. (8.3)
Since F
pos is assumed to be faithful and G0
forg is obviously faithful, (8.3) will
follow from the following trivial observation.
If F:U→Vis a faithful functor, V=F(U),
and f1∈Mor(U) is a monomorphism, then
F(f1) is a monomorphism in V.
(8.4)
In order to show this, assume that f1∈MorU(X, Y ) is a monomorphism
and f∗
2, f ∗
3∈MorV(Z∗, F (X)) such that F(f1)◦f∗
2=F(f1)◦f∗
3. Since
Vis the F-image of U, there exist Z∈Ob(U) and f2, f3∈MorU(Z, X )
such that Z∗=F(Z), f∗
2=F(f2), and f∗
3=F(f3). Since F(f1◦f2) =
F(f1)◦F(f2) = F(f1)◦f∗
2=F(f1)◦f∗
3=F(f1)◦F(f3) = F(f1◦f3) and
Fis faithful, f1◦f2=f1◦f3. Using that f1is a monomorphism in U, we
obtain that f2=f3. Hence, f∗
2=F(f2) = F(f3) = f∗
3, showing that F(f1)
is a monomorphism. This proves (8.4) and, consequently, (8.3).
Although X0=G0
forg(hX;≤Xi) = G0
forg(F
pos(X)) is only a set for X∈
Ob(A), we shall use the ordering ≤0
Xinduced by ≤Xon it as follows: for
x, y ∈X,
hx, ≤Xi ≤0
Xhy, ≤Xidef
⇐⇒ x≤Xy, that is, x0≤0
Xy0def
⇐⇒ x≤Xy. (8.5)
The least element and the largest element of hX;≤Xi=F
pos(X) will be
denoted by 00
X=h0hX;≤Xi,≤Xiand 10
X=h1hX;≤Xi,≤Xi. By (8.5),
00
Xresp. 10
Xare the least resp. greatest element of hX0,≤0
Xi. (8.6)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 28
Next, denoting the cometic functor FC
com by F
com, see Definition 3.2, we let
D:= FC
com(C).
By (8.3) and Theorem 3.6,
all morphisms of Dare injective maps; (8.7)
this is why we can apply Lemma 7.2 soon. Since we have three functors
already, it is worth defining their composite,
G
prod := F
com ◦G0
forg ◦F
pos,from Ato D.
For X∈Ob(A), the cometic projection from Definition 3.4 allows us to
define a relation νXon the set G
prod(X) = F
com(X0), as follows: for eligible
triplets c1, c2∈G
prod(X) = F
com(X0),
hc1, c2i ∈ νX
def
⇐⇒ πcom
X0(c1)≤0
Xπcom
X0(c1). (8.8)
Clearly, νXis a quasiorder. The set of least elements of hG
prod(X); νXiwill
be denoted by Z(G
prod(X)). Similarly, U(G
prod(X)) will stand for the set of
largest elements. (8.6) and (8.8) make it clear that
Z(G
prod(X)) = {c∈G
prod(X) : πcom
X0(c) = 00
X}, and
U(G
prod(X)) = {c∈G
prod(X) : πcom
X0(c) = 10
X}.(8.9)
It also follows from (8.8) that these sets are nonempty, because
~v triv(00) = h1
1
1X0,00
X,00
Xi ∈ Z(G
prod(X)), and
~v triv(10) = h1
1
1X0,10
X,10
Xi ∈ U(G
prod(X)).
Note the notational difference: 1
1
1X0is the identity morphism on the set X0,
which is the support set of hX;≤Xi, while 10
Xis the top element of the
ordered set hX0;≤0
Xi, which is isomorphic to hX;≤Xi=F
pos(X). Since the
ordered set F
pos(X)∈Pos+
01 consists of at least two elements, we obtain that
00
X6= 10
Xand the distinguished eligible triplets
~v triv(00
X)∈Z(G
prod(X)) and ~v triv(10
X)∈U(G
prod(X)) are distinct. (8.10)
Hence, for X∈Ob(A), Definition 7.1 allows us to consider the quasi-colored
lattice
L(G
prod(X), νX) =
hL(G
prod(X), νX), λG
prod(X),νX;γG
prod(X),νX;G
prod(X), νXi.(8.11)
We are going to turn the assignment given in (8.11) functorial. For fin
Mor(A), f0= (G0
forg ◦F
pos)(f) and G
prod(f) are only maps between two sets.
However, (8.5) and (8.8), respectively, allow us to guess that these maps are
isotone; these properties are conveniently formulated in the form (8.12) be-
low and (8.13) later. We claim that for X, Y ∈Ob(A) and f∈MorA(X, Y ),
f0= (G0
forg ◦F
pos)(f) : hX0;≤0
Xi→hY0;≤0
Yiis an isotone map. (8.12)
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 29
In order to show this, assume that x1, x2∈Xsuch that x0
1≤0
Xx0
2. By (8.5),
x1≤Xx2. Since f=F
pos(f) is an isotone map, f(x1)≤Yf(x2). Hence, by
(8.5) again, (f(x1))0≤0
Y(f(x2))0. Thus, applying (8.2),
f0(x0
1)=(f(x1))0≤0
Y(f(x2))0=f0(x0
2),
which proves (8.12). Next, we are going to show that for X, Y ∈Ob(A) and
f∈MorA(X, Y ),
G
prod(f) : hG
prod(X); νXi→hG
prod(Y); νYiis an isotone map. (8.13)
So let X, Y ∈Ob(A) and f∈MorA(X, Y ). Since πcom is a natural trans-
formation by Theorem 3.6 and C, which is the domain of F
com =FC
com, is a
subcategory of Set, the diagram
G
prod(X) = F
com(X0)G
prod(f)
−−−−−→ G
prod(Y) = F
com(Y0)
πcom
X0
y
πcom
Y0
y
X0f0
−−−−→ Y0
(8.14)
commutes. That is, for every eligible triplet c∈G
prod(X),
πcom
Y0(G
prod(f)(c)) = f0(πcom
X0(c)). (8.15)
Assume that hc1, c2i ∈ νX. By (8.8), πcom
X0(c1)≤0
Xπcom
X0(c2). By (8.12), this
gives that f0(πcom
X0(c1)) ≤0
Yf0(πcom
X0(c2)). Combining this inequality with
(8.8) and (8.15), we obtain that hG
prod(f)(c1), G
prod(f)(c2)i ∈ νY, proving
(8.13).
Our next task is to show that, for every f∈MorA(X, Y ),
G
prod(f)(Z(G
prod(X))) ⊆Z(G
prod(Y)) and
G
prod(f)(U(G
prod(X))) ⊆U(G
prod(Y)). (8.16)
Let c∈Z(G
prod(X)). By (8.9), πcom
X0(c) = 00
X. Since f=F
pos(f) belongs to
Mor(B)⊆Mor(Pos+
01), fis 0-preserving. Hence, by (8.2) and (8.15),
πcom
Y0(G
prod(f)(c)) = f0(πcom
X0(c)) = f0(00
X) = (f(0hX,≤Xi))0= (0hY ,≤Yi)0= 00
Y.
By (8.9), this means that G
prod(f)(c)∈Z(G
prod(Y)). This proves the first
half of (8.16); the second half follows in the same way.
Now, we are in the position to define a functor ELift :A→Latsd
5as
follows. For X∈Ob(A) and f∈MorA(X, Y )⊆Mor(A), we let
ELift(X) := L(G
prod(X); νX),see (8.11),
ELift(f) := the unique {0,1}-preserving lattice homomorphism
that Lemma 7.2 associates with G
prod(f);
(8.17)
it follows from (8.7), (8.10), (8.13), and (8.16) that Lemma 7.2 is applica-
ble. We are going to show that ELift is a functor from Ato Latsd
5. By
Lemma 6.4, Definition 7.1, and (8.11), we have that ELift (X)∈Ob(Latsd
5).
By Lemma 7.2, G
prod(f)∈Mor(Latsd
5). If f= 1X∈MorA(X, X ), then
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 30
G
prod(f) is the identity map since G
prod is a functor, and it follows from
(7.4) and the uniqueness part of Lemma 7.2 that ELift(f) is the identity
map 1ELift(X). Finally, assume that X, Y, Z ∈Ob(A), f1∈MorA(Y, Z ), and
f2∈MorA(X, Y ). We have to show that ELift(f1◦f2) = ELift (f1)◦ELift(f2).
By (7.4) and the uniqueness part of Lemma 7.2, it suffices to show that
ELift(f1◦f2)(ap) = (ELift (f1)◦ELift(f2))(ap) (8.18)
for all eligible triplets p∈G
prod(X), and similarly for bp. It suffices to deal
with ap. By (7.4) and (8.17), we have the following rule of computation:
ELift(f)(ap) = aG
prod(f)(p). (8.19)
We know that G
prod, as a composite of three functors, is a functor. Therefore,
G
prod(f1◦f2) = G
prod(f1)◦G
prod(f2). Using this equality and (8.19), we have
ELift(f1◦f2)(ap) = aG
prod(f1◦f2)(p)=a(G
prod(f1)◦G
prod(f2))(p)
=aG
prod(f1)(G
prod(f2)(p)) =ELift (f1)(aG
prod(f2)(p))
=ELift(f1)(ELift (f2)(ap)) = (ELift(f1)◦ELift (f2))(ap).
Thus, (8.18) holds, and ELift :A→Latsd
5is a functor, as required.
Clearly, the composite of faithful or totally faithful functors is a faithful
or totally faithful functor, respectively. By Theorem 3.6, F
com is totally
faithful. So is G0
forg. Therefore, G
prod =F
com ◦G0
forg ◦F
pos is faithful, and
it is totally faithful if so is F
pos. Hence, it follows from (8.19) that ELift is
faithful. Furthermore, if F
pos is totally faithful and X6=Y∈Ob(A), then
the same property of G
prod gives that {ap:p∈G
prod(X)}is distinct from
{ap:p∈G
prod(Y)}. Hence, it follows from Remark 6.3 and (8.17) that
ELift(X)6=ELift (Y). Consequently, ELift is totally faithful if so is F
pos.
Finally, we are going to prove that ELift lifts F
pos with respect to Princ.
The isomorphism provided by Corollary 8.2 will be denoted by ζX. That is,
ζX:hG
prod(X); νXi/ΘνX→ hPrinc(L(G
prod(X), νX)); ⊆i
(8.17)
= (Princ ◦ELift )(X),
defined by q/ΘνX7→ con(aq, bq),
(8.20)
is an order isomorphism. For X∈Ob(A), the map from hG
prod(X); νXi=
hF
com(X0); νXito hX0;≤0
Xi, defined by q7→ πcom
X(q), see Definition 3.1 or
around (8.14), is isotone by (8.8). By Theorem 3.6 (or Lemma 3.5), this map
is surjective. Furthermore, for p0, q0∈X0(that is, for p, q ∈X), if p0≤0
Xq0,
then h~v triv(p0), ~v triv(q0)i ∈ νXby (8.8) and, in addition, p0=πcom
X(~v triv(p0))
and q0=πcom
X(~v triv(q0)). So, the ordering ≤0
Xequals the πcom
X-image of νX.
Thus, using a well-known fact about orders induced by quasiorders, the map
hG
prod(X); νXi/ΘνX→ hX0;≤0
Xi,defined by q/ΘνX7→ πcom
X0(q),
is an order isomorphism. So is its inverse map,
hX0;≤0
Xi→hG
prod(X); νXi/ΘνXdefined by p07→ ~v triv(p0)/ΘνX.
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 31
Since hX;≤Xi → hX0;≤0
Xi, defined by x7→ x0=hx, ≤Xi, is also an order
isomorphism by (8.5), the composite
ξX:hX;≤Xi→hG
prod(X); νXi/ΘνX,defined by p7→ ~v triv(p0)/ΘνX,
(8.21)
of the two isomorphisms is also an order isomorphism. So we can let
κX:= ζX◦ξX,which is an order isomorphism (8.22)
from F
pos(X) = hX;≤Xito (Princ ◦ELift)(X) by (8.20) and (8.21). As the
last part of the proof, we are going to show that κ:F
pos →Princ ◦ELift
is a natural isomorphism. By (8.22), we have to show only that it is a
natural transformation. In order to do so, assume that X, Y ∈Ob(A) and
f∈MorA(X, Y ). Besides f=F
pos(f) and f0= (G0
forg ◦F
pos)(f), we will use
the notation h:= (Princ ◦ELift )(f). We have to show that the diagram
F
pos(X) = hX;≤Xif=F
pos(f)
−−−−−−−−−−−−−→ F
pos(Y) = hX;≤Yi
κX
y
κY
y
(Princ ◦ELift)(X)h=(Princ◦ELift )(f)
−−−−−−−−−−−−−→ (Princ ◦ELift)(Y)
(8.23)
commutes. First, we investigate the map h. For a triplet q∈G
prod(X), we
have that ELift(f)(aq) = aG
prod(f)(q)by (8.19). Analogously, ELift (f)(bq) =
bG
prod(f)(q). Therefore, applying the definition of Princ to the {0,1}-lattice
homomorphism ELift(f) : ELift(X)→ELift (Y), see (4.1) and (4.2), we have
that
h(con(aq, bq)) = con(aG
prod(f)(q), bG
prod(f)(q)). (8.24)
Consider an arbitrary p∈F
pos(X). By (8.20), (8.21), and (8.22),
κX(p) = ζX(ξX(p)) = ζX(~v triv(p0)/ΘνX) = con(a~vtriv (p0), b~v triv (p0)). (8.25)
Hence, (8.24) yields that
(h◦κX)(p) = con(aG
prod(f)(~v triv(p0)), bG
prod(f)(~v triv(p0))). (8.26)
On the other hand, using (8.25) for Yand f(p) instead of Xand p,
(κY◦f)(p) = κY(f(p)) = con(a~v triv (f(p)0), b~v triv (f(p)0)). (8.27)
We are going to verify that (8.26) and (8.27) give the same principal con-
gruence. Motivated by (C1), we focus on the colors of the respective or-
dered pairs that generate these two principal congruences. By the construc-
tion of our quasi-colored lattices, see Figure 6 and (6.19), these colors are
c1:= G
prod(f)(~v triv (p0)), in (8.26), and c2:= ~v triv (f(p)0), in (8.27). By (3.1),
(8.2), and (8.15),
πcom
Y0(c1) = πcom
Y0(G
prod(f)(~v triv (p0))) (8.15)
=f0(πcom
X0(~v triv(p0)))
(3.1)
=f0(p0)(8.2)
=f(p)0(3.1)
=πcom
Y0(~v triv(f(p)0)) = πcom
Y0(c2).
COMETIC FUNCTORS AND PRINCIPAL LATTICE CONGRUENCES 32
Hence, (8.8) yields that hc1, c2i ∈ νYand hc2, c1i ∈ νY. Thus, we conclude
from (C1) that (8.26) and (8.27) are the same principal congruences, which
means that the diagram given in (8.23) commutes. This proves that ELift
lifts F
pos with respect to Princ, as required. The proof of Theorem 4.7 is
complete.
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E-mail address:czedli@math.u-szeged.hu
URL:http://www.math.u-szeged.hu/~czedli/
University of Szeged, Bolyai Institute, Szeged, Aradi v´
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HUNGARY 6720
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