Representation of algebraic distributive lattices with $\aleph\_1$ compact elements as ideal lattices of regular rings

0.22 MB·English

by  Friedrich Wehrung

REPRESENTATION OF ALGEBRAIC DISTRIBUTIVE LATTICES
5 WITH ℵ COMPACT ELEMENTS AS IDEAL LATTICES OF
1
0
REGULAR RINGS
0
2

n FRIEDRICHWEHRUNG
a
J
Abstract. Weprovethefollowingresult:
2
2 Theorem. Every algebraic distributive lattice D with at most ℵ1 compact
elements isisomorphic tothe ideal lattice of a von Neumann regular ring R.

]
M (By earlier results of the author, the ℵ1 bound is optimal.) Therefore,
D is alsoisomorphic to the congruence lattice of a sectionally complemented
G modularlatticeL,namely,theprincipalrightideallatticeofR. Furthermore,
ifthe largestelementof D iscompact, then onecan assumethat Risunital,
.
h respectively,thatLhasalargestelement. Thisextends severalknownresults
t of G.M. Bergman, A.P. Huhn, J. T˚uma, and of a joint work of G. Gra¨tzer,
a
m H.Lakser,andtheauthor,anditsolvesProblem2ofthesurveypaper[10].
Themaintoolusedintheproofofourresultisanamalgamationtheorem
[ forsemilatticesandalgebras(overagivendivisionring),avariantofpreviously

known amalgamation theorems forsemilattices and lattices, due to J. T˚uma,
1
andG.Gra¨tzer,H.Lakser,andtheauthor.
v
7
6
3
1 Introduction
0
5 It is a well-knownand easy fact that the lattice of ideals of any (von Neumann)
0 regular ring is algebraic and distributive. In unpublished notes from 1986, G.M.
/
h Bergman [2] proves the following converse of this result:
t
a Bergman’s Theorem. Every algebraic distributive lattice D with countably many
m
compact elements is isomorphic to the ideal lattice of a regular ring R, such that if
: the largest element of D is compact, then R is unital.
v
i
X On the negative side, the author of the present paper, in [20], using his con-
struction in [19], provedthat Bergman’s Theoremcannot be extended to algebraic
r
a distributive lattices with ℵ many compact elements (or more). This left a gap at
2
the size ℵ , expressed by the statement of the following problem:
1
Problem 2 of [10]. Let D be an algebraic distributive lattice with at most ℵ
1
compact elements. Does there exist a regular ring R such that the ideal lattice of R
is isomorphic to D?
Inthispaper,weprovideapositivesolutiontothisproblem,seeTheorem5.2. Of
independent interest is an amalgamation result of ring-theoretical nature, mostly
inspired by the lattice-theoretical constructions in [17] and [13], see Theorem 4.2.
This result is the main tool used in the proof of Theorem 5.2.
1991 Mathematics Subject Classification. 16E50,16D25,06A12,06C20.
Keywordsandphrases. Ring,lattice,semilattice,Boolean,ideal,simple,diagramofalgebras.
1
2 F.WEHRUNG
Once the Amalgamation Theorem (Theorem 4.2) is proved, the representation
result follows from standard techniques, based on the existence of lattices called
2-frames in [6], or lower finite 2-lattices in [5]. Such a technique has, for example,
been used successfully in [14, 15], where A.P. Huhn proves that every distributive
algebraiclattice D with at most ℵ compactelements is isomorphic to the congru-
1
ence lattice of a lattice L. Theorem 5.2 provides a strengthening of Huhn’s result,
namely, it makes it possible to have L sectionally complemented and modular, see
Corollary5.3. Note that we already obtained L relatively complemented with zero
(thus sectionally complemented), though not modular, in [13].
We do not claim any originality about the proof methods used in this paper.
Most of what we do amounts to translations between known concepts and proofs
in universalalgebra,lattice theory, and ring theory. However,the interconnections
between these domains, as they are, for example, presented in [10], are probably
not well-established enough to trivialize the results of this paper.
1. Basic concepts
Lattices, semilattices. References for this section are [3, 11, 12].
LetLbealattice. WesaythatLiscomplete,ifeverysubsetofLhasasupremum.
An element a of L is compact, if for every subset X of L such that the supremum
of X, X, exists, a ≤ X implies that there exists a finite subset Y of X such
that a≤ Y. The unit of a lattice is its largest element, if it exists.
W W
We say that L is algebraic, if it is complete and every element is the supremum
W
of compact elements. If L is an algebraic lattice, then the set S of all compact
elementsofLisclosedunderthejoinoperation,anditcontains0(theleastelement
of L) as an element. We say that S is a {∨,0}-semilattice, that is, a commutative,
idempotent monoid (the monoid operation is the join).
If S is a {∨,0}-semilattice, an ideal of S is a nonempty, hereditary subset of S
closedunderthe joinoperation. The setIdS ofallideals ofS, partiallyorderedby
containment, is an algebraic lattice, and the compact elements of IdS are exactly
theprincipalideals(s]={t∈S |t≤s},fors∈S. Inparticular,thesemilatticeof
allcompactelementsofIdS isisomorphictoS. Conversely,ifLisanalgebraiclat-
ticeandifS isthesemilatticeofallcompactelementsofL,thenthemapfromLto
IdS that with every element x of L associates {s∈S |s≤x} is an isomorphism.
It follows that algebraic lattices and {∨,0}-semilattices are categorically equiva-
lent. The class of homomorphisms of algebraic lattices that correspond, through
this equivalence, to {∨,0}-homomorphisms of semilattices, are the compactness
preserving, -complete homomorphisms of algebraic lattices. (We say that a ho-
momorphism f: A → B of algebraic lattices is -complete, if f[X] = f( X),
W
for every(possibly empty) subset X ofA.) We observethat if B is finite, then any
W W W
homomorphism from A to B is compactness-preserving.
A {∨,0}-semilattice S is distributive, if its ideal lattice IdS is a distributive
lattice. Equivalently, S satisfies the following statement:
(∀a,b,c) c≤a∨b⇒(∃x,y)(x≤a and y ≤b and c=x∨y) .
For a lattice L, the(cid:0)set ConL of all congruences of L, endowed with co(cid:1)ntainment,
is an algebraic lattice. Its semilattice of compact elements is traditionally denoted
by Con L, the semilattice of finitely generated congruences of L.
c
We say that L is
REPRESENTATION OF ALGEBRAIC DISTRIBUTIVE LATTICES 3
— modular, if x∧(y∨z)=(x∧y)∨z for all x, y, z ∈L such that x≥z;
— complemented, if it has a least element, denoted by 0, a largest element,
denotedby 1,andfor alla∈L,there existsx∈L suchthata∧x=0 and
a∨x=1;
— sectionally complemented, if it has a least element, denoted by 0, and for
all a, b ∈ L such that a ≤ b, there exists x ∈ L such that a∧x = 0 and
a∨x=b.
We denote by 2 the two-element lattice.
Rings, algebras. All the rings encountered in this work are associative, but not
necessarily unital. A ring R is regular (in von Neumann’s sense), if it satisfies the
statement (∀x)(∃y)(xyx = x). If R is a regular ring, then the set of all principal
right ideals of R, partially ordered by containment, is a sectionally complemented
modular lattice, see, for example, [7, Page 209].
IfRisaring,thenwedenotebyIdRthesetofalltwo-sidedidealsofR,partially
orderedbycontainment. ThenIdRisanalgebraicmodularlattice,whichturnsout
tobedistributive ifRisregular. We denotebyId Rthe semilatticeofallcompact
c
elements of IdR, that is, the finitely generated two-sided ideals of R. It is to be
noted that Id can be extended to a functor from rings and ring homomorphisms
c
to {∨,0}-semilattices and {∨,0}-homomorphisms, and that this functor preserves
direct limits.
If K is a division ring, a K-algebra is a ring R endowed with a structure of
two-sided vector space over K such that the equalities
λ(xy)=(λx)y, (xλ)y =x(λy), (xy)λ=x(yλ)
hold for all x, y ∈ R, and λ ∈ K. Such a structure is called a K-ring in [4,
Section 1]. Most of the rings that we shall encounter in this work are, in fact,
algebras.
2. Embedding into V-simple algebras
Definition 2.1. A unital, regularring R is V-simple, if R is isomorphicto allits
R
nonzero principal right ideals, and there are nonzero principal right ideals I and J
of R such that I⊕J =R .
R
It is obvious that if R is V-simple, then it is simple. The converse is obviously
false, for example, if R is a field.
Notation. Let κ be an infinite cardinal number, let U be a two-sided vector space
over a division ring K. For example, if I is any set, then the set K(I) of all I-
families with finite support of elements of K is endowed with a natural structure
of two-sided vector space over K.
Let N (U) be the subset of the algebra End (U) of right K-vector space endo-
κ K
morphisms of U defined by
N (U)={f ∈End (U)|dim imf <κ}.
κ K K
It is obvious that N (U) is a two-sided ideal of End (U). We define a K-algebra,
κ K
E (U), by
κ
E (U)=End (U)/N (U).
κ K κ
The idea behind the proof of Lemma 2.2 and Proposition 2.3 is old, see, for
example, Theorem 3.4 in [1]. For convenience, we recall the proofs here.
4 F.WEHRUNG
Lemma2.2. Letκbeaninfinitecardinal number,letU beatwo-sidedvectorspace
of dimension κ over a division ring K. Then E (U) is a unital, regular, V-simple
κ
K-algebra.
Proof. Note that E (U) is nontrivial, because κ ≤ dim U. Since the endomor-
κ K
phismringEnd (U)isregular,soisalsothequotientringE (U)=End (U)/N (U),
K κ K κ
see Lemma 1.3 in [8].
Furthermore, the principal right ideals of R = End (U) are exactly the ideals
K
of the form
I ={f ∈R|imf ⊆X},
X
for a subspace X of U. If X and Y are subspaces of U, then X ∼= Y implies that
IX ∼= IY. If X is a subspace of dimension κ of U, then X can be decomposed as
X =X ⊕X , where dim X =dim X =κ, hence
0 1 K 0 K 1
[I ]=[I ]+[I ]=2[I ],
X X0 X1 X
where [I] denotes the isomorphism class of a right ideal I. Since X ∼= U, [IX] =
[I ]= [R]. However, if X is a subspace of U of dimension < κ, then the image of
U
I in E (U) is the zero ideal. The conclusion follows. (cid:3)
X κ
Proposition 2.3. Let K be a division ring. Every unital K-algebra has a unital
embedding into a unital, regular, V-simple K-algebra.
Proof. Let R be a unital K-algebra. Put κ=ℵ +dim R, where dim R denotes
0 K K
the right dimension of R over K, and U = R(κ), the R-algebra of all κ-sequences
with finite support of elements of R. We put
S =E (U).
κ
Since the dimension of U over K equals κ, it follows from Lemma 2.2 that S is a
unital, regular,V-simple K-algebra.
Define a map ϕ: R→End (U), by the rule
K
ϕ(a): U →U, x7→ax,
for all a ∈ R. It is easy to see that ϕ is a unital ring homomorphism from R to
End (U).
K
Let a ∈ R\{0}. If he | ξ < κi denotes the canonical basis of U over R, then
ξ
the range of ϕ(a) contains all the elements ae , for ξ < κ, thus its dimension over
ξ
K is greaterthanorequalto κ. Hence,ϕ(a) does notbelong to N (U). Therefore,
κ
the map ψ from R to S defined by the rule
ψ(a)=ϕ(a)+N (U),
κ
for all a∈R, is a unital K-algebra embedding from R into S. (cid:3)
3. Amalgamation of algebras over a division ring
ThefollowingfundamentalresulthasbeenprovedbyP.M.Cohn,seeTheorem4.7
in [4]. We also refer the reader to the outline presented in [16, Page 110], in the
section “Regular rings: AP”.
Theorem3.1. LetK beadivision ring, letR,A, andB beunitalK-algebras, with
R regular. Let α: R֒→A and β: R֒→B be unital embeddings. Then there exist a
unital, regular K-algebra C, α′: A֒→C, and β′: B ֒→C, such that α′◦α=β′◦β.
REPRESENTATION OF ALGEBRAIC DISTRIBUTIVE LATTICES 5
By combining Theorem 3.1 with Proposition 2.3, we obtain immediately the
following slight strengthening of Theorem 3.1:
Lemma 3.2. Let K be a division ring, let R, A, and B be unital K-algebras,
with R regular. Let α: R ֒→ A and β: R ֒→ B be unital embeddings. Then there
exist a unital, regular, V-simple K-algebra C, unital embeddings α′: A ֒→ C, and
β′: B ֒→C, such that α′◦α=β′◦β.
The following example partly illustrates the underlying complexity of Theo-
rem3.1,byshowingthateveninthecasewhereAandBarefinite-dimensionalover
R,onemaynot beabletofindafinite-dimensionalsolutionC totheamalgamation
problem:
Example 3.3. Let K be any division ring. We construct unital, matricial ex-
tensions A and B of the regular K-algebra R = K2 such that the amalgamation
problem of A and B over R has no finite-dimensional solution.
Proof. We put A= M (K) (resp., B = M (K)), the ring of all square matrices of
2 3
order 2 (resp., 3) over K, endowed with their canonical K-algebra structures. We
define unital embeddings of K-algebras f: R֒→A and g: R֒→B as follows:
x 0 0
x 0
f(hx,yi)= , and g(hx,yi)= 0 x 0 , (1)
0 y  
(cid:18) (cid:19) 0 0 y
for all x, y ∈K.  
Suppose that the amalgamation problem of A and B over R (with respect to
f and g) has a solution, say, f′: A ֒→ C and g′: B ֒→ C, where C is a finite-
dimensional, not necessarily unital K-algebra and f′, g′ are embeddings of unital
K-algebras. For a positive integer d, we denote by ed , for 1 ≤ i,j ≤ d, the
i,j
canonical matrix units of the matrix ring M (K). So, (1) can be rewritten as
d
f(hx,yi)=e2 x+e2 y, and g(hx,yi)=(e3 +e3 )x+e3 y, (2)
1,1 2,2 1,1 2,2 3,3
for all x, y ∈ K. Put u = f′(e2 ), for all i, j ∈ {1,2}, and v = g′(e3 ), for all
i,j i,j i,j i,j
i, j ∈{1,2,3}. Then apply f′ (resp. g′) to the first (resp., second) equality of (2).
Since f′◦f =g′◦g, we obtain that the equality
u x+u y =(v +v )x+v y,
1,1 2,2 1,1 2,2 3,3
holds, for all x, y ∈K. Specializing to x, y ∈{0,1} yields the equalities
u =v +v , (3)
1,1 1,1 2,2
u =v . (4)
2,2 3,3
However, the elements u of C, for i, j ∈ {1,2}, satisfy part of the equali-
i,j
ties defining matrix units in C, namely, u u = δ u , for all i, j, k, l ∈
i,j k,l j,k i,l
{1,2} (δ denotes here the Kronecker symbol). Similarly, the elements v of C,
− i,j
for i, j ∈ {1,2,3}, satisfy the equalities v v = δ v , for all i, j, k, l ∈
i,j k,l j,k i,l
{1,2,3}. Therefore,by (3) and (4), dim (u C)=dim (v C)+dim (v C)=
K 1,1 K 1,1 K 2,2
2dim (v C) = 2dim (u C) = 2dim (u C). But then, since C is finite-
K 3,3 K 2,2 K 1,1
dimensional, dim (u C) = 0, so dim (f′A) = dim (g′B) = 0, a contradic-
K 1,1 K K
tion. (cid:3)
For a further discussion of Example 3.3, see the comments following the state-
ment of Problem 1 in Section 6.
6 F.WEHRUNG
4. The amalgamation theorem
Definition 4.1. Let K be a division ring. A K-algebra R is V-Boolean, if it is
isomorphic to a finite direct product of V-simple K-algebras.
Theorem 4.2. Let K be a division ring, let R , R , and R be unital K-algebras,
0 1 2
withR regular, let S bea finiteBoolean lattice. For k ∈{1,2},let f : R →R be
0 k 0 k
a homomorphism of unital K-algebras and let ψ : IdR →S be a unit-preserving
k k
-complete homomorphism, such that ψ ◦ Idf = ψ ◦ Idf . Then there exist
1 1 2 2
a unital, regular, V-Boolean K-algebra R, homomorphisms of unital K-algebras
W
g : R →R, for k ∈{1,2}, and an isomorphism α: IdR→S such that g ◦f =
k k 1 1
g ◦f and α◦Idg =ψ for k ∈{1,2} (see Figure 1).
2 2 k k
IdR
(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)EEα(cid:15)(cid:15) YY3333333
g(cid:0)(cid:0)1(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)@@R^^>>>>>g>>2 (cid:11)(cid:11)y(cid:11)(cid:11)yI(cid:11)y(cid:11)d(cid:11)ψg11yyy<<SOO bbEEEIψd2g3E323E3E333
R R IdR ψ IdR
1f^^==1===== (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)f(cid:1)2(cid:1)@@ 2 Id1bbEEfE1EEEEE 0 yyyyyIydyyf<<2 2
R IdR
0 0
Figure 1
Proof. We adapt to the context of regular rings the lattice-theoretical proof of
Theorem 1 of [13].
We put ψ =ψ ◦Idf =ψ ◦Idf . We start with the following case:
0 1 1 2 2
Case 1. S ∼=2.
We put I = {x ∈ R | ψ (R xR ) = 0}, for k ∈ {0,1,2}. Since ψ is a
k k k k k k
-complete homomorphism, I is the largest ideal of R whose image under ψ is
k k k
zero. Furthermore, since ψ is unit-preserving, I is a proper ideal of R .
k k k
W Next, we put R = R /I , and we denote by p : R ։ R the canonical pro-
k k k k k k
jection.
For k ∈ {1,2}, the equivalence x ∈ I ⇔ f (x) ∈ I holds for all x ∈ R , thus
0 k k 0
there exists a unique unital embedding f : R ֒→ R such that p ◦f = f ◦p ,
k 0 k k k k 0
see Figure 2.
R
g (cid:1)(cid:1)@@ ^^=== g
1(cid:1)(cid:1) ==2
(cid:1) =
fk (cid:16)(cid:1)0 (cid:1) .=N
R0 // Rk R1]]<< (cid:2)AAR2
< (cid:2)
p0 (cid:15)(cid:15)(cid:15)(cid:15) (cid:31)(cid:127) (cid:15)(cid:15)(cid:15)(cid:15)pk f1<<<<. N (cid:16)0 (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)f2
R //R R
0 k 0
f
k
Figure 2
REPRESENTATION OF ALGEBRAIC DISTRIBUTIVE LATTICES 7
Since R is regularand f ,f areunital embeddings, there exist, by Lemma 3.2, a
0 0 1
unital,regular,V-simpleK-algebraRandembeddings g : R ֒→R,fork ∈{1,2},
k k
such that g ◦f =g ◦f , see Figure 2.
1 1 2 2
We put g =g ◦p , for k ∈{1,2}, see Figure 3.
k k k
//R oo
g (cid:1)(cid:1)@@ ^^=== g
g1 1(cid:1)(cid:1)(cid:1) ===2 g2
(cid:16)(cid:1)0 (cid:1) .=N
R R
OOOO1]]<<< f f (cid:2)(cid:2)AA OOOO2
p1 <<<<1. N (cid:16)0 (cid:2)(cid:2)2(cid:2)(cid:2)(cid:2) p2
R1^^<<< ROOOO0 (cid:2)(cid:2)@@R2
f1<<<<p0(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)f2
R
0
Figure 3
Therefore,
g ◦f =g ◦p ◦f =g ◦f ◦p =g ◦f ◦p =g ◦f .
1 1 1 1 1 1 1 0 2 2 0 2 2
Since R is V-simple, it is simple, thus, since S ∼= 2, there exists a unique isomor-
phism α: IdR → S. To verify that α◦Idg = ψ , for k ∈ {1,2}, it suffices to
k k
verify that (Idg )(I)=0 iff ψ (I)=0, for every I ∈IdR . We proceed:
k k k
(Idg )(I)=0 iff g [I]=0
k k
iff g ◦p [I]=0
k k
iff p [I]=0
k
(because g is an embedding)
k
iff I ⊆I
k
iff ψ (I)=0,
k
which concludes Case 1.
Case 2. General case, S finite Boolean.
Without loss of generality, S = 2n, with n < ω. For i < n, let π : S ։ 2
i
be the projection on the i-th coordinate. We apply Case 1 to the maps π ψ , for
i k
8 F.WEHRUNG
k ∈{1,2}, see Figure 4.
IdR(i)
(cid:11)EE YY33
(cid:11) 3
(cid:11)(cid:11)(cid:11) αi 333
(cid:11) 3
(cid:11) (cid:15)(cid:15)
g1(cid:127)(cid:127),i(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)R?? (i)__?????g?2?,i (cid:11)(cid:11)x(cid:11)(cid:11)Ix(cid:11)dπ(cid:11)x(cid:11)igψ1,1xixx<< 2OO ccFFIπFdiψg232F3,F3i3F333
R1f__??1????? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)f(cid:127)2(cid:127)??R2 IdRI1dccFFfF1FFFFFψ0 xxxxxIdxxfxI;;2dR2
R IdR
0 0
Figure 4
We obtain a unital, regular, V-simple K-algebra R(i), K-algebra homomorphisms
g , for k ∈ {1,2}, an isomorphism α : IdR(i) → 2, such that g ◦f = g ◦f
k,i i 1,i 1 2,i 2
and α ◦Idg =π ◦ψ , for k∈{1,2}, see Figure 4.
i k,i i k
Now we put R = R(i), with the componentwise ring structure. So R is a
i<n
unital, regular,V-Boolean K-algebra. For k ∈{1,2},we define a unital homomor-
L
phism g : R →R by the rule
k k
g (x)=hg (x)|i<ni, for all x∈R .
k k,i k
It is immediate that g ◦f =g ◦f .
1 1 2 2
Furthermore, observe that IdR(i) ∼=IdR, via the isomorphism that sends
i<n
a finite sequence hI |i < ni to I . Define an isomorphism α: IdR →S by
i Q i<n i
the rule
L
α I =hα (I )|i<ni, for all I ∈IdR.
i i i
!
i<n
M
For k∈{1,2} and I ∈IdR , we compute:
k
α◦(Idg )(I)=α (Idg )(I)
k k,i
!
i<n
M
=hα ◦(Idg )(I)|i<ni
i k,i
=hπ ◦ψ (I)|i<ni
i k
=ψ (I),
k
so α◦Idg =ψ . (cid:3)
k k
5. The representation theorem
We first state a useful lemma, see [5] and [6]:
Lemma 5.1. There exists a lattice F of cardinality ℵ satisfying the following
1
properties:
(i) F is lower finite, that is, for all a∈F, the principal ideal
F[a]={x∈F |x≤a}
is finite.
(ii) Every element of F has at most two immediate predecessors.
REPRESENTATION OF ALGEBRAIC DISTRIBUTIVE LATTICES 9
Theorem 5.2. Let D be an algebraic distributive lattice with at most ℵ compact
1
elements, and let K be a division ring. Then there exists a regular K-algebra R
satisfying the following properties:
(i) IdR∼=D.
(ii) If the largest element of D is compact, then R is a direct limit of unital,
regular, V-Boolean K-algebras and unital embeddings of K-algebras. In
particular, R is unital.
Proof. A similar argument has already been used, in different contexts, in such
various references as [6, 14, 15, 13].
We first translate the problem into the language of semilattices. This amounts,
by defining S as the {∨,0}-semilattice of compact elements of D, to verifying the
existence of a regular K-algebra R satisfying the following condition
(i’) Id R∼=S,
c
along with (ii). We do this first in the case where the largest element of D is
compact, that is, where S has a largest element. It is proved in [10] that S is a
directlimit offinite Boolean{∨,0}-semilatticesand{∨,0,1}-homomorphisms,say,
hS,ϕ i =limhS ,ϕii ,
i i∈I −→ i j i≤j inI
where I is a directed partially ordered set, and hS ,ϕii is a direct system
i j i≤j inI
of finite Boolean {∨,0}-semilattices and {∨,0,1}-homomorphisms (in particular,
ϕi: S → S and ϕ : S → S, for i ≤ j in I). Furthermore, one can take I
j i j i i
countably infinite if S is finite, and |I|=|S| if S is infinite. In particular,|I|≤ℵ .
1
Let F be a lattice satisfying the conditions of Lemma 5.1. Since |F| ≥ |I| > 0,
there exists a surjective map ν : F ։I. Since F is lower finite, it is well-founded,
0
sowecandefine inductivelyanorder-preserving,cofinalmapν: F →I,byputting
ν(x)=any element i of I such that ν (x)≤i and ν(y)≤i, for all y <x,
0
for all x ∈ F. This is justified because F is lower finite, and this does not use
part (ii) of Lemma 5.1. As a conclusion, we see that we may index our direct
system by F itself, that is, we may assume that I satisfies the conditions (i), (ii)
of Lemma 5.1.
Weshallnowdefineinductivelyunital,regular,V-BooleanK-algebrasR ,unital
i
homomorphisms of K-algebras fi: R → R , and isomorphisms ε : Id R → S ,
j i j i c i i
for i≤j in I.
Let ̺: I →ω be the natural rank function, that is,
̺(i)=sup{̺(j)|j <i}+1,
for all i∈I. For all n<ω, we put
I ={i∈I |̺(i)≤n}.
n
By induction on n < ω, we construct V-Boolean K-algebras R (note then that
i
IdR = Id R ), maps ε : IdR → S , and unital homomorphisms of K-algebras
i c i i i i
fi: R →R , for all i≤j in I , satisfying the following properties:
j i j n
(a) fi =id , for all i∈I .
i Ri n
(b) fi =fj ◦fi, for all i≤j ≤k in I .
k k j n
(c) ε is a lattice isomorphism from IdR onto S , for all i∈I .
i i i n
10 F.WEHRUNG
(d) The following diagram is commutative, for all i≤j in I :
n
Idfi
IdR −−−−j→ IdR
i j
εi εj
 
Si −−−−→ Sj
y ϕi y
j
For n=0, it suffices to construct a V-Boolean K-algebra R such that IdR ∼=
0 0
S . This is easy: if p is the number of atoms of S , take any V-simple K-algebra
0 0
R, and put R =Rp.
0
Suppose having done the constructionon I , we show how to extend it to I .
n n+1
Let i ∈ I such that ̺(i) = n + 1. Denote by i and i the two immediate
n+1 0 1
predecessors of i in I. Note that i and i do not need to be distinct.
0 1
IdR
CC [[6i
(cid:8) 6
(cid:8) 6
(cid:8) ε 6
(cid:8) i 6
(cid:8) 6
(cid:8)(cid:8) (cid:15)(cid:15) 6
R Idg S Idg
{g{{0{{{{{== i aaCCCCCgCC1C (cid:8)(cid:8)(cid:8)u(cid:8)u(cid:8)ϕu(cid:8)(cid:8)ii0ε0i0uuu:: i ddIIϕIii1ε6i161I6I6I666
R R IdR IdR
fiii000∧aaCCiC1CCCCC {{{{{f{i{i10{∧== i1i1 Idif0iddHi00HH∧HiH1HHHH vvvvIvdvvfvivi10:: ∧i1i1
R IdR
i0∧i1 i0∧i1
Figure 5
By Theorem 4.2, there exist a unital, regular, V-Boolean K-algebra R , unital
i
homomorphisms of K-algebras g : R → R , for k < 2, and an isomorphism
k ik i
ε : IdR →S such that the following equalities hold:
i i i
g ◦fi0∧i1 =g ◦fi0∧i1; (5)
0 i0 1 i1
ε ◦Idg =ϕikε , for k ∈{1,2}, (6)
i k i ik
see Figure 5. If i =i , we may replace g by g : the diagrams of Figure 5 remain
0 1 1 0
commutativeand(5), (6)remainvalid. Thuswe may definefi0 =g andfi1 =g ,
i 0 i 1
and (5), (6) are restated as
fi0 ◦fi0∧i1 =fi1 ◦fi0∧i1; (7)
i i0 i i1
ε ◦Idfik =ϕikε , for k ∈{1,2}. (8)
i i i ik
Atthispoint,wehavedefinedfj,ifi∈I \I andj isanimmediatepredecessor
i n+1 n
of i. If i ∈ I \I and j < i, then the only possibility is to put fj = fiν ◦fj ,
n+1 n i i iν
where ν < 2 is such that j ≤ i . For this to be possible, we need to verify that if
ν
j ≤i ∧i ,thenfi0◦fj =fi1◦fj. Thisfollowsfrom(7),alongwiththe following
0 1 i i0 i i1
sequence of equalities:
fi0 ◦fj =fi0 ◦fi0∧i1 ◦fj
i i0 i i0 i0∧i1
=fi1 ◦fi0∧i1 ◦fj
i i1 i0∧i1
=fi1 ◦fj.
i i1