This document summarizes a thesis on congruence distributive varieties with the compact intersection property. It begins with an introduction discussing congruence lattices of algebras and the observation that varieties where this is well understood often have the property that compact congruences intersect compactly. The thesis will characterize locally finite, congruence-distributive varieties with this property. Basic definitions and theorems on congruences, compact elements, and directed systems are provided. The main results are that compact intersection is equivalent to finite subalgebras of subdirectly irreducible algebras being subdirectly irreducible, and that compact intersection is also equivalent to the meet-preserving property for embeddings of finite algebras.

1. P. J. ’AFÁRIK UNIVERSITY, Ko²ice
Faculty of Science
Institute of Mathematics
Congruence distributive varieties with compact
intersection property
(Rigorózna práca)
Ko²ice, 2011 Filip Krajník

2. Contents
Introduction 1
1 Basic denitions and denotations 2
2 Congruence intersection property 6
2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Some special varieties 13
4 Conclusion 18
References 19

3. Introduction
It is well known that a lattice is algebraic if and only if it is isomorphic to the
congruence lattice of some algebra. Much less is known about congruence lattices of
algebras of a specic type.
Let K be a class of algebras and denote by Con K the class of all lattices isomorphic
to Con A (the congruence lattice of an algebra A) for some A ∈ K. There are many
papers investigating Con K for various classes K. However, the full description of Con K
has proved to be a very dicult (and probably intractable) problem, even for the most
common classes of algebras, like groups or lattices.
The present work is motivated by the observation that in most relevant cases when
Con K is well understood, the algebras in K have a special property: the compact
intersection property (CIP), i.e. the intersection of any two compact congruences of
A ∈ K is compact. This seems quite natural. Algebraic lattices are determined by
their sets of compact elements. There is a considerable evidence that the diculty in
describing congruence lattices is connected with the fact that the compact congruences
form a join-semilattice, which in general is not a lattice. For instance, there are several
renement properties, which are trivial in lattices, but very nontrivial in semilattices
([12], [7], [8]).
There are nice results using the above intersection property. Let us mention the
following two. Every algebraic distributive lattice in which the compact elements are
closed under intersection is isomorphic to the congruence lattice of a lattice (E. T.
Schmidt [11].) Similarly, every algebraic distributive lattice in which the compact ele-
ments are closed under intersection is isomorphic to the congruence lattice of a locally
matricial algebra (P. R uºicka [10]).
In our work we characterize locally nite, congruence-distributive varieties V with
the property that compact congruences of any A ∈ V are closed under intersection. Our
assumptions on V are essential in our arguments, but there are examples suggesting
that our results have a broader validity.
1

4. 1 Basic denitions and denotations
Let L be a lattice. An element a ∈ L is called compact i ∀X ⊆ L, if a ≤ X, then
a ≤ Y for some nite Y ⊆ X. Let K be a class of an algebras. Let Con A be the
congruence lattice of an algebra A ∈ K and denote Conc A ⊆ Con A a join-semilattice
of all compact congruences of algebra A.
An element a ∈ L is called strictly meet-irreducible i a = X implies that a ∈ X,
for every subset X of L. Note that the greatest element of L is not strictly meet-
irreducible. Further, let M(L) denote the set of all strictly meet-irreducible elements.
Let SI(K) denote the class of all subdirectly irreducible members of K.
If f is a function, then dom(f) stand for its domain. By Ker(f) we denote the binary
relation on dom(f) given by (x, y) ∈ Ker(f) i f(x) = f(y). We denote A ≤ B if A ∈ K
is subalgebra of an algebra B ∈ K. For a subset B ⊆ A let B denote the subalgebra
of A generated by B. If P is an ordered set and x ∈ P then ↑x = {y ∈ P | y ≥ x}.
By f X we mean the restriction of f to X. If B ≤ A and θ is a congruence on A, we
dene θ B to be θ ∩ B2
, the restriction of θ to B.
Theorem 1.1. [4]
If L is an algebraic lattice, then for all a ∈ L, a = X, where X =
{b | a ≤ b, b ∈ M(L)} and ∀x, y ∈ L where x y, ∃z ∈ M(L), z ≥ y, z x.
Theorem 1.2. [4]
If A ∈ K and θ is a strictly meet-irreducible element of Con A, then the quotient algebra
A/θ is subdirectly irreducible and its congruence lattice is isomorphic to
↑θ = {α ∈ Con A | θ ⊆ α}.
Hence M(Con A) is the set of all θ ∈ Con A such that the quotient algebra A/θ is
subdirectly irreducible. Equivalently, α ∈ M(Con A) if and only if α = Ker(f) for some
surjective homomorphism f : A → S, with S ∈ SI(K).
Let (P, ≤) be a directed set. Let (Ap, p ∈ P) be a family of objects indexed by P
and fp,q : Ap → Aq be a homomorphism for all p ≤ q, where
1. fp,p is the identity of Ap
2. fp,r = fp,q ◦ fq,r for all p ≤ q ≤ r
2

5. The (Ap, fp,q) is called a directed system over P. The disjoint union of the family
(Ap, p ∈ P) is the set p∈P Ap = p∈P {(x, p) | x ∈ Ap}. Directed limit of the directed
system (Ap, fp,q) is dened as the disjoint union of the Ap modulo a certain equivalence
relation ∼
lim
→
Ap :=
p∈P
Ap/ ∼
If x ∈ Ap, y ∈ Aq, then
x ∼ y ⇔ ∃r ∈ P : fp,r(x) = fq,r(y).
Hence if fp,q are set inclusions, then
lim
→
Ap =
p∈P
Ap.
Note that varieties of algebras are closed under a directed limits.
For every homomorphism f : A → B (A, B ∈ K) we dene the mapping
Conc f : Conc A → Conc B
by the rule that Conc f(α) is the congruence generated by the set
{(f(x), f(y)) | (x, y) ∈ α}, for every α ∈ Conc(A). It is well known, that Conc is
a functor from K to S (category of a (0,∨)-semilattices), thus
1. Conc f ◦ g = Conc f ◦ Conc g
2. Conc idA = idConc A for every A ∈ K
Let A = (Ap, ϕp,q) be a directed system in K, then ConcA = (Conc Ap, Conc ϕp,q) is a
directed system in S. We also know that Conc preserves directed limits, i.e.
Conc lim
→
Ap = lim
→
Conc Ap
Further let A = (Ap, ϕp,q) and B = (Bp, ψp,q) be a directed systems over P and
hp : Ap → Bp be an isomorphisms for every p ∈ P.
Ap Eϕp,q
hp
c
Aq
Bp
Eψp,q
Bq
c
hq
Diagram 1
3

6. If diagram 1 comutes, then
lim
→
Ap lim
→
Bp.
Let P be an ordered set, denote A := {Ap | p ∈ P} and F := {fp,q | p, q ∈ P, p ≤ q}.
Then a triple (P, A, F) is called ordered diagram of sets. For every such triple we dene
its inverse limit as
lim
←
Ap := {a ∈
p∈P
Ap | aq = fp,q(ap) for every p, q ∈ P, p ≤ q}.
If P is an antichain, then inverse limit is the direct product.
Denition 1.1.
An ordered diagram of sets (P, A, F) is called admissible if the following conditions are
satised:
1. for every p ∈ P and every u ∈ Ap there exists
a ∈ lim
←
Ap
such that ap = u.
2. for every p, q ∈ P, p q there exists
a, b ∈ lim
←
Ap
such that ap = bp and aq = bq.
Theorem 1.3. [9]
Let V be a locally nite congruence distributive variety. Let L be a nite distributive
lattice and let P = M(L). For every p ∈ P let Ap ∈ V and for every p ≤ q let fp,q :
Ap → Aq be a homomorphism such that (P, A, F) is an admissible ordered diagram of
sets and, moreover, for all p ∈ P, the sets {Ker(fp,q) | p ≤ q} and M(Con Ap) coincide.
Then
A := lim
←
Ap
is an algebra whose congruence lattice is isomorphic to L.
Sketch of a proof. Since every nite distributive lattice is determined by its or-
dered set of meet irreducible elements, it suces to prove that the ordered sets P and
M(Con A) are isomorphic. Denote hp : A → Ap as natural projections and dene a
map ϕ : P → M(Con A) by ϕ(p) = Ker(hp). This is the required isomorphism.
4

7. Further, let K, L ∈ S ((0, ∨)-semilattices). Let ϕ : K → L be a (0, ∨)-
homomorphism, then dene a map ψ : L → K
ψ(β) = {α | ϕ(α) ≤ β}.
The following is well known
1. ψ preserves ∧ and the largest element.
2. ϕ(α) = {β | α ≤ ψ(β)}.
3. ϕ(α) ≤ β ⇔ α ≤ ψ(β).
The construction also works for innite complete lattice. Such pair (ϕ, ψ) is known as
Galois connection. Note that if K = Conc A, L = Conc B and ϕ = Conc f, for some
A, B ∈ K, f : A → B then ψ(β) = {(x, y) ∈ A | (f(x), f(y)) ∈ β}. If A is a subalgebra
of B and f : A → B is the inclusion, then ψ(β) is the restriction of β ∈ Con B to A.
5

8. 2 Congruence intersection property
Let V be a locally nite congruence distributive variety. Let MSI(K) denote the
class of all partially ordered sets that are isomorphic to a subset of M(Con A) for a
subalgebra A of an algebra B ∈ SI(K).
Theorem 2.1. [6]
If maximal cardinality of a set P ∈ MSI(V) is 1, then the compact elements of Con A
are closed under intersection.
We generalize this theorem, we nd necessary and sucient conditions of compact-
ness of intersection of two compact congruences of an algebra. Denote
NSI := {M(Con A) | A ≤ B ∈ SI(V), A - nite}.
Theorem 2.2.
The following conditions are equivalent
1. Intersection of two compact congruences of A is compact for every A ∈ V.
2. Finite subalgebra of subdirectly irreducible alegbra in V is subdirectly irreducible.
3. If Q ∈ NSI, then Q is down directed.
4. If Q ∈ NSI, then Q has a least element.
5. For every embedding f : A → B of algebras in V with A nite, the mapping Conc f
preserves meets.
Proof.
(2)⇔(3)⇔(4) trivial
(1)⇒(3)
Let T ≤ S ∈ SI(V). We show that for all β1, β2 ∈ M(Con T) there exists β ∈ M(Con T)
such that β ⊆ β1 ∩ β2. Let A := F(ℵ0) denote the free algebra in V with ℵ0 as free
generating set. Choose a surjective map h0 : X0 → T, where X0 ⊆ ℵ0 is nite and
large enough. Since A is free, h0 can be extended to a homomorphism h : A → T.
Further, we consider the natural homomorphisms g1 : T → T/β1, g2 : T → T/β2, then
Ker(gih0) ∈ M(Con X0 ).
6

9. c
X0
r
rrrrj
T
¨
¨¨¨¨%
T/β2T/β1
g1 g2
h0
Figure 1
Since Con X0 is nite and distributive, there is the smallest element γi in a set
{α ∈ Con X0 | α Ker(gih0)}. Let αi ∈ Con A is the congruence generated by γi.
Choose a surjective map d : X0 → X0 /γi, Ker(d) = γi. Since A is free, d can
be extended to a homomorphism l : A → X0 /γi, thus Ker(l X0 ) = γi. Since
αi ⊆ Ker(l), αi X0 ⊆ Ker(l X0 ) = γi, and thus αi X0 = γi. Congruences
α1, α2 are compact, then α1 ∩ α2 is compact too. It means that there exists a nite set
Y ⊆ ℵ0, X0 ⊆ Y such that α1 ∩ α2 is generated by α1 ∩ α2 Y .
Let f : A → S be a surjective homomorphism such that f Y = h Y , then
Ker(f X0 ) = Ker(h X0 ) ⊆ Ker(gih0). Thus γi Ker(f X0 ) and hence αi
Ker(f). Since Ker(f) ∈ M(Con A), then α1 ∩ α2 Ker(f) and thus
α1 ∩ α2 Y Ker(f Y ) = Ker(h Y ).
Therefore there exists δ ∈ M(Con Y ) such that
δ ≥ Ker(h Y ), δ α1 ∩ α2 Y .
Let b0 : Y → Y |δ := W be the natural map, it can be extended to a ho-
momorphism b : A → W. Moreover for all y ∈ Y there exists x0 ∈ X0 such
that (y, x0) ∈ Ker(h). Therefore (y, x0) ∈ Ker(b0), so b0(y) = b0(x0). It shows that
b0( X0 ) = b( Y ) = W.
Since Ker(b0 X0 ) = δ X0 ⊇ Ker(h X0 ), there exists a homomorphism
k : T → W such that kh X0 = b0 X0 . Further since b0( X0 ) = W ∈ SI (V),
Ker(k) ∈ M(Con T). Further, α1 ∩ α2 Y Ker(b0) implies that α1 ∩ α2 Ker(b)
and thus α1, α2 Ker(b).
7

10. X0 Eh X0
b0 X0
c
T
¨
¨¨¨
¨¨¨
¨¨¨%
k
W
Figure 2
Since αi are generated by γi for i=1,2, we have γi Ker(b) and thus
γi Ker(b0 X0 ).
By denition γi it means Ker(b0 X0 ) ⊆ Ker(gih0). For every (x, y) ∈ Ker(k) we
have x , y ∈ X0 such that h(x ) = x, h(y ) = y. Thus (x , y ) ∈ Ker(b0 X0 ), so
(x , y ) ∈ Ker(gih0). It means that gi(h0(x )) = gi(h0(y )), hence gi(x) = gi(y). We have
proved that Ker(k) ≤ Ker(gi) = βi for i=1,2.
(3)⇒(1)
Let A ∈ V and suppose that α1, α2 ∈ Con A are compact, but α1 ∩ α2 is not compact.
There exists nite subalgebra Y ≤ A such that αi Y generates αi (i = 1, 2). De-
note γi := αi Y . Since Con Y is nite distributive lattice, there exists ∨-irreducible
δ1, δ2, . . . , δn, ε1, ε2, . . . , εm ∈ Con Y such that γ1 = n
j=1 δj, γ2 = m
k=1 εk. Let
¯δj ∈ Con A is generated by δj, ¯εj similarly. Since δj ⊆ γ1 ⊆ α1 ∈ Con A, ¯δj ⊆ α1.
Moreover n
j=1
¯δj ⊇ n
j=1 δj = γ1 = α1 Y , thus α1 = n
j=1
¯δj, α2 = m
k=1 ¯εk similarly.
Hence by distributivity α1 ∩ α2 = i,j(¯δj ∩ ¯εk). Since α1 ∩ α2 is not compact, ¯δj ∩ ¯εk is
not compact for some j, k.
Let β ∈ Con A be generated by ¯δj ∩ ¯εk Y , thus β ¯δj ∩ ¯εk, so there exists surjective
homomorphism h : A → S ∈ SI(V) such that β ⊆ Ker(h), ¯δj ∩ ¯εk Ker(h). Let
T := h(Y ) ⊆ S, then Con T is isomorphic to L := {α ∈ Con Y | Ker(h Y ) ⊆ α}.
Since δj, εk are ∨-irreducible in Con Y , there exists
η1 = max{α ∈ Con Y | δj α},
η2 = max{α ∈ Con Y | εk α}.
Clearly η1, η2 ∈ M(Con Y ). If δj ⊆ Ker(h Y ), then ¯δj ⊆ Ker(h), which contradicts our
denition of homomorphism h and thus δj Ker(h Y ). Hence Ker(h Y ) ⊆ η1, thus
η1 ∈ L and similarly η2 ∈ L. Since η1, η2 ∈ M(Con Y ), we have η1, η2 ∈ M(L). For
every ρ ∈ M(L),we have
ρ ⊇ Ker(h Y ) ⊇ β Y ⊇ ¯δj ∩ ¯εk Y ⊇ δj ∩ εk.
8

11. Either ρ ⊇ δj or ρ ⊇ εk, by ∧-irreducibility of ρ. In the case ρ ⊇ δj we have ρ η1, and
from ρ ⊇ εk we deduce ρ η2. Hence, η1 and η2 do not have a common lower bound
in L, so L is not down directed.
(5)⇒(4)
Let A ≤ B ∈ SI(V) , A nite. Suppose that M(Con A) has not a small element. Let
α1, α2, . . . , αn is the minimal elements of M(Con A), then
Conc f(α1 ∧ α2 ∧ . . . αn) = Conc f(∆) = ∆
and
Conc f(α1) ∧ Conc f(α2) ∧ Conc f(αn) = ∆.
(4)⇒(5)
Suppose that Conc f : Conc A → Conc B not preserve an intersections. Then there is a
γ ∈ M(Conc B) such that
γ ≥ Conc f(α ∧ β),
γ Conc f(α) ∧ Conc f(β),
for some α, β ∈ Conc A = Con A (A is nite). Hence
γ Conc f(α), γ Conc f(β).
A/γ is subalgebra of subdirectly irreducible alegbra. Denote α∗
= α ∨ γ A and β∗
=
β ∨ γ A. By distributivity α∗
∧ β∗
= (α ∧ β) ∨ γ A = γ A. If α∗
= γ A, then
Conc f(α) ≤ Conc f(α∗
) = Conc f(γ A) ≤ γ.
Hence α∗
= γ A and β∗
= γ A, equivalently α γ A and β γ A, so M(Con A/γ)
has not a least element.
Actually, this result is not completely new, as the equivalence of the rst two con-
ditions was proved in [1] using the concept of equationally denable principal meets.
However, we provide a direct proof which does not refer to polynomials and, we believe,
provides an insight helpful in describing Con U, where U is a congruence distributive
variety of algebras and for every A ∈ U the set Conc A is closed under intersection.
9

12. 2.1 Examples
Stone algebra. A bounded distributive lattice with pseudocomplementation L is
called a Stone algebra if and only if it satises the Stone identity:
∀a ∈ L : a∗
∨ a∗∗
= 1
Let B0 denote the two-element Boolean algebra and B1 denote the three-element chain
{0, e, 1} (0e1) as a distributive lattice with pseudocomplementation.
B1 :
s1 = 0∗
se
s0 = 1∗
= e∗
B0 :
s
s
1 = 0∗
0 = 1∗
Con(B1) :
s
u
u
Con(B0) :
s
u
Figure 3
Up to isomorphism, B0 and B1 are the only subdirectly irreducible Stone algebras,
hence NSI consists of one-element chain and two-element chain and thus intersection
of two compact congruences of A is compact for every Stone algebra A.
The varieties M3, N5. Let M3 denote the variety generated by the 5-element
lattice M3. The subdirectly irreducible algebras in M3 are two-element chain and M3.
One of the subalgebras of lattice M3 is 3-element chain. Since congruence lattice of
3-element chain is 4-element lattice D2, NSI contains element, which doesn't have a
least element. Hence there is A ∈ M3 such that there exists two compact congruences
of A, whose intersection is not compact.
Further let N5 denote the variety generated by the lattice N5. The subdirectly
irreducible algebras in N5 are two-element chain and N5. One of the subalgebras of
lattice N5 is 3-element chain. Hence like M3, there is A ∈ N5 such that there exists
two compact congruences of A, which intersection is not compact.
10

13. M3:
s
s
s
d
d
dds
s
d
d
dd
N5:
s
e
e
es
s
s
s
¡
¡
¡
dd
Figure 4
D2:
s
u
d
d
ddu
s
d
d
dd
Distributive lattices with pseudocomplementation. Let Bn denote the variety
of distributive lattices with pseudocomplementation satisfying the identity
(x1 ∧ · · · ∧ xn)∗
∨ (x∗
1 ∧ · · · ∧ xn)∗
∨ · · · ∨ (x1 ∧ · · · ∧ x∗
n)∗
= 1,
for n ≥ 1. Then B1 is the class of Stone algebras. Moreover, if B−1 is the trivial
class, B0 is the class of Boolean algebras and Bω is the class of all distributive lattices
with pseudocomplementation, then Bn, −1 ≤ n ≤ ω is a complete list of varieties of
distributive lattices with pseudocomplementation. Moreover,
B−1 ⊂ B0 ⊂ B1 ⊂ · · · ⊂ Bn ⊂ · · · ⊂ Bω.
Subdirectly irreducible members of Bn are Bn, Bn−1, . . . , B−1 and subalgebras of Bn are
isomorphic to Bn, Bn−1, . . . , B−1. Therefore intersection of two compact congruences of
A is compact for every A ∈ Bn, −1 ≤ n ≤ ω.
Moreover, T.Katri‡k [3] has proved that algebraic lattice A is a congruence lattice
of a distributive lattices with pseudocomplementation i the join-semilattice of all com-
pact elements of A forms a dual Heyting algebra H, where D(H) := {x ∈ H | x∗
= 0}
is relatively complemented.
Specially, algebraic lattice A is a congruence lattice of a Stone algebra i the
join-semilattice of all compact elements of A forms a dual relative Stone algebra S,
where D(S) := {x ∈ S | x∗
= 0} is relatively complemented.
Finite subdirectly irreducible algebras with constants. Let A be a subdirectly
irreducible algebra generating a congruence distributive variety HSP(A). Enrich the
type of A by dening every element a ∈ A as a constant (nullary operation). Denote
the resulting algebra as A∗
. Then V =HSP(A∗
) satises the Theorem 2.2. Indeed, every
subdirectly irreducible member of V belongs to HS(A∗
) (Jónsson's lemma). Since A∗
has no proper subalgebras, we have HS(A∗
) =H(A∗
). And it is easy to see that members
of H(A∗
) do not have proper subalgebras. So subdirectly irreducible algebras in V have
no proper subalgebras, so the condition (2) of the Theorem 2.2 is trivially satised.
11

14. 3 Some special varieties
Let V be locally nite and congruence distributive variety such that
1. Conc F is the two-element chain for every F ∈ SI(V).
2. F doesn't have one-element subalgebra.
Variety of bounded distributive lattices is an example of such variety.
Theorem 3.1.
The following conditions are equivalent
1. L Conc A for some A ∈ V.
2. L is a directed limit of a system
({Bp | p ∈ P}, {ϕp,q | ϕp,q : Bp → Bq, p, q ∈ P, p ≤ q}), where
(a) Bp is a nite Boolean lattice for every p ∈ P.
(b) ∀c ∈ M(Bq), ψp,q(c) := {a ∈ Bp | ϕp,q(a) ≤ c} ∈ M(Bp).
(c) ϕp,q is a Boolean embedding.
3. L is a Boolean algebra.
Proof.
(1)→(2)
Let P be the family of all nite subsets of A ordered by set inclusion. Let Ap be the
subalgebra of A generated by p ∈ P. For every p, q ∈ P, p ≤ q, we put Bp = Conc Ap,
ϕp,q = Conc ep,q, where ep,q is the inclusion Ap → Aq. So L is a directed limit of (0, ∨)-
semilattices. By the Theorem 2.2 ϕp,q is a lattice embedding. An element c ∈ Bp is a
co-atom if and only if c ∈ M(Bp), hence Bp is a Boolean lattice for every p ∈ P.
Let c be a co-atom of Bq and let b1, b2, . . . , bn be all co-atoms of Bp, hence
i∈{1,...,n} ϕp,q(bi) = ϕp,q( i∈{1,...,n} bi) = ϕp,q(0) = 0 ≤ c. Thus there exists j ∈
{1, . . . , n} such that ϕp,q(bj) ≤ c, so bj ≤ ψp,q(c). Since c ∈ M(Bq), Aq/c ∈ SI(V).
Since Ap/ψp,q(c) ≤ Aq/c, so Ap/ψp,q(c) is not a one-element algebra. Hence ψp,q(c) = 1,
i.e. ψp,q(c) = bj.
Further ϕp,q(0) = 0 by denition. If ϕp,q(1) 1, then ϕp,q(1) ≤ c for some co-atom
c ∈ M(Bq). Hence ψp,q(c) = 1, but we know that ψp,q(c) = 1, so ϕp,q(1) = 1. Since ϕp,q
12

15. is a lattice homomorphism, which preserves 0 and 1, it also preserves complements.
Hence ϕp,q is a Boolean embedding.
(2)→(3)
L is a directed limit of Boolean lattices and all ϕp,q are a Boolean endomorphisms, then
L is a Boolean algebra.
(3)→(2)
Every Boolean algebra is a directed limit of its nite subalgebras, so (a) and (c) holds
trivially. Let p, q ∈ P, p ≤ q. Let b1, b2, . . . , bn be all co-atoms in Bp and let c be a
co-atom in Bq, then
0 = ϕp,q(0) = ϕp,q(
i=1,...,n
bi) =
i=1,...,n
ϕp,q(bi) ≤ c ∈ M(Bq).
Hence ∃bi : ϕp,q(bi) ≤ c, so ψp,q(c) ≥ bi. If ψp,q(c) ≥ 1, then ϕp,q(1) ≤ c, with contradicts
ϕp,q(1) = 1. Hence ψp,q(c) = bi, so (b) hold.
(2)→(1)
Choose F ∈ SI(V) arbitrarily. For every p ∈ P let Ap be the direct product
Π{F | o ∈ M(Bp)}. Let p, q ∈ P, p ≤ q. Let b1, b2, . . . , bn be the co-atoms of Bp and let
c1, c2, . . . , cm be the co-atoms of Bq, (m, n ∈ N). Let fp,q be a map Ap → Aq dened
by fp,q(a1, . . . , an) = (d1, . . . , dm), where di = aj such that bj = ψp,q(ci), it is clear that
fp,q is homomorphism.
Further let αk be the k-th projection on Ap → F (k = 1, . . . , n) and βl is a projection
on Aq (l = 1, . . . , m),
Let hp be the isomorphism Conc Ap → Bp dened by hp(Ker(αk)) = bk and let hq
be the isomorphism Conc Aq → Bq dened by hq(Ker(βl)) = cl
Conc Ap EConc fp,q
hp
c
Conc Aq
Bp
Eϕp,q
Bq
c
hq
Diagram 2
We show that diagram 2 comutes. We know that all maps are lattice homomorphisms,
so it suces to prove that ϕp,q(hp(β)) = hq(Conc fp,q(β)) for every β ∈ M(Conc Ap),
13

16. that is β = Ker(αj) for some j. By the denition
ϕp,q(bj) = {a ∈ Bq | bj ≤ ψp,q(a)}
D
.
Denote Ij := {i | i ∈ {1, . . . , n}, ψp,q(ci) = bj}. Since for every ci ∈ D, i ∈ Ij, then
ϕp,q(bj) ≤ i∈Ij
ci. If ϕp,q(bj) i∈Ij
ci, then
∃k ∈ {1, . . . , n} Ij, ϕp,q(bj) ≤
i∈Ij
ci ∧ ck.
Further ψp,q is ∧ - homomorphism, hence
bj ≤ ψp,q(
i∈Ij
ci ∧ ck) =
i∈Ij
ψp,q(ci) ∧ ψp,q(ck) = bj ∧ ψp,q(ck),
so ψp,q(ck) = bj, with contradicts k /∈ Ij, so
ϕp,q ◦ hp(Ker(αj)) = ϕp,q(bj) =
i∈Ij
ci.
Further we know that Conc fp,q(Ker(αj)) is a congruence generated by
{(fp,q(x), fp,q(y)) | (x, y) ∈ Ker αj}. Let fp,q(x) = (d1, . . . , dm), fp,q(y) = (e1, . . . , em).
We have ((d1, . . . , dm), (e1, . . . , em)) ∈ Conc fp,q(Ker(αj)) if and only if di = ei for every
i ∈ Ij. Thus
hq ◦ Conc fp,q(Ker(αj)) = hq(
i∈Ij
Ker(βi)) =
i∈Ij
ci.
So we have proved that Diagram 2 commutes.
Further, let A be the directed limit of ({Ap}, {fp,q}). Commutativity of Diagram 2
means that
lim
→
Conc Ap = lim
→
Bp = L.
Since Conc preserves directed limits, we have
Conc A = Conc lim
→
Ap = lim
→
Conc Ap = L.
Now suppose that V is locally nite and congruence distributive variety such that
1. Conc F is the two-element chain for every F ∈ SI(V).
14

17. 2. There exists F ∈ SI(V) such that F has a one-element subalgebra D = {u}.
Variety of distributive lattices is an example of such variety.
Denition 3.1. A Generalized Boolean algebra B is a distributive lattice with least
element 0 such that for any b ∈ B, the interval 0, b is a Boolean algebra.
Theorem 3.2.
The following conditions are equivalent
1. L Conc A, A ∈ V.
2. L is a directed limit of a system
({Bp | p ∈ P}, {ϕp,q | ϕp,q : Bp → Bq, p, q ∈ P, p ≤ q}), where
(a) Bp is a nite Boolean lattice for every p ∈ P.
(b) For every co-atom c ∈ Bq, ψp,q(c) is 1 or co-atom in Bp.
(c) ϕp,q is a 0-preserving lattice embedding.
3. L is a generalized Boolean algebra.
Proof.
(1)→(2)
As in Theorem 3.1 we argue that Bp is a nite Boolean lattice for every p ∈ P,
bj ≤ ψp,q(c) (which implies ψp,q(c) = bj or ψp,q(c) = 1), that ψp,q is a lattice embedding
and ϕp,q(0) = 0.
(2)→(3)
Without loss of generality we can assume that for all p ≤ q, Bp is a 0-sublattice of
Bq. That is, ϕp,q is the set inclusion and so L = p∈P Bp. Further, L is directed limit
of nite Boolean lattices and ϕp,q is a lattice endomorphism preserving 0. Hence L
must be a distributive lattice with 0. We need to show that every interval 0, v in L is
complemented. So let 0, v be an interval in L and let u ∈ 0, v , then there is r ∈ P
such that u, v ∈ Br. Since Br is Boolean, ∃u ∈ Br such that u ∧ u = 0 and u ∨ u = v.
The same holds also in L.
(3)→(2)
Let B be a generalized Boolean algebra and let G ⊆ B (G nite). Denote v = G,
then 0, v is a Boolean algebra. Denote AG the Boolean subalgebra of 0, v
generated by G. Generalized Boolean algebra is directed limit of its nite subalgebras
15

18. AG, ∀G ⊆ B ((a), (c) holds). Let G ⊆ H ⊆ B. Let b1, b2, . . . , bn be all co-atoms in AG
and let c be a co-atom in AH, then
0 = ϕG,H(0) = ϕG,H(
i=1,...,n
bi) =
i=1,...,n
ϕG,H(bi) ≤ c ∈ M(AH).
Hence ∃bi : ϕG,H(bi) ≤ c, so ψG,H(c) ≥ bi, so (b) hold.
(2)→(1)
We proceed similarly as in Theorem 3.1, but choose F ∈ SI(V) which has the 1-element
subalgebra {u}.
Let p, q ∈ P, p ≤ q. Let b1, b2, . . . , bn are co-atoms of Bp and c1, c2, . . . , cm are co-
atoms of Bq, (m, n ∈ N). Let Ap := F × F × · · · × F
n-times
, Aq := F × F × · · · × F
m-times
and fp,q
is an homomorphism Ap → Aq dened by fp,q(a1, . . . , an) = (d1, . . . , dm), where
1. di = aj, if bj = ψp,q(ci).
2. di = u, if 1 = ψp,q(ci).
Let αk is a projection on Ap (k = 1, . . . , n) and βl is a projection on Aq (l = 1, . . . , m),
then hp(Ker(αj)) = bj and hq(Ker(βi)) = ci. Denote
Ij := {i | i ∈ {1, . . . , n}, ψp,q(ci) = bj},
K := {i | i ∈ {1, . . . , n}, ψp,q(ci) = 1}.
Similarly as in previous case, we can prove that
ϕp,q(hp(Ker(αj))) =
i∈Ij∪K
ci = hq(Conc fp,q(Ker(αj))), for every j.
So the Diagram 2 commutes. Hence
Conc A = lim
→
Conc Ap = lim
→
Bp = L,
where A is the directed limit of ({Ap}, {fp,q}).
16

19. 4 Conclusion
We say that a variety V has the compact intersection property (CIP) if, for every
A ∈ V, the compact congruences of A are closed under intersection. (That is, Conc A is
a lattice). It seems that we only have a good description of Con V when V is congruence
distributive and V has CIP. We found necessary and sucient conditions, when locally
nite congruence distributive variety V has CIP.
It seems that these conditions are very helpful in describing the class Con V. So, our
aim in the further research is a systematic description of classes Con V for congruence
distributive varieties CIP. In this present work we study the following two simplest
cases.
Let also V is such that every subdirectly irreducible algebra is simple and no simple
algebra has a one-element subalgebra. Then we proved that the following conditions
are equivalent:
1. L ∈ Conc V
2. L is a Boolean algebra
Further if there is a simple algebra with a one-element subalgebra, then we proved
that the following conditions are equivalent:
1. L ∈ Conc V
2. L is a generalized Boolean algebra
We hope that the study of particular cases with also lead to general results. For
instance, our conjecture is that for a locally nite congruence-distributive variety V
with CIP, the class Con V is determined by all possible diagrams of the form Conc D,
where D is a diagram in V consisting of subdirectly irreducible algebras and proper
embeddings between them.
17

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