I nternat. J. Math. & Math. Sci.
Vol. 3 No. 3 (1980) 461-476
461
REPRESENTATION OF CERTAIN CLASSES OF DISTRIBUTIVE
LATTICES BY SECTIONS OF SHEAVES
U. MADDANA SWAMY and P. MANIKYAMBA
Mathematics Department
Andhra University
Waltair 530 003
INDIA
(Received March 13, 1979 and in revised form July 9, 1979)
ABSTRACT.
Epstein and Horn ([6]) proved that a Post algebra is always a
P-algebra and in a P-algebra, prime ideals lie in disjoint maximal chains.
In this paper it is shown that a P.igebra L is a Post algebra of order n -> 2,
if the prime ideals of L lie in disjoint maximal chains each with n-i elements.
The main tool used in this paper is that every bounded distributive lattice
is isomorphic with the lattice of all global sections of a sheaf of bounded
distributive lattices over a Boolean space.
Also some properties of P-algebras
are characterized in terms of the stalks.
KEY WORDS AND PHRASES.
Post Algebra, P-algebra, B-algebra, Heyting Algebra,
Stone Lattice, Boolean Space,
Sheaf of
Distributive Lattices Over a Boolean
Space, Prime Ideals Lie in Disjoint Maximal Chains, Regular Chain Base.
1980 Mathemai Subject
Classification
Codes:
Primary 06A35, Secondary 18 F 20.
U. MADDANA SWAMY AND P. MANIKYAMBA
462
INTRODUCTION.
i.
Epstein ([S]) proved that in a Post algebra of order n > 2prlme ideals
lle in disjoint maximal chains each with n
i elements.
He has also proved
that if L is a finite distributive lattice and prime ideals of L lle in disjoint maximal chains each with n-I elements, then L is a Post algebra of order
Epstein and Horn ([6]) have shown that a Post algebra is always a P-
n.
It is
algebra and in a P-algebra prime ideals lie in disjoint maximal chains.
the main theme of this paper that a P-algebra L is a Post algebra of order
n
>- 2, if the prime ideals of L lle
in disjoint maximal chains each with n-i
elements.
The main tool used in this paper is the fact that every bounded distributive
lattice is isomorphic with the lattice of all global sections of a sheaf of
bounded distributive lattices over a Boolean space ([iS] and [9]).
It is
well known that a P-algebre is always a (double) Heyting algebra, a (double)
L-algebra, a pseudocomplemented lattice, a Stone lattice and a normal lattice.
We characterize these properties of P-algeBras in detail in terms of the stalks
of the corresponding sheaf.
We give another characterization of Post algebras
by regular chain bases.
Throughout this paper, by L we mean a (momtrivial) bounded distributive
lattice
(L, V/k, 0, I) and B
elements of L.
x,yL,
For any
a B,
we denote the largest
B(L) the Boolean algebra of complemented
we denote the complement of a by
zL such that
-+ y and the largest element
x
x y.
If, for every x,yE L, x
i
y
respectively.
(x y) V (y---x)
< y (if it
exists) by
y (xy) exists, then we say that L is a
Heyting algebra (respectively B algebra).
and x
For any
xAz < y (if it exists) by
aB such that xa
/
a’.
Dually, we define x
/
If in a Heyting algebra (B-algebra), (x
/
y
y) V (y
/
x)
i) for every x,yL, then we ay that L is an L-algebra
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
(respectively BL-algebra).
For any x t L, if x
- -
0 exists, then we say
that x has the pseudocomplement and we usually write x
exists for each
x L,
L is denoted by L
d
,
for x
then we say that L is pseuodcomplemented.
If both L and L
d
463
O.
If x
The dual of
are Heyting algebras (B-algebras,
L-algebras, BL-algebras), then we say that L is a double Heytlng algebra
L is
(respectively double B-algebra, double L-algebra, double BL-algebra).
said to be a P-algebra if L is a BL-algebra.
Epstein and Horn proved that
L is a P-algebra if and only if L is a double L-algebra ([6], theorem 3.4).
For the elementary properties of these types of lattices, we refer to ([2])
and ([6]).
By a sheaf of bounded distributive lattices we mean a triple
(,,X)
satisfying the following:
and X are topological spaces
i)
X is a local homeomorphism
ii)
iii)
iv)
Each
the maps (x,y)
x
v)
-I (p), p E X
xvy and (x,y)
w(y)}
/ w(x)
the maps
is a bounded distributive lattice;
p
into
> O(p)
>
x^y
from
V= {(x,y)
E
are continuous and
and i
p
l(p) from X--+ Q are continuous,
where O(p) and l(p) are the smallest and largest elements of w
-i
(p)
respectively.
call the sheaf space X the base space and
We
P
for
the sheaf
-i
(p), p %X and call
(,,X)
For any sections
and we call
[(o,0)[
the stalk at p.
we mean a continuous map
and z we write
the proection map.
I(,)[
By a (global) section of
X --+
such
that
for the closed set
the support of o and Ite
[0
We write
for
(,0)
{pE
o
xlo(p)
For the
preliminary results on sheaf theory, we refer to the pioneering work of
Hofmann ([ 8 ] ).
i.
+ z(p)}
U. MADDANA SWAMY AND P. MANIKYAMBA
464
By Spec L, we mean the (Stone) space Y of all prime ideals of L with
the topology for which {Y
the hull-kernel topology; i.e.
base, where for any xE L, Yx
{P Spec L/x
P}.
x
IxE L}
is a
Throughout this paper X
denotes Spec B which is a Boolean space, ie., a compact, Hausdorff and totally
disconnected space.
X
Since a
a
is a Boolean isomorphism of B onto the
Boolean algebra of all clopen subsets of X, we identify a and X
simply a for X
e
For any p
a
be the quotient lattice L/e
p
p
where
is the congruence on L given by
P
(x,y) 6
and
x
X
and write
letbe
p> xa
ya for some a 6B-p,
the disjoint union of all
p,
p
X.
For each x E L, define
(p)
containing x. Topologize
(x), the congruence class of
P
P
with the largest topology such that each x, x E L, is continuous. Define
X
by
/
+
X by (s)
p if
E
The following theorem is the main tool used
p
in this paper and is due to Subrahmanyam
THEOREM I.i
(,,X)
([15]) (see also [9]).
described above is a sheaf of bounded distributive
lattices in which each stalk has exactly two complemented elements, viz.,
0(p) and l(p).
if p
1.2
For any a
1.3
For any x,y
1.4
x
B, p
X, a(p)
l(p) if p
a and
(p)
0(p)
a.
x
B,
L and a
/a
/a
if and only if xa
is an isomorphism of L onto the lattice
global sections of the sheaf
,,X).
ya.
(x, ) of all
We identify x with x and write simply
x for x.
1.5
For any prime ideal P of L, there exists a unique p
that {x(p)/x P} is a prime ideal of
ideal of
P
p
X such
On the other hand if Q is a prime
p E X, then {xL/x(p) 6 Q} is a prime ideal of L.
This
465
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
correspondence exhibits the set of all prime ideals of L as the disjoint union
Moreover, if P and Q are prime
of the sets of prime ideals of the stalks.
p
ideals of distinct stalks
and
q,
then P and Q are incomparable, when they
are regarded as prime ideals of L.
D’
Throughout this paper, by a stalk
sheaf
(,,X)
p E X, we mean the stalks of the
described above at p.
P SEUDOCOMPLEMENTED LATTICES.
2.
It is well known that a bounded distributive lattice is a Heyting algebra
if and only if it is relatively pseudocomplemented; i.e., each interval Ix,y],
x < y E L is pseudocomplemented ([i]).
Also the class of all distributive
pseudocomplemented lattices and the class of all Heyting algebras are
mentation and (x,y)
L.
[I]), when
see [i] and
equationally definable
(x
+
we regard the pseudocomple-
y) as unary and binary operations respectively in
Thanks to the referee for suggesting a simpler proof of the following.
THEOREM 2.1.
L is pseudocomplemtnetd if and only if each stalk
,
x
is pseudocomplemented and the pseudocomplementation x
YD’ p’E
X
is continuous
is precisely x (p) for
and in this case, the pseudocomplement of x(p) in
all x E L.
PROOF.
Suppose L i pseudocomplemented.
all x and p,
(x(p))p
show that the map x
>
exists and is equal to x (p).
x
is continuous.
hypothesis implies that the map f
x
/
O
If L is a Heyting algebra, then each 8
binary operation (x,y)
then (x
(y
/
/
yl)A
x
I) y A a
a.
(x
Similarly,
-
(x
Then it is easy to
For the converse, if x E L, the
defined by f(p)
(x(p))*
is a global
y for some y and it is clear that y
Therefore, f
section of
Then it is easily seen that for
+
y).
a
For, if a
a
E B,
is compatable with the
B and (x,y) and
Xl) xa -< Xl#,a Yl a _< Yl’
we have (y
yl a -< (x x I) a
/
/
x
(xl,Y I)
so that (x
and hence
/
8
x
I)
a <
466
(x
U. MADDANA SWAMY AND P. MANIKYAMBA
/
xI, y
/
Hence the following theorem and its proof are analogous
8a.
yl
to the above.
L is a Heytin algebra if and only if each stalk
THEOREM 2.2.
,
(s,t)
is a Heytlng algebra, and the operation
continuous and in this case x(p)-+ y(p)
for I x,y
3.
(s
/
X,
p
ofV
t)
into
is equal to
p
P
X
is
(x y)(p)
L.
NORMAL LATTICES.
(Cornish [4]).
DEFINITION 3.1.
L is said to be normal if any two
distinct minimal prime ideals of L are comaximal and L is said to be
relatively normal if each interval [x,y], x
For any x,y t L, let
x t L, we write (x)
only if (x
,
Y)L
if and only if
,
L
for
,
,
(x,y)
(x,0) L.
,
(x y,z)
For any
Cornish ([4]) proved that L is normal if and
for all x,y
, V(y,z)
(x,z) L
L
L is normal.
be the ideal {zt L/ x^z < y} of L.
L
(X)L V(Y)L
y
L, and that L is relatively normal
for all x,y,z
L
L where V stands
for the join operation in the lattice of all ideals of L.
THEOREM 3.2.
(Speed [IS]).
A pseudocomplemented distributive lattice is
normal if and only if it is a Stone lattice.
THEOREM 3.3.
(Balbes and Horn [I]):
A Heyting algebra is relatively
normal if and only if it is an L-algebra.
THEOREM 3.4.
PROOF.
(p)
(u
exists a
L is normal if and only if each stalk
Suppose L is normal and p
,
V)p
t
t
tlV t 2
l(p) Vt2(p), l(p)
t
,
Since, (xy^t)(p)
B-p such that xhyA t a
(ya) L and hence
t(p)
L.
t
X. Let u,v
*
(U)p
Conversely, suppose each stalk
and t
P
p
I
2(p)
P
p %X, is normal.
so that u
x(p)N y(p) t(p)
0, so that t
for some t
,
JD,
(x a)
,
L
(V)p
X is normal.
,
% (xyAa)
and t
2
Hence
Let x,y
x(p)
0(p) there
,
(xAa)
V
L
L
(ya)
Now
L
p
is normal.
L and
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
(x
z
For each p
hY)L
,
,
y(p))p (x(p))p V CY(P))p,
(x(p) h
L, such that a
B-p, t and s
there exists a
a h(tV S), t
hZ
hX ha
By the compactness of X, it follows that there exists a I,
n
B and tl, t2,.., t n, Sl,..., s
V a
L such that
i,
alh z
i
n
0.
Shy ha
a
X, since z(p)
467
n
a2,...,
a
h
i
n
(t
i
V
si), t ih x ha i
0
S
n
(Sih i)
i.
ha
i=l
n
V (t
i
i=l
hX
Now, Put
ha
i
hX)
V (t
ih a
i=l
t
i)
and
n
(z
V
then, z
a
i=l
t
ha
n
V
s
ih
y
n
V
0
V
i)
(a h(t
i
iffil
(Sih aih y)
v
i
S
i))
Hence (,by)
S by.
and
t V S
,
,
,
(X) L V(y) L
L
Hence L is normal.
and the otherslde inclusion is obvious.
The proof of the following theorem is analogus to that of the above.
L is relatively normal if and only if each stalk
THEOREM 3.5.
p,
X,
p
is relatively normal.
DEFINITION 3.6
and denoted by L
(x) L**
A
{uL / u
,
(Speed [12]).
A
and (ii) {pX
PROOF.
h
,
A
L
,
,
A
and x v y is dense in L.
z
B-p and hence z
P
} is open for each x
dense in
p
Therefore
follows that y(p)
P
L.
and x
Let p
h
a
-P
,
A
if and only if (1)
There exists y
Clearly, x(p)
X.
L, such that ((x(p) v y(p))
some a
L, there
0 and x v y fs dense; i.e., (x A
y
h
z(p)
6 A
Now
h
0(p), then
0, so that z(p)
lattice
(Y)L
if and only if, for each x
x(p) is dense in
Suppose L
L}
0 for every v
v
L, such that x
THEOREM 3.7.
,
(x)
*
L such that
L, there exists y
if, for each x
Speed ([12] proved that L
exists y
L is said to be a distributive
0(p).
,
{0}.
Y)L
X
for each p
L.
L such that x
y(p)
0(p).
x v y)
h
z
h
h
0
Also, if
0 for
a
Hence x(p) V y(p) is
suppose x(p) is dense in
0(p) and hence there exists a
y
B-p such that y
It
h
a
0.
U. MADDANA SWAMY AND P. MANIKYAMBA
468
We claim that x(q) is dense in
z E L, such that x(q)
O(q).
B-q, z(q)
^b
for all q E a.
For, if q E a and z(q) E
q,
0(q), then there exists b E B- q such that
z(q)
0; so that (xvy)
x^z^b
a
^
q
0 and since
0, and hence z ^a^b
^z ^a^b
Conversely, suppose (1) and (li) hold and x E L.
For each p 6 X, by (i) and (li), there exists yEL and a E B-p such that
0 and (xVy)(q) is dense in
x^y^a
q
for all q E a.
By the usual
yl,y 2,...,yEL, a l,...,an E B
compactness argument, there exists
such that
n
V
a
i=l
i
i
a
i
0 for i
^ aj
j
x^Yi ^a
0 and (x v
is dense
yi)(p)
n
Now put
for all p
(Yi ^ a l)"
V
y
i=l
is dense in L.
((x v y)
0(p)
For, (x v y)
^
z)(p)
p E a. and therefore z
^
X, since the stalk
For any
P
.
Hence L E A
STONE LATTICES.
4.
P
is a Stone lattice if and only if
,
(x(p))p
{0(p)}).
^
0 and x v y
y
0 for some z E L, then, for all p E a i,
z
(x(p) v y(p))
0.
Then x
^,
z(p) and hence z(p)
0(p) for all
has exactly two complemented elements,
.)p
is dense (i.e.
if x(p)
o(p)
Hence, by theorem 2.1, 3.2, and 3.4, we have the
following.
THEOREM 4.1.
Suppose L is pseudocomplemented.
Then the following are
equivalent.
(i)
L is a Stone lattice
(ii)
L is normal
(iii)
Each stalk
(iv)
Each stalk
(v)
Each stalk
p,
P
t
P
p E X, is a normal
p E X, is a Stone lattice
p E X, is dense.
The following theorem is a consequence of theorem 2.2, 3.3 and 3.5.
then
469
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
THEOREM 4.2.
Let L be a Heyting algebra.
Then the following are
equivalent.
(i)
(ii)
(iii)
(iv)
L is an L-algebra
L is relatively normal
Each stalk O
Each stalk
P
p,
p E X, is relatively normal
p E X, is an L-algebra.
Since L is an L-algebra if and only if it is relatively Stone lattice
(Theorem 4.11 of [ i]) (i.e., each interval is a Stone lattice) in view of
theorem 4.1, one may suspect that if L is an L-algebra, then each stalk is
relatively dense and hence a chain.
This is not true (see 4.4
below),
though the converse is proved in the following.
THEOREM 4.3.
If L is a Heyting algebra and each stalk is a chain, then
L is an L-algebra.
PROOF.
If each stalk is a chain, then by theorem 1.5, the prime ideals
of L lie in disjoint maximal chains and hence L is relatively normal lattice
and hence the theorem follows from theorem 2.3.
EXAMPLE 4.4.
Let B
4
be the 4-element Boolean algebra and A be the
distributive lattice obtained by adjoining an external element to B
smallest element.
4
as the
Then A is an L-algebra which is not a chain (Thanks to
the referee).
Epstein and Horn ([ 6]) proved that L is a Stone lattice if and only if L
is pseudosupplemented and
0x^
y
(0
x) ^
d
(0 y) for all x, y 6 L.
Now, these two necessary and sufficient conditions for L to be a Stone lattice
can be viewed in terms of the stalks as follows.
THEOREM 4.5.
L
d
is pseudosupplemented if and only if
each x 6 L and in this case
PROOF.
xl
0
Follows from Lemma 5.2.
--x for all x
L.
Ixl
is open for
U. MADDANA SWAMY AND P. MANIKYAMBA
470
X, let (p) be the smallest ideal of L containing p.
For any p
The
proof of the following theorem is simple.
THEOREM 4.6.
X
For any p
the stalk
p
is dense if and only if (p) is
a prime ideal of L.
It can be easily seen that each stalk
x
Ixl n IYl
^ Yl
p
X, is dense if and only if
Hence from theorem 4.5 and 4.6 and
[73 ), we have the following.
lemma 2.9 of
THEOREM 4.7.
x
for all x,y E L.
p,
E L and each stalk
REMARK 4.8.
xl
L is a Stone lattice if and only if
X is dense.
p
P
is open for all
Swamy and Rama Rao ([ I0] proved that a lattice L is a Stone
lattice if and only if L is isomorphic to the lattice of all global sections
of a sheaf of dense bounded distributive lattices over a Boolean space in
which each section has open support (see also [ 9] ).
that when L is a Stone lattice, then our sheaf
It can be verified,
(,,X)
coincides with the
sheaf constructed in ([ i03 ).
5.
P-ALGEBRAS.
The following results interpret B-algebras in sheaf theoretic terminology.
LEMMA 5.1.
{p
y exists in B if and only if
Then x
X / x(p) < y(p)} is closed and in this case
PROOF.
such that x
^
a < y.
Hence a
If
x
then there exists a
X.
X, x(p)
For any p
if and only if p
p
L.
Let x,y
x
x=> y
=> y.
<
x_
y
{p
X / x(p) < y(p)}.
y(p) if and only if there exists a
exists in B, then, for any p
X, x(p)
B-p
< y(p)
Conversely, if {p 6 X / x(p) < y(p)} is closed,
a if and only if x(p) < y(p) for all
B such that p
- - -> y.
The proof of the following is easy.
LEMMA 5.2.
For any x,y E L,
l(x,y) s
a largest element a of B such that x
^
a
open if and only if there exists
y
^
a.
471
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
The following theorem is a consequence of the above lemmas.
THEOREM 5.3.
The following are equivalent.
i)
L is a dual B-algebra
2)
For any x,y
3)
For any x,y E L, {a E B / x
4)
L is a B-algebra
5)
{p
6)
l(x,Y)
L, {a
y a}
B / x v a
^
y
a
is a principal filter of B.
a} is a principal ideal of B.
^
X / x(p) -< y(p)} is closed for every x,y
is open for every x,y
THEOREM 5.4.
L.
L.
Suppose L is a B-algebra.
Then the following are equivalent.
i)
L is a P-algebra; i.e.
2)
Each stalk is a chain
3)
For every x,y
4)
For every x,y E L, there exists a E B such that x
PROOF.
L is a BL-algebra
L, there exists a E B such that x
2< 3
is proved in
([ iS]
and
3-->
^
a
-< y and y
/ a >
^
y and y
4 is clear.
I
-> 2
a’
a’
-
< x.
x.
follows
from lemma 5.1.
6.
POST ALGEBRAS.
The following definition is slightly different from that of Chang and
Horn ([3]).
DEFINITION 6.1.
By a generalized Post algebra, we mean the lattice C (Z,C)
of all continuous maps of a Boolean space Z into a discrete bounded chain C
where, the operations are pointwise.
THEOREM 6.2.
The following are equivalent
i)
L is a generalized Post algebra.
2)
There exists a chain C in L such that the natural map c
is an isomorphism for all p
3)
> c(p)
C +
p
E X.
There exists a chain C and, for each p
such that for any c 6 C and x 6 L, {p6 X/
X, an order isomorphism a
ap(C)
P
x(p)} is open in X.
C
/
P
U. MADDANA SWAMY AND P. MANIKYAMBA
472
PROOF.
I
Let L
C (Z,D) where Z is a Boolean space and D is a discrete bounded
chain.
2
It is well known that a
is a Boolean isomorphism of the algebra
Xa
of all clopen subsets of Z onto B, the centre of L, where
We identify
istic function on a.
Xa with
a.
Xa
is the character-
Also the Stone space X is
homeomorphic with Z.
For any d E D, let d
Let C be the set of all constant maps of Z into D.
Then C is a
denote the constant map which maps every element of Z onto d.
chain in L.
X.
Let p
homomorphism.
If
d
dl,
and hence d
d
I
Now, let x
2.
L.
Z + D is continuous, x
it follows that
23:
(x,d)
8
(d)
P
{p6X / c(p)
3------i:
x
x(p)} which
P
by (8(x))(p)
C is also bounded.
-i
ep
(
L and a
P
-I
(d) for some d E D,
^
^
x
-i
(d)
d
^
x
p
C
-
Let X
P
{pX /
C and x E L.
Spec B.
P
P
(x(p))
(c)
P
(d),
P
(c)
is an isomorphism
L
Define 8
Let c E C.
L and p 6 X.
{x /
X.
-i
is an
/
/
C (X,C)
Then
c}
x(p)} is open by (3) and
Clearly 8 is a homorphlsm and one-one since e
(f(p)) for every p
B, then
a
2
B-p
a for some a
^
is bounded and
-p
Now, we will show that 8 is onto.
by o(p)
is a
P
is open.
{pX /
hence 8(x) is continuous.
d
L/e
P
is an isomorphism.
(x(p)) for each x
(x))- { c}
C+
B-p and since x
X, then, for any c
We’first observe that since
of C onto
x
I
If C is a chain in L and the natural map
isomorphism for every p
is so.
then d
Then if p.
Hence 8
P
_
d2(p)
-i
since x
x(p)}
dl(p)
D such that
2
p
Clearly, the natural map
Let f 6 t (X,C).
Define o:
We will show that o is a section.
-i
P
X
/
Let
473
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
s
-i
(xCa))
{pa
I
a
P
x(p)}
(fCP))
{p6X / f(p)
pX /
c}
e
c6C
Since f is continusous, it follows that o
-i
(x(a)) is open.
L} is a base for the topology on
and x
and clearly
o
id
x.
Therefore,
P
x(p)}.
(c)
Since
is continuous
that
it follows
{x(a) / a B
L and also 8(x)
x for some x
f.
Hence 8 is an isomorphism and therefore L is a generalized Post algebra.
THEOREM 6.3.
Let n > 2 and L a P-algebra.
Then the following are
equivalent.
I)
L is a Post algebra of order n.
2)
Spec L is the disjoint union of maximal chains each with n-i elements.
3)
Each stalk is a chain with n elements.
i >2 is proved in ([ 5] ).
PROOF.
Since L is a P-algebra, by theorem 4.4, each stalk
P
p
X, a chain.
by theorem 0. (5), Spec L is the disjoint union of the chains Spec
has n-i elements and therefore
X, Spec
P
p
C
e
A.
for some e
has n elements and hence
p
X.
A each with
If Spec L is the disjoint union of all maximal chains C
n-I elements, then, for any p
p,
Also,
Hence Spec
(2)--(3).
Now, suppose each stalk is a chain with n elements and Cn is the n< n. For any p 6 X let
{0(p)
element chain 1 < 2 <
Xlp (p) <
p
X2p(P)
Xnp(P)
ep: Cn--’.p by =p(i) Xip(P)
<...<
isomorphism.
5.3
a.
Let i 6 C
n
where
Xlp,X2p ,...,xnp L.
for each +/-
x 6 L and p
Since L is a B-algebra and
Cn.
Clearly,
p
B-p such that
X.jp(p) <XkP(P)
there exists b 6 B-p such that x. (q) <
Xkp (q)
Cn and
q 6 b.
and hence
3P
Xip(q)
Xiq(q)
for all i
Deflne for any p % X
X such that e (i)
x(p) so that there exists a
Xip(P)
q
l(p)
Xip(q)
p
is an order
x(p).
ie.,
x(q) for all
for all j < k, by theorem
for all j < k and q
Then p 6 a
^
b
b 6 B and
P
U. MADDANA SWAMY AND P. MANIKYAMBA
474
for any q 6 a
^
eq(i)
b,
Xlp(q)
Xiq(q)
x(p)} is open for each i
C
Hence {p6X /
x(q).
ep(1)
and x E L.
n
By a chain base C for L we mean a chain C with 0 in L
DEFINITION 6.4.
such that L is generated by the centre B and C; i.e., every x E L can be
n
(a
V
written in the form
i=l
for some a
i
i
6 B and c
E C
i
A chain base C in L is said to be regular, if, for
DEFINITION 6.5.
Cl # 2
c
^
i
E C and a
B, c
I
< c
It is proved in ([ 15]
and a
2
^
c
-< c
2
I
0.
imply a
that a bounded distributive lattice L is a
generalized Post algebra if and only if there exists a regular chain base
for L.
Now, we characterize chain bases and regular chain bases in terms of
It is also proved in ([ 15]
the stalks.
the natural map
all p
X.
p
C
/
Jp’
that if C is a chain base for L,
by (c)
defined
c(p) is an eplmorphlsm for
P
We prove the converse in th4 following.
THEOREM 6.6.
Let C be a chain in L and 0 E C.
p C-+P
Then
is
an epimorphism for each p 6 X, if and only if C is a chain base for L.
PROOF.
Suppose
For each p
P
X, there exists c
C such that
P
so that there exists a E B-p such that c
exists a partition
x
^
a_i
al,a2,...,an
so that x-- x
^
X and let x E L.
is an epimorphism for each p
1
x
^
of B and
n
V a.
i=l
^
P
P
(Cp)
x
a
^
x(p),
x(p) ie., c (p)
P
a. Therefore, there
Cl,C2,...,c n
C such that c
i
^ ai=
n
n
(x
V
i=l
^
V _(c
i
i=l
a
^ ai)._
Hence C is a chain base for L.
The following theorem is a straight forward verification.
THEOREM 6.7.
Then the following are equivalent.
Let C be a chain in L.
i)
The natural map
2)
For any c
c
I # 2
p: C
/
p
C and a
X.
is one for all p
B
c
I
< c
2
and a
^
c
2
< c
I
imply a
0.
475
REPRESENTATION OF DISTRIBUTIVE LATTICES BY SHEAVES
3)
c E C and 0 # a E B, a
^ c I # a ^ c 2.
I # 2
By summarizing the above results, we have the following
For any c
HEOREM 6.8.
Suppose L is a bounded distributive lattice.
Then the
following are equivalent.
l)
L is a generalized Post algebra
2)
There exists a chain C in L such that the natural map
isomorphism for all
3)
p
C
/
P
:C/
P
is an
X.
There exists a chain C and for each p
P
p
such that for any
X, an order isomorphism
C and x
L
{pX /
p
(c)
x(p)} is
open in X.
4)
L has a regular chain base.
REMARK.
The equivalence of (1) and (4) is established in ([IS]) by
using the Boolean extension techniques.
REFERENCES
[I]
2]
Balbes, R. and Horn, A., "Stone Lattices".
537-545.
Duke Math.J. 37 (1970)
Birkhoff, G., "Lattice Theory", Third Edition, AS. Colloq. Publ. (1967).
[ 3]
Chang, C.C. and Horn, A., "Prime Ideal Characterization of Generalized
Post Algebras", Proc. Syrup. in Pure Maths. Vol. 2, (1961) 43-48.
[4]
Cornish, W. H., "Normal Lattices", J. Aust. Math. Soc. 14 (1972), 200-215.
[ 5]
Epstein, G., "Lattice Theory of Post Agebras".
(1960)
5]
Trans. Amer. Math. Soc. 95
300-317.
Epstein, G. and Horn, A., "P-algebras an Abstraction of Post Algebras",
Algebra Universalis, Vol. 4__ (1974) 195-206.
[7 ]
Epstein, G., and Horn, A., "Chain Based Lattices", Pacific J. Maths.
Vol. 55, No. i (1975).
[ 8]
Hofmann, K. H., "Representation of Algebras by Continuous Sections".
Bull. Amer. Math. Soc. 78 (1972) 291-373.
[9]
Maddana Swamy, U., Representation of Algebras by Sections of Sheaves".
Doctoral Thesis, 1974, Andhra University, Waltalr, India.
476
103
[11]
U. MADDANA SWAMY AND P. MANIKYAMBA
Maddana Swamy, U., and Rama Rao, V.V., "Triple and Sheaf Representation
of Stone Lattices:, Alg. Univ. Vol. 5 (1975) 104-113.
Monterio Antonio, "Axiomes Independents Pour Les Algebras De Brouwer".
De Revista de la union Matema.ti.ca Argentina y da la Asociacion.
Fisca Argentiana, Vol. XVII de Homenaje a BEPPO LEVI, 149-160,
Buenos Aires, 1955.
12]
Speed, T. P., "Some Remarks on a Class of Distributive Lattices".
J. Aust. Math. S oc. 9 (1969) 289-296.
Speed, T. P., "On Stone Lattices", J. Aust. Math. Soc., 9 (1969) 297-307.
14]
Stone, M. H., "Topological Representations of Distributive Lattices and
Brouwerian Logic". Casopis. Pest. Math.
67 (1937) 1-25.
z
[15]
Subrahmanyam, N. V., "Lectures on Sheaf Theory". Lecture Notes, Dept.
of Mathematics, Andhra University, Waltair, India (1978) (unpublished).