I nternat. J. Math. & Mh. Sci.
(1982)21-30
Vol. 5 No.
21
SOME REMARKS ON CERTAIN CLASSES OF SEMILATTICES
P.V. RAMANA MURTY and M. KRISHNA MURTY
Department of Mathematics
Andhra University
Waltair 530003
India.
(received June 28, 1979 and in revised form March 7, 1980)
ABSTRACT.
In this paper the concept of a
,-semilattice is introduced as a
generalization to distributive *-lattice first introduced by Speed [i].
shown that almost all the results of Speed can be extended to a more
of distributive -cmilattices.
It is
eneral
class
In pseudocomplemented semilattices and distri-
butive semilattices the set of annihilators of an element is an ideal in the
sense of Gr’tzer [2].
But it is not so in general and thus we are led to the
definition of a weakly distributive semilattice.
In 2 we actually obtain the
interesting corollary that a modular ,-semilattice is weakly distributive if and
only if its dense filter is neutral.
In 3 the concept of a sectionally pseudo-
complemented semilattice is introduced in a natural
way.
It is proved that iven
a sectionally pseudocomplemented semilattice there is a smallest quotient of it
which is a sectionally Boolean algebra.
Further as a corollary to one of the
theorems it is obtained that a sectionally pseudocomplemented semilattice with a
dense element becomes a ,-semilatticeo
Finally a necessary and sufficient con-
dition for a *-semilattice to be a pseudocomplemented semilattice is obtained.
22
P.V.R. MURTY AND M.K. MURTY
KEY WORDS AND. PHRASES. Pseudocomplemented semilattice, -semllattice, weakly
dis’tt@tive semiatce, Nal filter, Sectionally pseudocomplemented semilattice.
1980
i.
MATHEATICS SUBJECT CLASSIFICATION CODES:
06
INTRODUCTION.
Speed [i] has introduced the concept of a distributive *-lattice.
In this
paper we observe that the condition he has placed on a distributive lattice to become a distributive ,-lattice can be placed on any meet semilattice with 0 an@
accordingly in this paper we call such semilatticesas ,-semiltices (see definition
I).
It is shown that almost all the results of Speed can be extended to a more
general class of distributive
,-semilattices.
The concept of a filter in join
semilattice and hence (dually) that of an ideal in a meet semilattice is due to
Gr’tzer [2].
In pseudocomplemented semilattices and distributive semilattices
the set of annihilators of an element is an ideal in the sense of
Gtzer.
But it is not so in general and thus we are led to the definition of a weakly
distributive semilattice.
In 2 we prove as a corollary
to theorem
4 that a mod-
ular @-semilattice is weakly distributive if and only if its dense filter is
neutral.
Further we show in Theorem 3 that the filter congruence induced by the
dense filter on a modular
R on L defined by R
,-.qemilattice
{(x.y)
L is equal to the well known congruence
(x)*
( L x L
(y)*}.
In 3 we study sectionally pseudocomplemented semilattices the concept of
which is introduced in [3] and
[4].
Actually we prove in Theorem 8 that given a
sectionally pseudocomplemented semilattice there is a smallest quotient of it
which is a sectionally Boolean algebra.
Further we show in Theorem i0 that given
a sectionally pseudocomplemented semilattice L there is a Boolean algebra S such
that if L has a dense element then L/R is a Boolean algebra isomorphic to S, and
obtain as a corollary that a sectionally pseudocomplemented semilattice with a
dense element becomes a ,-milattice.
Finally we obtain a necessary and suffi-
cient condition for a ,-semilattice to be a pseudocomplemented semilattice.
2.
RESULTS ON MODULAR,-SEMILATTICES.
REMARKS ON CLASSES OF SEMILATTICES
DEFINITION I.
Let S be a meet semilattice with 0" then for any subset A of
S, A * stands for {x E S
(a) *
23
An element s E S
x ^a
0 for all a 6A}.
If A
{a} then we denote A
is called a dense element of S if and only if
(a)
by
{0}.
D denotes the set of all dense elements of S and it can be seen that D is a filter
of S (if D is nonempty).
The concept of a distributive ,-lattice is due to Speed [i].
However the
condition he has placed on a distributive lattice in his definition may as well
be placed on any meet semilattice with 0 as in the following"
DEFINITION 2.
A meet semilattice with 0 is said to be a
only if for any aE L there exists
DEFINITION 3.
a’ L
(a’)
such that (a)
(Gratzer [2] and Katrinak [5]).
,-semilattice if and
A nonempty subset I of a
meet semilattice S is called an ideal of S if and only if
(i)
xI
and a <x (a 6 S) implies a
I
(ii) x,yE I implies the existence of z in I such that
DEFINITION 4.
I(a,b S) implies either
and x
DEFINITION 6.
semilattice if (a)
a or b I.
A meet semilattice (S, ^) is called a distributive semilattice
x>_y^ z imply the existence of yl,Zl
if and only if x,y,z 6S and
Yl--> ’ Zl_>Z
<z and V < z.
An ideal I of a meet semilattice S is said to be a prime
ideal if and only if a ^b
DEFINITION 5.
"
such that
Yl ^Zl"
A meet semilattice L with O is called a weakly distributive
is an ideal for every a6L.
The notion of weak distibutivity
in semilattices is a natural generalization of the notion of 0-distributivity
introduced by Varlet [6].
Clearly every pseudocomplemented semilattice and every
distributive meet semilattice with 0 is weakly distributive.
Also one can oh-
serve that in a weakly distributive semilattice S, a proper filter U is an ultra
filter of S if and only if S-U is a minimal prime ideal of S.
It will be shown in this article that almost all the results of Speed [I]
for distributive *-lattices (see Theorems 1,2 and 6 of this paper) can be ex-
tended to weakly distributive *-semilattices and proofs are given only for those
results which can not be obtained from those of Speed.
24
P.V.R. MURTY AND M.K. MIfRTY
From now onwards in this article, L denotes a weakly distributive semilattice.
M
Let
-
For any subset A of L write
be the set of all minimal prime ideals of L.
{M
h(A)
A
M} and M (A)
{M6
MI 4 ,M}.
Then {h (A)
A_c
called the hull-kernel topology and {, {x}
of hl is a topology on
a base for the above topology.
Also {h(a)
la6.L}
L} of subsets
=
X
x 6L} is
is a base for some topology on
called the dual hull-kernel topology.
LEMMA i.
(i)
In L the following hold:
A prime ideal M is minimal if and only if (x)
(ii)
MX
(iii)
h(x)
h((x)
(iv)
(z)
(x) i (Y)
(v)
(x
(vi)
(x)
h((x)
^ y)
**
(x) **
I
topology on
C (y)
h((y)
for the hull-kernel topology and
]d
for the dual hull-kernel-
In L, the following conditions are equivalent.
L is a /-semilattice
(2) T h
Td"
hl is compact in the hull-kernel topology.
(i)(2) and (2)(3) are straightforward.
PROOF.
I
h(x) i h(y)
then we have the Theorem"
THEOREM i.
(3)
nT"
h(z)
h(x)
(y)
If we write
(I)
for every x EM
M #
Now h(x)
is compact.
set of
t 6 (x)
I.
that
h(xlh(tl), ...I-h(tn
(z)
Thus
THEOREM 2.
In L the following are equivalent.
(2)
For any
x
AX’
for some t l,...,t
(3),(i).
L is a ,-semilattice.
x
so that by compactness of h(x) it follows
such that t.< z for i< i< n.
(I)
L there exists x’6 L such that
0 and
Assume that
is a closed subset of ,%{and hence is a compact sub-
.o (h(x)h(t)
there exists z 6. (x)
(x)
=&i’ x
(3)(i)"
[x), [x’) -c D
n (x)
Now h((x) )
and hence
h(z) and hence
Q.E.D.
25
REMARKS ON CLASSES OF SEMILATTICES
0 and r
^
x’
xl E (t
0 then x,
0 and hence
r
Assume (2).
(i)(2) is obvious.
PROOF.
zED.
Thus t
c,
(x)*.
f for some f E F}
^
A meet semilattice (S,^) is called a
(see Rhodes [7]).
S such that x
THEOREM 3.
x,y,z
x
^ Yl
z
^
S and x e y
z imply the existence of
^
Yl ^ Zl"
I
{(x,y)
R where R
In a modular *-semilattice S we have
(y)*}.
PROOF.
Clearly
modularity of S, y
^
implies y
x
2
^
x
Y2 ^
y
^ Y2
(y)*.
eD !
R.
Now let
(x)*
x’
0
y
y implies y
By Theorem 2, x 2 e x’ and x 2 e y
^
f
is a congruence on S called the congruence induced by the
modular semilattice if and only if
x
^
Q.E.D.
DEFINITION 7.
(x)*
t
{(x,y)Ix ^
filter F.
YI’ Zl
x
^
^
O.
If (S,^) is a meet semilattice and F is a filter of S then
y
If t
r)* so that there exists z -> x, x’ and z
^
r
^
Clearly (x’)**
x for some
^
x
x imply x
^
Y2
e x’.
eD
and thus
^ x2 ^ Y2
Evidently (x
^
y
x
x
^
2
Similarly x
2ED.
Y2 ED’
As also
^ y)*
(x)*.
for some x
^
we have
x’
e x’.
2
0
By
y
^
(x,y)
x
x
since
Q.E.D.
R.
It can be observed that if S is not modular then
need not be equal to R
In the following theorem a necessary and suffi-
even if S is pseudocomplemented.
cient condition for a modular *-semilattice to be weakly distributive is obtained.
THEOREM 4.
D
If S is a *-semilattice which is directed above such that
R
then S is a weakly distributive semilattice if and only if for all x,y6 S
Assume that S is weakly distributive.
PROOF.
y
^
0.
d
Then there exists
t for some dD.
(z
Now we have (z)*
^
x,y
(a’)**.
(a)*
so that x
^
Thus we have (x
[x) v D and y
^
y
a
^
a’ E [y) v D.
zS
Let t E S and let x
such that x
z, y
([x) n [y)) v D.
^
Hence
By (I) there exists e ED such that z
(x)* and (y
using
^
e
_
t and
z and z
^
-<
t
^
e
t’
Conversely suppose (I) holds
0 which implies that (x)** and (y)**
a
a’)*
d
^
t)* so that there exists eD such that z
and hence ([x) v D) N ([y) v D)
and
...(I)
([x) v D) n ([y) v D)
([x) n [y)) v D
^
a)*
R
a’
(y)*.
Therefore, x
([x) v D) n ([y) v D
-< a’ for some z >- x, y.
(a)*
^
a’
P.V.R. MURTY AND M.K. MIIRTY
26
Since z
^
e
a
^
<- a
COROLLARY I.
a’
^
O,
we have z
^
z
0 and so
a
S is a weakly distri-
Then
Let S be a modular *-semilattice.
Q .E .D.
(a)*.
butive semilattice if and only if D is neutral in the lattice of filters of S.
Let L be a meet semilattice with 0 and D #
THEOREM 5.
,
then the following
are equivalent.
L/
(i)
x
^
is pseudocomplemented;
0 implies the existence of dD such that x
z
PROOF.
^
d
(I)(2):
(y)**.
(x)*
t
L is a *-semilattice satisfying the condition
(2)
Since x
Let X L and let
^
y
0 so that there exists
0.
Therefore r
*-semilattice.
Clearly
Now if x
-
^
z
(x)*.
such that r
0 and thus r
t
^
D
z’ where (z)**
Suppose r
^
d
r
_<
(z)*
(z’)*.
We now claim that
^
(z’)
^
x
0 and
d and hence r
(y)**.
(y)** and hence (x)*
(x)
0 then
y
^
d _<
OD(y).
((x))*
0 we have (y)**
y
^
^
where (z)**-- (z’)*.
Q.E.D.
If L is a modular similattice with 0 and D
*-semilattice if and only if L/
PROOF.
t
Thus L is a
(2)(i).
COROLLARY 2.
^
#
then L is a
is pseudocomplemented.
By Theorem 3 and the above Theorem 5.
Q.E.D.
Theorem 2 of 5 in [i] can as well be generalized to modular *-semilattices
as in the following:
THEOREM 6.
If L is a modular *-semilattice, then the following are equiva-
lent:
(i)
L is a Boolean algebra
(2)
L is a disjunctive semilattice, that is, a,b L and a < b implies the
existence of
(3)
cL
such that a
c
0 and b
^
c
# 0.
1 is the on’ly dense element of L.
PROOF.
Therefore L/
Clearly
D
(i)(2)
and
(2)(3).
Now assume (3).
L and by Corollary 2, L is pseudocomplemented.
modular *-semilattice, D #
3.
^
By (3)
id.
Since L is a
9.
SECTIONALLY PSEUDOCOMPLEMENTED SEMILATTICES.
In this article a further study is made of the concept of a sectionally
Q.E.D.
REMARKS ON CLASSES OF SEMILATTICES
27
v
Pseudocomplemented semilattice introduced by Katrink [3] and [4].
(see [3] and [4]).
DEFINITION 8
A meet semilattice L with 0 is called a
sectionally pseudocomplemented semilattice if and only if for every a 6 L the
[0,a]
interval
Every
is a pseudocomplemented semilattice.
pseudocomplememnted semilattice is a sectionally pseudocomplemented
L and X 6 [0,a] the pseudocomplement of x in this
semilattice where for a
terval [o,a] is given by x
,A
n-
a.
Throughout this article L stands for a sectionally pseudocomplemented semilattice and for any
x
x 6 [0,a] the pseudocomplement of x
LFMA 2.
For any x,y,z 6 L we have
,
(XA y AZ) yAz
Since v A (XA Z)
PROOF.
we have y
(x
in
[0,a]
is denoted by
a
y
A
A
A
Z)
(x AZ)
*yAZ
*z
(XA y
<
X A Z
^
0
y
*Z
A
< yA
only if ((x ha)
THEOREM 7.
(y ha)
If
@
,a
(y)
yh
<
a),
@(y)
h
<
<_ y
h
(x
z)
h
*Z
*@ (a)
L, the L/@
((x
h
h
is a sectionally pseudo-
*
a)
a)
@((x ha)
,
a)
(y)= @(0) then (Xhy ha,0).
*a
y ha) 6
(0).
SO
Suppose
I
@.
Therefore
O.E.D.
and L
LI->
are sectionally pseudocomplemented semilattices,
2
L
2
is said to be a ,-homomorphism if and only if
(f(x)) *f(a) for all x, a 6 L, and x <_ a.
Note that if f:
LI+
L
2
is a
*-homomorphism then kernel f is a ,-congruence and every filter congruence on L
is a ,-congruence.
DEFINITION II.
,
that ((xhy ha) yha
a)
@((x ha)
If L
y,z and
<
*yAm
0
Z
L whenever (x y) (
,
@(yha)
A
on L is said to be a *-congruence if and
and hence by the above lemma (y h(Xh a)
then a meet homomorphism f:
*
z)
^
Suppose @(x) < @(a). Clearly @(x)
DEFINITION I0.
f(x
y
for all a
6
,y
(X A
, z y)
(XA Z)
^ X^ y A
, yAZ
Z
NOW (XA y AZ)
is a ,-congruence on
(a) such that @(x)
@
Z
and y A
and hence (x
complemented semilattice where (@(x))
PROOF.
Z
AZ) *yAZ
DEFINITION 9. A meet congruence
,a
,
(XA Z)
If in L every section
[O,a]
is a Boolean algebra for all
then we call L as a sectionally Boolean algebra, also called as a (dual)
P.V.R. MURTY AND M.K. MURTY
28
[8].
semi-Boolean algebra in the sense of Abbott
In the following theorem we now show that given a sectionally pseudocomplemented semilattice there is a smallest quotient of it which is a sectionally Boolean
alzebra.
If we define @ on L by (x,y)
THEOREM 8.
0 if and only if (x
for all a 6 L, then @ is a ,-congruence on L such that L/ is a sectionally
Boolean algebra and if for any ,-congruence
L, L/ is a sectionally Boolean
on
algebra then
R(see theorem 3, 2).
First observe that @
PROOF.
,-congruence @
Also observe for
any
on L
((x ^a)
If a6 L then S
THEOREM 9.
*
(a)
*a)
((x ^a)
Q.E.D.
denotes the Boolean algebra of closed elements of [0,a].
a
{x
If we write S
(
S
a6L
a
x(a)
x(b)^ a whenever a
<_
b},
then S is a Boolean algebra and there is a ,-homomorphism f" L /S (whose kernel
is @) such that for each a 6 L we have S
L
f(x
*a
*b
f(a))(b)
are isomorphic.
Let x 6 L and a 6 L with x
*
f((x ^a) a )(b) (since x
)(b)
((xAB)
Aa) **a
(x
by f(x)(a)
/S
[0,f(a)]
and
AaAB)
**b
(xAb)
f(a)
(f(x))
(b).
_<
A
a(L
Now define
a
If b
_< a.
(f(x)(b)) * b
((f(x))’
f(a)(b)
A
@.
Thus f is a ,-homomorphism and ker f
be seen that the map xf(x) is a homomorphism of S a into [0,f(a)].
kernel of the above map is
prove that it is onto.
0
so that the map is a monomorphism.
Let y 6 S such that y
If b 6 L, then observe that y(a)^ b
_<
y(b).
Now y(b)
_<
(y(b) ^a)**b
REMARK.
verse limit of
f(a).
_<
y(b^ a)
y(b)
A
** b
It can
Now we shall
Te claim that f(y(a))
y.
,,,
y(b)^ a.
(a Ab)
A
Clearly
Clearlv (y(b)^ a)
(y(b)
Aa
^b)
b
**b
O.E.D.
Thus the map is an isomorphism.
If L is also directed above then it can be seen that S is the in-
{S
THEOREM i0.
morphic to S.
y(b) Af(a)(b)
L then
((xAa)*aA b) **b ((XAaAb=a Ab)=
a)
(a^b) **b
b
S
r
It can be seen that S is in fact a subalgebra of
PROOF.
f"
a
a
ab’
L} where
ab Sb /S a
(a
<
b) defined by
If L has a dense element then L/ is a Boolean
ab (x)
alebra
x ^a.
iso-
REMARKS ON CLASSES OF SEMILATTICES
Let d E L be dense so that L/@
PROOF.
and by theorem 9 we have S
S so that L/@
[0,f(d)]
d
_-- [O,f(d)].
29
under the map @[x] (x Ad)
Sd
Since f(d)(b)
**d
b for all b we have
_-- S.
Q.E.D.
If L has a dense element then L is a ,-semilattice.
COROLLARY 3.
Follows from Theorem i in [9].
PROOF.
a sectionally
Finally it is natural to ask when does a ,-semilattice become
Since every section of a ,-semilattice is also
pseudocomplemented semilattice.
a *-semilattice we will prove the following:
A necessary and sufficient condition for a ,-semilattice S to
TEOREM II.
x’
be pseudocomplemented is that (i) for each x E S there exists a greatest
,
(x)
such that (x)
x’’
and (2) x -< y implies
_<
y’’.
If S is a pseudecomplemented semilattice, then
PROOF.
x’
complement of x as
suppose that S is a ,-semilattice satisfying (i) and (2).
we have x N
Then (x
X
^ y’
AV’.
x’’
,
(y)
and since (x)
,
(x)
Since
takin the pseudo-
it can be seen that (i) and (2) are satisfied.
for any x,y 6 S we have (x)
and hence (x
((xAy’)’’) *
Now (x Ay’)" _< X"A y’ (by(2)).
if and only if
(x’’)
we have
Ay’
x’
(xAy’) *
Hence
x" Ay’
S is pseudocomplemented.
x’
x’
v’.
x’’’.
(xVV Ay’)
SO
Conversely
First observe that
Since (x’)
Suppose x
Now we claim that (x
x"
S
we have
that x <_
X’’Ay’
x"
<_
(x)
o.
^y
^ y’
_<
(xAy’)".
y’ and hence
0 .E .D.
ACKNOWLEDGEMENT:
We thank the referee for his valuable suggestions and comments which helped
in shapinz the paper into the present form.
P.V.R. MURTY AND M.K. MURTY
30
REFERENCES
i.
SPEED, T.P. Some Remarks on a class of Distributive lattices, J. Aust. Math.
Soc., IX (1969), 289-296.
2.
GRTZER,
G.
Lattice Theory.
First Concepts and Distributive lattices, W.H.
Freeman and Company, 1971.
v
Notes on Stone lattices I, Mat. fyz. Casop SAV 16 (1966), 128-142.
3.
KATRINAK, T.
4.
KATRINAK, T. Charakterisierung der Verallgemeinerten Stoneschen Halbverbande,
Mat. Casop. 19 (1969),235-247.
5.
KATRINAK, T.
6.
VARLET, J.C. A generalization of the notion of pseudocomplementedness, Bull
Soc. Roy. Sci. Liege 36 (1968), 149-158.
7.
RHODES, J.B.
8.
ABBOTT, J.C.
9.
SPEED, T.P. A Note on commutative semigroups ,J. Aust. Math. Soc., Vol. VIII
(1968), 731-736.
v
v.
Pseudokomplementare Halbverbande, Mat. Casop. 18 (1968), 121-143.
Modular and distributive semilattices.
Vol. 201 (1975), pp 31-41.
Semi-Boolean algebra, Mat. Vestnik
Trans. Amer. Math. Soc.,
(19) (1967), 177-198.
Lattice Theory, American Mathematical Society, Colloquium
Publications (Third edition 1967).
i0.
BIRKHOFF, G.
ii.
CORNISH, W.H. Pseudocomplemented modular semilattices, J. Aust. Math. Soc.
Vol. XVIII, (1974), 239-251.