On distributive lattices

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A N N A LES SO C IETA T IS M A TH EM ATICAE PO LO N AE Series I : COM M ENTATIONES M ATH EM ATICAE X V I I I (1971) R O C Z N IK I PO LS K IE G O T O W A R Z Y S T W A M ATEM ATYCZNEGO

Séria I : P R A C E M A TEM ATYCZN E X V I I I (1974)

H

i d e g o r o

N

a k a n o

and S

t e p h e n

B

o m b e r g e r

(Detroit, Mich.)

On distributive lattices

In this paper we generalize Glivenko’s theorem and then show how this simplifies the proof that the clusters of a linear lattice form a complete Boolean algebra.

Let 8 be a linear lattice. A set of 8 is called a m anifold if is not empty.

A manifold M is said to be a cluster if M = M 11, where M L — {x\

\x\a

\y\

= 0 for every y e M }. In [1] it is proved that the clusters form a Boolean algebra by the order of inclusion. In [ 2 ] we gave two different proofs.

In these proofs we used stars: a manifold M of 8 is called a star if x e M and 0 < y < \x\ imply y e M . The purpose of this paper is to simplify the proof of [ 1 ] by stars instead of ideals.

According to [1], a Brouwerian lattice A is a lattice such that for any a , b e L , {ж|ала?<&} contains a greatest element which is denoted by b : a. In a Brouwerian lattice with 0, 0 : a is denoted by a.*

G ^livenko ’ ^s ^theorem . I f L is a Brouwerian lattice with 0, then the *correspondence a -> u** is a lattice-epimorphism o f L onto the Boolean* algebra o f elements o f L such that a = a **.

In [1], p. 45, it is shown that a Brouwerian lattice is a distributive lattice. Thus, if A is a Brouwerian lattice with 0, then L is a distributive lattice with 0 and x л у = 0 if and only if x < y. We now generalize* Glivenko’s theorem by these conditions.

In a previous paper [2], we proved the

L ^emma . Let A be a lattice with 0. I f there is a map x ->■ x' from A into A such that

(i) x" — x,

(ii)

^{х а}

у = 0 i f and only i f x < y r, and (iii) (®v y)' = x ' a y',

then A is a Boolean algebra.

Using this lemma we can prove the

T ^heorem . Let L be a distributive lattice with 0. I f there is a map x from L into L such that

( 1 ) ха у = 0 i f and only i f x < y',

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1 5 2 H . N a k a n o a n d S. R o m b e r g e r

then A — {x'\ x e L } is a Boolean algebra and a function f : A defined by f (x) — x" is a lattice-epimorphism.

P ro o f. Since x' < x we have

( ² ) x' a x — ⁰

by (1). This implies

(3) x < x".

Since y < x implies x ’ a y < x' a x which implies x' A y = 0 by ( 2 ), we have x' < y ’ by (1). Therefore

( 4 ) y < x implies x' ^ y'.

B y (3) and (4), we conclude

(5) x'" = x'.

Since x v y > x, у, we obtain (xv у)' by (4). Since (x' A y')A ( X V y) = (x' A y' A X ) v (x' А у' А у) = 0

by ( 2 ), we obtain x' a y' < (xv y)' by (1). Therefore

( ⁶ ) (xv y)' = x'A y'.

According to (5), x e A if and only if ж" = x.

I t is easily shown that A is a lattice under the same order as L with meet “ a ” and join “v ”, where for a, be А, а д b = ал b and a v b

— (av b)". Also 0 ^ x implies x' ^ 0r by (4). So

0 " = 0 "л O' = 0,

the last equality following from (2). Thus O e l and A is a lattice with 0 and satisfies (i) by (5), (ii) by (1), and (iii) by the following. For a , be A, we have

(a v b)' = (av b)'" = (av b)' = a ' a b' = a' a b’

by ( 6 ). Thus by the lemma, A is a Boolean algebra.

Let X, y e L . B y (4) x < у implies y' < x' which in turn implies x"

< y". Thus

( 7 ) x ^ у implies f (x) < № •

B y ( 6 ) and ( 6 ),

f ( x v y) = (xv у)" = ( x ' a у')' = ( x "' a у'")'

= (®"v y")" = x" v y" = f ( x ) v f ( y ) .

Since xa y < x, у, we have that f ( x

^a

y) < f ( x ) , f ( y ) by (7).

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Distributive lattices

1 5 3

Thus

( 8 ) f(xA ² / K f(x) д f{y).

On the other hand, by setting 0 = (

х а

y)'

^a

{

x

"

a

y " ) we have

z a х а

у = 0 ,

^{z a}

x' = 0 , and ZAy' = 0 by ( ² ). Thus

so that

za x ^< y', z ^< x", and

z a

x < y'

^a

y" — 0 , which implies 0 < x' and thus

2 < x' ^a x" ^{= 0 .}

z < y "

Consequently,

²^{: =}

⁰ which implies x"

^а

у"

< ( ж л

у)". Therefore

f{XA у) = ^{( Х А} у ) " > X" ^A y" = f ( x ) Af(y)

B y ( ⁸ ) and the above f ( xA y) = f (x) д/(у) and / is a lattice-homo

morphism. B y (5) / is trivially onto and thus / is a lattice-epimorphism.

As an application, let 8 be a linear lattice. As defined in [ 2 ], a manifold M is called a star if x e M and 0 < у < \x\ implies y e M . For any system of stars (À€ Л) it is obvious that { J M x and П Mx are also stars.

ЛеЛ ЛеЛ

Therefore the stars of 8 form a distributive lattice under the inclusion order with { 0 } as the zero of this lattice. According to the lemma in [ 2 ], we have M n N = {0} if and only if M c N 1 for M and N stars. Thus the clusters of 8 (0 Ф M <= 8 is called a cluster if M = M 11) form a Boolean algebra and the m ap/(ilf) = M 11- is a lattice-epimorphism by our gener

alization of Glivenko’s theorem.

References

[1] G. B irk h o ff, Lattice theory, Third Edition, Amer. Math. Soc. (Colloquium Pub

lications) 25 (1967).

[2] H. N ak an o and S. R o in b e rg e r, Cluster Lattices, Bull. Acad. Sci. Polon., Sér.

Sci. math, astronom. phys. 19 (1971), p. 5-7.

W A Y N E ST A T E U N IV E R S IT Y

On distributive lattices

H

N

and S

B

(Detroit, Mich.)

On distributive lattices

In this paper we generalize Glivenko’s theorem and then show how this simplifies the proof that the clusters of a linear lattice form a complete Boolean algebra.

Let 8 be a linear lattice. A set of 8 is called a m anifold if is not empty.

A manifold M is said to be a cluster if M = M 11, where M L — {x\

\y\

= 0 for every y e M }. In [1] it is proved that the clusters form a Boolean algebra by the order of inclusion. In [ 2 ] we gave two different proofs.

In these proofs we used stars: a manifold M of 8 is called a star if x e M and 0 < y < \x\ imply y e M . The purpose of this paper is to simplify the proof of [ 1 ] by stars instead of ideals.

According to [1], a Brouwerian lattice A is a lattice such that for any a , b e L , {ж|ала?<&} contains a greatest element which is denoted by b : a. In a Brouwerian lattice with 0, 0 : a is denoted by a*.

G livenko ’ s theorem . I f L is a Brouwerian lattice with 0, then the correspondence a -> u** is a lattice-epimorphism o f L onto the Boolean algebra o f elements o f L such that a = a **.

In [1], p. 45, it is shown that a Brouwerian lattice is a distributive lattice. Thus, if A is a Brouwerian lattice with 0, then L is a distributive lattice with 0 and x л у = 0 if and only if x < y*. We now generalize Glivenko’s theorem by these conditions.

In a previous paper [2], we proved the

L emma . Let A be a lattice with 0. I f there is a map x ->■ x' from A into A such that

(i) x" — x,

(ii)

у = 0 i f and only i f x < y r, and (iii) (®v y)' = x ' a y',

then A is a Boolean algebra.

Using this lemma we can prove the

T heorem . Let L be a distributive lattice with 0. I f there is a map x from L into L such that

( 1 ) ха у = 0 i f and only i f x < y',

then A — {x'\ x e L } is a Boolean algebra and a function f : A defined by f (x) — x" is a lattice-epimorphism.

P ro o f. Since x' < x we have

( 2 ) x' a x — 0

by (1). This implies

(3) x < x".

Since y < x implies x ’ a y < x' a x which implies x' A y = 0 by ( 2 ), we have x' < y ’ by (1). Therefore

( 4 ) y < x implies x' ^ y'.

B y (3) and (4), we conclude

(5) x'" = x'.

Since x v y > x, у, we obtain (xv у)' by (4). Since (x' A y')A ( X V y) = (x' A y' A X ) v (x' А у' А у) = 0

by ( 2 ), we obtain x' a y' < (xv y)' by (1). Therefore

( 6 ) (xv y)' = x'A y'.

According to (5), x e A if and only if ж" = x.

I t is easily shown that A is a lattice under the same order as L with meet “ a ” and join “v ”, where for a, be А, а д b = ал b and a v b

— (av b)". Also 0 ^ x implies x' ^ 0r by (4). So

0 " = 0 "л O' = 0,

the last equality following from (2). Thus O e l and A is a lattice with 0 and satisfies (i) by (5), (ii) by (1), and (iii) by the following. For a , be A, we have

(a v b)' = (av b)'" = (av b)' = a ' a b' = a' a b’

by ( 6 ). Thus by the lemma, A is a Boolean algebra.

Let X, y e L . B y (4) x < у implies y' < x' which in turn implies x"

< y". Thus

( 7 ) x ^ у implies f (x) < № •

B y ( 6 ) and ( 6 ),

f ( x v y) = (xv у)" = ( x ' a у')' = ( x "' a у'")'

= (®"v y")" = x" v y" = f ( x ) v f ( y ) .

Since xa y < x, у, we have that f ( x

y) < f ( x ) , f ( y ) by (7).

Distributive lattices

Thus

( 8 ) f(xA 2 / K f(x) д f{y).

On the other hand, by setting 0 = (

y)'

{

"

y " ) we have

у = 0 ,

x' = 0 , and ZAy' = 0 by ( 2 ). Thus

so that

za x < y', z < x", and

x < y'

y" — 0 , which implies 0 < x' and thus

2 < x' a x" = 0 .

z < y "

Consequently,

0 which implies x"

у"

у)". Therefore

f{XA у) = ( Х А у ) " > X" A y" = f ( x ) Af(y)

B y ( 8 ) and the above f ( xA y) = f (x) д/(у) and / is a lattice-homo­

morphism. B y (5) / is trivially onto and thus / is a lattice-epimorphism.

As an application, let 8 be a linear lattice. As defined in [ 2 ], a manifold M is called a star if x e M and 0 < у < \x\ implies y e M . For any system of stars (À€ Л) it is obvious that { J M x and П Mx are also stars.

alization of Glivenko’s theorem.

References

[1] G. B irk h o ff, Lattice theory, Third Edition, Amer. Math. Soc. (Colloquium Pub­

lications) 25 (1967).

According to [1], a Brouwerian lattice A is a lattice such that for any a , b e L , {ж|ала?<&} contains a greatest element which is denoted by b : a. In a Brouwerian lattice with 0, 0 : a is denoted by a.*

G ^livenko ’ ^s ^theorem . I f L is a Brouwerian lattice with 0, then the *correspondence a -> u** is a lattice-epimorphism o f L onto the Boolean* algebra o f elements o f L such that a = a **.

In [1], p. 45, it is shown that a Brouwerian lattice is a distributive lattice. Thus, if A is a Brouwerian lattice with 0, then L is a distributive lattice with 0 and x л у = 0 if and only if x < y. We now generalize* Glivenko’s theorem by these conditions.

L ^emma . Let A be a lattice with 0. I f there is a map x ->■ x' from A into A such that

T ^heorem . Let L be a distributive lattice with 0. I f there is a map x from L into L such that

( ² ) x' a x — ⁰

( ⁶ ) (xv y)' = x'A y'.

( 8 ) f(xA ² / K f(x) д f{y).

x' = 0 , and ZAy' = 0 by ( ² ). Thus

za x ^< y', z ^< x", and

2 < x' ^a x" ^{= 0 .}

⁰ which implies x"

f{XA у) = ^{( Х А} у ) " > X" ^A y" = f ( x ) Af(y)

B y ( ⁸ ) and the above f ( xA y) = f (x) д/(у) and / is a lattice-homo

[1] G. B irk h o ff, Lattice theory, Third Edition, Amer. Math. Soc. (Colloquium Pub