Algebra Universalis, 22 (1986) 2 2 9 - 2 3 4

0002-5240/86/030229-06501.50 + 0.20/0 9 1986 Birkh~iuserVerlag, Basel

On the triple construction of distributive p-algebras P. V. RAMANA MURTY

Introduction In [1] Chen and Gr~itzer have obtained a triple construction for Stone lattices and in [9] W. C. Nemitz has obtained a construction for bounded implicative semilattices by means of admissible maps. Both the Stone lattices and bounded implicative semi-lattices are pseudo-complemented and distributive. In [1[ Chen and Gr~itzer have asked whether there is any connection between the two types of constructions. In [4] we have answered this question and have shown that Chen and Gr~itzer's construction for Stone lattices admits a Nemitz's type of construction as well. An answer to this question can also be found in [6; 3.7, Satz 5.7 and 5.9]. See also [8;4.1]. Let L be a distributive p-algebra. Then L admits the Nemitz construction iff O[aO(L)] is a comonomial congruence of D(L) for every closed element a. Following the lines of Chen and Gr~itzer in [7] Katrifi~ik has obtained a triple construction for p-algebras i.e., for pseudo-complemented distributive lattices by means of triples (B, D, q)) where B is a Boolean algebra, D is a distributive lattice with 1 and q~ is a join homomorphism of Bonto F(D), the lattice of filters of D such that 0q~ = 0 , lq~= 1 and acI)Na'~ is a principal filter= [d,). The aim of this paper is to establish a one-one correspondence between q~'s in the p-triples (B, D, q~) of Katrifi~k and certain maps f : B x D---* D and show that the construction of Katrifi~ik for p-algebras by means of p-triples admits a construction similar to that of Nemitz for bounded implicative semilattices by means of admissible maps. We begin with the following: D E F I N I T I O N 1. If A is a Boolean algebra and D is a distributive lattice with 1, then a mapping f : A x D--~ D is called a generalized admissible map if and

Presented by Ralph Freese. Received November 14, 1984. Accepted for publication in final form duly 10, 1986. 229

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only if it satisfies the following: i) f(a, f(b, d)) =f(a, d) whenever a <- b, ii) f(a, d N e) =f(a, d) N f ( a , e), iii) f(1, d) = d, iv) f(O, d) = 1, v) f(a, d U e) =f(a, d) Uf(a, e), vi) f(a, f(a', d))=f(a, 1 ) = f ( a ' , 1), vii) f(a, d) N f(b, d) <-f(a v b, d), viii) f (a, d) n f (a', d) = d n f (a, 1), (ix) f (a, f (b, d) ) <-f (a N b, d). In the above definition it can be seen that axioms (vi) to (ix) are all independent. T H E O R E M 1. If A is a Boolean algebra, D is a distributive lattice with 1 and f :A x D---~D is a generalized admissible map, then S = {(a, f(a, d) t a e A , d D} is a pseudo-complemented distributive lattice (where for (a, f(a, d)) and (b, f(b, e)) ~ S define (a, f(a, d)) <- (b, f(b, e)) if and only if a <- b and f(a, d) f(a, e)) whose Boolean algebra of closed elements is isomorphic to A and the dense filter to D. To prove this theorem we require the following: L E M M A 1. If f :A x D ~ D is a generalized admissible map and f(a, d)
Proof. f(a Nb, d ) = f ( a n b , f(a, d)) (by (i) of Definition i)<--f(aN b, f(a, e)) (by (ii) of Definition 1) =f(a n b, e) (again by (i) of Definition 1). Proof of Theorem 1. It can be easily seen that S is a partially ordered set with (0,f(0, 1)) as the least element and (1, f(1, 1)) as the greatest element. Also (a n b, f(a n b, d n e)) is the greatest lower bound of (a, f(a, d)) and (b, f(b, e)). We now claim that least upper bound of (a,f(a, d)) and (b,f(b, e)) is (a v b, f(a v b, (f( a' , e) n d) U ( f ( b' , d) n e))). We have f(a, d) = f(a, f(a, d)) =f(a, f(a, 1 ) n f ( a , d))=f(a, f(a', e)) N f(a, d)) (by (vi) of Definition 1 ) = f ( a , f ( a ' , e ) ) n f ( a , d ) = f ( a , f ( a ' , e ) n d ) (by (ii) of Definition 1 a t above) <-f(a, (f( , e) n d) U (f(b', d) N e)). Similarly f(b, e) <-f(b, (f(b', d) N e) Uf(a', e) N d)) thus showing that (a v b, f(a v b, (f(a', e) n d) U (f(b', d) f-t e))) is an upper bound of (a, f(a, d)) and (b, f(b, e)). Now if (a, f(a, d))<(c,f(c,g)) and ( b , f ( b , e ) ) < - ( c , f ( c , g ) ) then a<_c,b<_c and f(a,d) < f(a,g), f(b, e)<-f(b,g) so that by (viii) of Definition 1 above, we have d Nf(a, 1) =f(a, d) Nf(a', d) <-f(a, d) <--f(a, g). Also f(a v b, f(a'. e) n d) = f(a v b, f ( a' , e)) n f ( a v b, d) <-f(a' N b, e) Of(a v b, d) (by ix) of Definition 1 above) <_f(a' n b, g) (by Lemma 1 above, and the fact that f(b, e) <-f(b, g)).

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Since d N f(a, 1) <-f(a, g) and f(a, 1) = f(a', 1) we have d n f(a', e) <-f(a, g) so that f(a v b, f(a', e) n d) <-f(a v b, f(a, g)) <-f(a, g) (by (ix) of Definition 1 above) and hence f(a v b, f(a', e) n d) <-f(a' n b, g) n f ( a , g) <--f((a' N b) v a, g) (by (vii)) = f(a v b, g). Similarly we can show that f(a v b, f(b', d) n e)
b, h 2 U g ) ) = ( c v ( a n b ) , f ( c v ( a n b ) , (hlUg)n(h2Ug)))=(c,f(c,(hln h2) O g)) (since c >- a n b) = (c, f(c, g)). Therefore S is distributive (See [2; Lemma II.5.1]. If ( a , f ( a , d ) ) e S then (a',f(a', 1 ) ) e S is the pseudo-complement of (a, f(a, d)). Now (a, f(a, d)) n (a', f(a', 1)) = (0, f(0, d)) = (0, f(0, 1)), and if (a, f(a, d)) n (b, f(b, e)) = (O, f(0,1)) then a N b = 0 so that b - < a ' and f(b, e) <-f(b, 1). Hence (b, f(b, e)) <- (a', f(a', 1)). Therefore (a, f(a, d)) ~ S is closed if and only if (a, f(a, d)) = (a, f(a, d))** = (a, f(a, 1)) and it can be easily seen that the map a ~ (a, f(a, 1)) is an isomorphism of A onto the Boolean algebra of closed elements of S. Further (a, f(a, d)) ~ S is dense if and only if (a, f(a, d))* = (0, f(0, 1)) i.e., if and only if (a', f(a', 1)) = (0, f(0, 1)) i.e., if and only if a = 1. Thus for every d c D, (1, f(1, d)) is dense in S and the map d ~ (1, f(1, d)) is a lattice isomorphism of D onto the set of all dense elements of S. T H E O R E M 2. If L is a pseudo DL respectively denote the Boolean dense elements of L and if f :L** • then f is a generalized admissible f(a, d)<-f(a, e)".

complemented distributive lattice and L** and algebra of closed elements of L and the set of DL--+ DL is defined by f(a, d) = ( a n d) U a*, map satisfying "a N d---a n e if and only if

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Proof. A straight forward verification shows that f is indeed a generalized admissible map. If a n d-< a n e where a is a closed element and d and e are dense, then (a N d ) U a * --(a N e ) U a * so that f(a, d)<-f(a, e). Conversely if f(a,d)<-f(a,e), then a N S ( a , d ) = a N { ( a n d ) U a * } = ( a n d ) U ( a n a * ) = (a n d ) U O = a n d so that a nd<-a he. C O R O L L A R Y 1. If L is a pseudo-complemented distributive lattice, then a generalized admissible map satisfying "a N d <-a n e if and only if f(a, d)<-f(a, e)" is unique.

Proof. Let f : L * * x DL'--~DL (L** being the Boolean algebra of closed elements of L, Dc being the dense filter of L) be generalized admissible map satisfying the above stated condition. From the condition it follows that aNd=aNe if and only if f ( a , d ) = f ( a , e ) . Now a n f ( a , d ) = a N d since f(a, f(a, d)) =f(a, d) so that a n d <-a nf(a, d). Also a* = a* N 1 5 f ( a , d) for f(a*, 1) = f ( a * , f(a, d)). Thus f(a, d) is an upper bound of a n d and a*. If x is an upper bound of a N d and a* then x is dense. Now a Of(a, d) = a N d <-x so that by the condition in the theorem we have f(a, d) <-f(a, x). Also a* = a* n l ~ x so that f(a*, 1)<-f(a*, x). Thus f(a*, x)=f(a*, I). We have f(a, 1 ) = f(a*, 1) (since f is a generalized admissible map) so that f(a, d)<-f(a, I) = f(a*, 1) = f ( a * , x) and hence f(a, d) <-f(a, x) nf(a*, x) = x nf(a, 1)---xJ Thus f(a, d) = ( a n d) U a*. C O R O L L A R Y 2. If L is a pseudo-complemented distributive lattice, then L is isomorphic to SL = {(a, f(a, d)) l a ~ L**, d ~ DL} where f(a, d) = ( a n d) U a*. In [71 Katrifigk gave a construction for p-algebras i.e., for pseudocomplemented distributive lattices by means of triples (B, D, q~) where B is a Boolean algebra, D is a distributive lattice with 1 and q~ is a join homomorphism of B into F(D), the lattice of filters of D such that 045=0, 14~= 1 and aCbNa'eb is a principal filter Ida). This is a generalization of the construction of Stone lattices by Chen and Gr/itzer in [1]. We now establish a one-to-one correspondence between q~'s in the p-triples (B, D, cb) of Katrififik and the generalised admissible maps f : B x D---> D and show that the mapping (a, f(a, d)) (a, f (a, d) ) from S = { (a, f (a, d ) ) l a cA, d e D } onto {(a, d) l a ~ B, d e aqS, and d--< da} is actually an isomorphism between the constructed lattice (as in above Theoerem 1) which is in line with that of Nemitz's construction for bounded implicative semilattices and that of Katrififik's in [7]. DEFINITION 2. If A is a Boolean algebra, D is a distributive lattice with !

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and f a generalized admissible map from A x D into D, then we call (A, D, f ) a generalized admissible triple. We now prove the following interesting T H E O R E M 3. The generalized admissible triples correspond one-to-one to the p-triples of Katrifi~k in [7].

Proof. If {A, D , f ) is a generalized admissible triple, then define q~s:A---~ F(D) where F(D) denotes the lattice of filters of D, by a 4 ~ s = { d e D I f(a', d) =f(a', 1)}. Then a ~ s is a filter in D and f(a, d) eaCbs for any d e D (by (vi) of Definition 1). If a -< b, then aq)s _~ bq~s for if d e aq~s then f(a', d) = f(a', 1) and hence f(b', d ) = f ( b ' , f ( a ' , d)) (since b'<-a')=f(b', f(a', 1 ) ) = f(b', l). Then aq)s v bq)s~(a v b)q)s. If d e(a v b)q~s, then f(a' Nb', d)= f(a' N b', 1) so that f(a v b, d) n f ( a ' n b', d) =f(a v b, d) n f ( a ' N b', 1). Now f(a, d) n f ( b , d) <--f(a v b, d) (by (vii) of Definition 1) =f(a v b, d) Nf(a v b, 1 ) = f ( a v b , d ) n f ( a ' n b ' , l ) (since for any c e A we have f(c, 1)= f(c', 1)) =f(a v b, d) n f ( a ' nb', d) (from the observation made above) = d O f(a v b, 1) -
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T H E O R E M 4. Let (B, D, cI)} be a p-triple and let f be the corresponding generalized admissible map. Then S = {(a, f(a, d)) I a e B, d ~ D}, the distributive pseudocomplemented lattice from T h e o r e m 1, is isomorphic to L = {(a, d) I a 9 B, d 9 a ~ and d -< d~} (the ordering in L being defined by (a, d) --(b, e) if and only if a -< b and d <--epa (see [71).

Proof. Define a:S---) L by o(a, f(a, d)) = (a, f(a, d)). Since f is the generalized admissible map corresponding to 9 we have f(a, d ) = do, N da where [dpa ) = aq~ ~ [d) and [d,) = aq~ fq dqb so that f(a, d) 9 aq5 and f(a, d) <- d, and hence (a, f(a, d)) 9 L. Let (a, f(a, d)) <- (b, f(b, e)) in S so that f ( a , d) <--f(a, e). Now [dbp,) = aq) n [ d b ) = a ~ f3 bq~ n b ' @ = aq~ fq b'@ (since a -< b) ~ a@ fq a'q~ (since b' - a') = [d~) so that do <--dbOa and hence f (a, d) <-f (a, e) = eo~ N da
REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

C. C. CHEN and G. GRATZER,Stone lattices l, Can. J. Math., Vol. 21 (1969), 884-894. G. GRATZER, General lattice theory, Birkhauser Verlag, 1978. O. FraNK, Pseudocomplements in Semilattices, Duke Math. J. Vol. 29 (1962), 505-514. P. V. RAMANA MURT~ and V. V. RAtvtA RAG, Characterization of certain classes of pseudocomplemented semilattices, Algebra, Univ. Vol. 4 (1974), 289-309. P. V. RAMANAMURTY and M. KRISr~A MUgTY, On admissible semilattices, Alg. Univ. Vol. 6 (1976), 355-366. T. KATRI~AK, Die Kenn Zeichnung der distributiven pseudokomplementaren Halbverbiinde, J. Reine. angew math. 241 (1970), 160-174. T. KATRIg~AK, Uber eine Konstruction der distributiven pseudokomplementaren Verbiinde, Mathematische Nachrichten, Vol. 53 (1972), 85-99. T. KATRI~AKand P. MEDERLY, Constructions of p-algebras, Algebra Univ. 17 (1983), 288-316. W. C. NEMITZ, Implicative Semilattices, Trans. Amer. Math., Soc. Vol. 117 (1965), 128-!42. Andhra University Wattair India