Distributive lattices as a generalization of Brouwerian lattices

Discrete Mathematics 20 (1977) 197-299. @ North-Holland Publishing C.ornpany

NUTE

i 1,

DISTRXBU’TIVE LATTICES AS A GENERALIZATION OF BROUWERIAN LATTICES Hen ner KRUGER Institut f’r Informatik

und Praktische Mathematik,

Uniwrsit&

Kiel. D 2.3 Kid, (inmany

Received 10 Au:gust 1976 Revised 21 January I’.377

it is ~&en for granted rhat Brouwerian lattices are distributive. The purpose of this paper is to give an axiom system for Brouwerian lattices from which can be deduced a slightly modified a&m system for distributive lattices.

A rat&ice{M, fl, LI) 5 called a Brouwerian or subjunctive lattice, iff there exists a binary operation r-, and the following axioms hold:

In

af-lbSc

.a

agbbrc,

(1)

aflbsc

+=

acbrc.

(2)

such a lattice it can be seen that bsar-b

6)

which implies an(bUc)~(arb)L-l(arc). al”lbgc

=+

arbSa,-c

CiDJ (3)

holds on account of (I) and (arb)lla

~b.

(4)

‘Iknvm 1. A lastice (MT fl, U) is Brouwerian, ifi there exists a binary operation I-and the axioms (S), $I), and (3) hold.

Ptaof. it is sufficient to verify that (S), (2), and (3) implies (1). From (3) we have s b r c, and using (S) we have (1). aClbgc =$+ bra ‘J’hewem 2. A lattice (M, fi, U) is distributiue, ifi there exists a binary operation ra& the axioms (D), (2), and (3) hold.

Pd. In order to prove (5), (2), and (3) in a distributive lattice we substitute I’7for I-. In order to prove the converse we use a slight modificatioil of the proof that BrcPuwerian lattices are distributive ([I], p. 181). Lc=t9 : = (a fl B) U(a ll c). Then 9 and h fl c G 9. It follows immediately from [3), (I3X and (2) that dur-9, (ar-b)U(arc)Gffarq, aR[BiJc)c arb6arq and arc u r 9, a ll(b LIc) s 9. Obviously, distributivity holds. In lattices the operation r is uniquely determined by (S), (2), and (3). That does not hold for (D), (Z), and (3). It is sufficient to find aT:other binary operation I-+ satisfying (D), (2), and (3). In Brouwcrian lattices the axioms (D) and (2) with r+ substituted for t- hold, iff a 1 b G a P+ b d Q r-- b. Axiom (3), hclwever, is more restrictive. In @olean lattices there is no binary operation defineld by a Boolean expression, except a ll b and Q -3 b:= a’et 6, satisfying (O), (2), and (3). ham these theorems it can be seen that the difference between Brouwerian lattices and distributive lattices is only in correlation to the differeEice between the axioms (S) and CD).This relation is closely connected to the structure of lattices, In .ordered zwerch-lattices, a neither commutative nor associative gc:neralhation of lztices [2,3,4], we can define subjunctive zwerch-lattices by f:l) and (2). In subjunctive zwerch-lattices infimum(a, b) exists and (a r b) A a = inf(a, b), corresponding to (4) in Brouw-erian lattices, as well as many other properties similar to lattice theory like a A 6 s & and (a A 6 6 a + a A b = inf(a, 6)) hold, But (S) does not hold on account of ‘Sinf(u, 6) +S

a A b = inf(u, b).

Using the nondistributib e :nodular lattice of five elements in order to prove the independence of the axioms (Da, (2), and (3), we get arb: yes yes no

= b

arb:=ai-Ib no yes yes.

F\‘gbtethat it is possible to substitute the implications (l), {2), and (3) by ine+atities (and also by equaiities); indeed, it is known, that (2) can be replaced by (4). on the other hand, a G bf-(cU(a.!lb))

WI

ai-b~u~(ctifuflb))

0”)

and

can qhx

(1) and (3) respcztive!y (I am indebted to Y. Weber for pbis hint). ‘.

.,.

;

‘-.

_-’

Rekvevaces [ 1) H. Gericke, Therxie der Vertliinde, 81-Hochschultaschenbuch W38a, Mannheih (1963). [21 H. Krfiger, Subjunktionsbegrii?e in orthomodularen VerbHndcn und B&&‘sche R&hiefrerbPnde. ! Dissertatiorr, TW Ml&en, Mfinchen {IWZ). ! f3] H. Kr&pr, Zwercb-Assoziwivitiit und vc?rbandstihnticbe Algebren, rjayerischri Akademie der Wisxxenschsften, maWnat. Ktasse, Sittungsbcrichte (1973) 23-48. (41 H. K&get, Das Assoziativgesetz als Kommutativitiitsaxiom in Rocxie’schen Zwetchverbiinden, J. ,Reine Angew. Math, 285 (19%) 5.3-B.