Bulletin of the Section of Logic
Volume 12/3 (1983), pp. 117–120
reedition 2008 [original edition, pp. 117–121]
Jacek Hawranek
Jan Zygmunt
SOME ELEMENTARY PROPERTIES OF
CONDITIONALLY DISTRIBUTIVE LATTICES
The notion of a conditionally distributive lattice was introduced by B.
Wolniewicz while formally investigating the ontology of situations (cf. [2]).
In several of this lectures he has appealed for a study of that class of lattices.
The present abstract is a response to that request.
1. A characterization theorem
We shall consider only distributive lattices which have unit elements.
Definition 1. A lattice L (with the unit element 1) is conditionally
distributive, in short, c-distributive, iff the following clause holds:
(c) b + c 6= 1 ⇒ a(b + c) = ab + ac
for all a, b, c ∈ L.
It is easily seen that the class of c-distributive lattices is closed under
homomorphisms and sublattices, but it is not closed under direct products.
The two typical examples of non-c-distributive lattices are M5⊕ and N5⊕
whose diagrams are as follows:
•
•
•
@
• •@•
@
@•
M5⊕
•
#J
•#
J•
•
c
c•
N5⊕
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Jacek Hawranek and Jan Zygmunt
Theorem 1. For an arbitrary lattice L the following three conditions are
equivalent:
1. L is c-distributive;
2. L contains neither M5⊕ nor N5⊕
3. Every sublattice of L which does not contain the unit of L is distributive.
Theorem 1 is the counterpart of a well-known Birkhoff’s criterion for
distributivity. By inspection of its proof we get a characterization theorem
for conditionally modular lattices, that is those fulfilling the clause:
(cm) a ≥ c and b + c 6= 1 ⇒ a(b + c) = ab + c.
Theorem 2. For an arbitrary lattice L the following three conditions are
equivalent:
1. L is conditionally modular;
2. L does not contain N5⊕ ;
3. Every sublattice of L not containing the unit of L is modular.
Corollary. A lattice L is c-distributive iff it satisfies the clause:
(c0 ) a + b 6= 1 and a + c 6= 1 ⇒ a + bc = (a + b)(a + c).
Wolniewicz’s original definition of a conditionally distributive lattice
used both conditions (c) and (c0 ).
2. Conditionally prime filters
Definition 2. A subset H of a lattice L is a conditionally prime filter of
L, in short, c-prime filter, iff it is a filter of L satisfying the clause:
(cp) a + b ∈ F and a + b 6= 1 ⇒ a ∈ F or b ∈ F .
Theorem 3. If L is a c-distributive lattice, and F is a filter of L, and
a ∈ L − F , then there exists a c-prime filter G of L such that a 6∈ G ⊇ F .
Corollary. If L is a c-distributive lattice, and a 6≤ b (a, b ∈ L), then
there exists a c-prime filter F in L with a ∈ F and b 6∈ F .
Theorem 4. For an arbitrary lattice L the following four conditions are
equivalent:
Some Elementary Properties of Conditionally Distributive Lattices
119
1. L is c-distributive;
2. Every relatively maximal filter of L is c-prime;
3. Every filter of L is the intersection of all c-prime filters containing
it;
4. The lattice J (L) = hI(L), ⊆i of all ideals of L is c-distributive.
3. A representation theorem
Let 1 be a fixed set, and A be a family of subset of 1 such that
(i) ∅, 1 ∈ A;
(ii) 1 is closed under finite intersections;
(iii) X ∪ Y 6∈ A and X ∪ Y ⊆ Z ∈ A ⇒ Z = 1 for all X, Y, Z ∈ A.
Define a dual lattice L(A) = hA, ∧, ∨i setting for any X, Y ∈ A
X ∧Y =X ∩Y
X ∨Y =
Lemma.
clusion.
X ∪Y
1
if X ∪ Y ∈ A − {1}
otherwise.
L(A) is a c-distributive lattice whose lattice ordering is set in-
Theorem 5. If L is a c-distributive lattice, then L is isomorphic to some
dual lattice L(A).
Proof. Let F be the set of all c-prime filter of the lattice L in question.
Define a function f : L → P (F) setting f (a) = {F : a ∈ F and F ∈ F}.
Letting 1 = F and A = {f (a) : a ∈ L}, we see that f satisfies the following
conditions:
f (a) ∧ f (b) = f (a) ∩ f (b)
f (a) ∨ f (b) =
f (a) ∪ f (b) if a ∨ b 6= 1
1
otherwise,
and that it is an isomorphism of L onto L(A).
120
Jacek Hawranek and Jan Zygmunt
We end with the open problem of describing c-distributive lattices in
terms of ordered topological spaces (cf. [1]).
References
[1] A. Urquhart, A topological representation theorem for lattices, Algebra Universalis 8 (1978), pp. 45–58.
[2] B. Wolniewicz, On the lattice of elementary situations, Bulletin
of the Section of Logic, vol. 9, no. 3 (1980), pp. 115–121.
The Section of Logic
Institute Philosophy and Sociology
Polish Academy of Sciences
Institute of Philosophy,
Sociology and Logic
University of Wroclaw