Bulletin of the Section of Logic
Volume 18/3 (1989), pp. 105–111
reedition 2006 [original edition, pp. 105–111]
Boguslaw Wolniewicz
ON ATOMIC JOIN-SEMILATTICES
For motivation cf. [2] and [3]. In general terminology we follow [1].
1. Preliminaries
Henceforth let (L, ∨, 0, 1) be a bounded non-degenerate join-semilattice.
At(L) are its atoms, if any, and At(x) are the atoms contained in the
element x. F in(A) is the collection of finite subsets of A, including the
empty one. An element x is join-reducible if x = y ∨ z, for some y, z 6= x.
As usual, L is atomic if At(x) 6= ∅, for any x 6= 0. And L is atomistic
if for any x ∈ L there is an A ⊂ At(L) such that x = supA.
L is finitely atomic if it is atomic and At(x) is finite, for any x 6= 1. L
is finitely atomistic if for any x ∈ L : x = supA, for some A ∈ F in(At(L)).
And L is uniquely atomistic if it is atomistic, and for any x 6= 1, and any
A ⊂ At(L) : if x = supA, then A = At(x).
Evidently, the five classes of join -semilattices just defined are related
as in the diagram, arrows indicating implications:
finitely atomic
finitely atomistic
uniquely atomistic
PP
q
P
)
PP
atomistic
PP
PP
q
P
)
atomic
PP
(1) The following conditions are equivalent:
a) L is atomistic;
b) x = supAt(x), for any x ∈ L;
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Boguslaw Wolniewicz
c) At(x) = At(y) ⇒ x = y, for any x, y ∈ L.
Indeed, (a) ⇐ (b) is obvious. Conversely, x = supA, for some A ⊂
At(L), by hypothesis. Clearly, A ⊂ At(x), and x is an upper bound of
At(x). If y is another such bound, and y ≤ x, then y is an upper bound of
A, and so x ≤ y. Thus there is no upper bound of At(x) smaller than x.
Again, (b) ⇒ (c) is obvious by the foregoing. Conversely, suppose y
is an upper bound of At(x) such that y ≤ x. Then we have At(y) ⊂ At(x)
by the latter, and At(x) ⊂ At(y) by the former. Hence by the hypothesis
x = y, showing x to be the least upper bound of At(x).
(2) If L is atomistic and finitely atomic, then it is finitely atomistic, provided its unit is join-reducible.
Indeed, if x 6= 1, then At(x) is finite. So x = supAt(x). Otherwise
1 = x∨y, for some x, y 6= 1. Hence 1 = supAt(x)∨supAt(y) = sup(At(x)∪
At(y)), all the relevant suprema existing by finiteness. Thus 1 is a join of
a finite number of atoms too.
(3) If L is both finitely and uniquely atomistic, then it is finitely atomic.
Indeed, if x 6= 1, then x = supA, for some finite A ⊂ At(L); and
A = At(x).
Consequently,
(4) If L is uniquely atomistic and 1 is join-reducible, then L is finitely
atomistic iff L is finitely atomic.
(5) If L is atomistic and finitely atomic, then L is a lattice and inf {x, y} =
sup(At(x) ∩ At(y)).
Indeed, as A = At(x) ∩ At(y) is finite, we set z = supA. Clearly,
z is a lower bound of {x, y}. If z 0 is another, then z 0 ≤ x, z 0 ≤ y. Hence
At(z 0 ) ⊂ A, as in all posets, and so supAt(z 0 ) ≤ supA. Since L is atomistic,
the latter means that z 0 ≤ z, showing z to be the greatest lower bound of
A.
On Atomic Join-Semilattices
107
2. Uniqueness
(6) L is uniquely atomistic iff L is atomistic and for any A ⊂ At(L) such
that supA exists the following holds:
supA 6= 1 ⇒ At(supA) = A.
Indeed, implication ⇒ is obvious in view of uniqueness. Conversely,
tale any x 6= 1. L is atomistic, so x = supA, for some A ⊂ At(L). Hence
At(supA) = A by hypothesis, showing A to be unique.
(7) If L is uniquely atomistic, then for any A, B ⊂ At(L) : supB 6= 1 ⇒
(supA ≤ supB ⇒ A ⊂ B), provided both suprema exist.
Indeed, if supB differs from 1, so does supA. So A = At(supA) and
B = At(supB), by (6). As in all posets, supA ≤ supB implies At(supA) ⊂
At(supB). Hence A ⊂ B.
(8) If L is a uniquely atomistic lattice, and A, B are sets of atoms such that
their suprema exist and differ from 1, then A ∩ B = ∅ iff supA ∧ supB = 0.
Indeed, as in any join-semilattice, sup(A ∩ B) ≤ supA, provided both
suprema exist. Consequently, sup(A ∩ B) ≤ supA ∧ supB = 0, showing
that sup(A ∩ B) = 0. Hence A ∩ B = ∅. Conversely, if supA ∧ supB 6= 0,
then there is an atom u belonging both to At(supA) and At(supB). By
hypothesis both suprema differ from 1. Hence by (6): u ∈ A ∩ B, i.e.
A ∩ B 6= ∅.
3. Conditional Distributivity
(9) If L is finitely atomic and uniquely atomistic, then for any x, y ∈ L :
x ∨ y 6= 1 ⇒ At(x ∨ y) = At(x) ∪ At(y).
Indeed, as x, y 6= 1 in view of the hypothesis, At(x) ∪ At(y) is finite.
Thus sup(At(x) ∪ At(y)) = supAt(x) ∨ supAt(y) = x ∨ y = supAt(x ∨ y),
the first identity valid in all posets, and the rest holding as L is atomistic.
Hence At(x) ∪ At(y) = At(x ∨ y) by uniqueness.
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Boguslaw Wolniewicz
Call a lattice (L, ∨, ∧) conditionally distributive if x ∧ (y ∨ z) = (x ∧
y) ∨ (x ∧ z), provided y ∨ z 6= 1.
(10) If L is finitely atomic and uniquely atomistic, then L is a conditionally
distributive lattice.
Indeed, L is a lattice by (5). As At(x ∧ y) = At(x) ∩ At(y) in any
lattice whatever, so by (9) we get for any x, y, z ∈ L such that y ∨ z 6=
1 : At(x ∧ (y ∨ z)) = At(x) ∩ At(y ∨ z) = At(x) ∩ (At(y) ∪ At(z)) =
At(x ∧ y) ∪ At(x ∧ z) = At((x ∧ y) ∨ (x ∧ z)). Hence we get the rest by using
the identity x = supAt(x), as L is atomistic.
(11) If L is finitely atomistic and L is a conditionally distributive lattice,
then L is uniquely atomistic.
Indeed, take any x 6= 1, and any finite set of atoms such that x =
supA. Clearly A ⊂ At(x). Set A = {x1 , . . . , xk } and suppose there is an
y ∈ At(x) such that y 6∈ A. Then, on the one hand, we have y ≤ supA, i.e.
y ∧ supA = y. On the other, however, we get by conditional distributively:
y ∧ supA = y ∧ (x1 ∨ . . . ∨ xk ) = (y ∧ x1 ) ∨ . . . ∨ (y ∧ xk ) = 0 ∨ . . . ∨ 0 = 0, as
all the meets y ∧ xi are meets of distinct atoms. This contradiction shows
that A = At(x), i.e. that L is uniquely atomistic.
Combining (10) and (11) we get eventually as a corollary:
(12) If L is a finitely atomistic lattice, then L is conditionally distributive
iff L is uniquely atomistic.
4. Coherent Sets
Call a set A ⊂ L coherent if for any B ∈ F in(A) : supB 6= 1.
Clearly, every proper ideal of L is a coherent set, and every ideal
generated by a coherent set is proper.
(13) If A is a coherent set, and B ∈ F in(A), then A ∪ {supB} is coherent
too.
Indeed, the ideal I(A) generated by A is proper, and supB ∈ I(A).
Hence A ∪ {supB} is coherent, being a subset of the coherent set I(A).
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On Atomic Join-Semilattices
(14) Every maximal ideal R of L is a maximal coherent set in L. In fact,
maximal ideals coincide with maximal coherent sets.
Indeed, being maximal, R is proper, hence coherent. Suppose R ⊂ R0 ,
with R0 coherent. Then I(R0 ) is proper, and so I(R0 ) = R = R0 , showing R
to be maximal coherent. Conversely, if R is maximal coherent, then I(R) is
proper, hence coherent too. But R ⊂ I(R), so I(R) = R, as R is maximal.
Thus R is an ideal. Now suppose R ⊂ I, where I is a proper ideal. So I is
coherent; hence I = R, showing R to be maximal also as an ideal.
(15) If R is a maximal ideal of L, then for any x ∈ L:
x 6∈ R ⇒
_
x ∨ z = 1.
z∈R
Indeed, if x 6∈ R, then R ∪ {x} is not coherent by (14). Hence supB =
1, for some B ∈ F in(R ∪ {x}). Clearly, B = A ∪ {x}, for some A ∈ F in(R).
Setting z = supA, we have x ∨ z = 1 and z ∈ R, as R ∪ {z} is coherent by
(13) and R is maximal. (This strengthens “Proposition 1” in [2].)
Let us say that z is then an impedance to x in R.
5. Separability and Complementation
Let R be all maximal ideals of L. Set r(x) = {R ∈ R : x ∈ R}, for any
x ∈ L. Call x, y ∈ L inseparable if r(x) = r(y). Call x, y ∈ L incompatible
if x ∨ y = 1, and let {x}⊥ be all the elements incompatible with x.
Clearly, r(0) = R, and {0}⊥ = {1}.
(16) r(x) = r(y) iff {x}⊥ = {y}⊥ , for any x, y ∈ L.
Indeed, suppose {x}⊥ 6= {y}⊥ . Then either x∨z1 = 1 and y∨z1 6= 1, or
vice versa, for some z1 ∈ L. Assuming the former, we get: z1 ∈ R ⇒ x 6∈ R,
for all R ∈ R; and y ∨z1 ∈ R1 , for some R1 ∈ R. Hence x 6∈ R1 and y ∈ R1 ,
for some R1 ∈ R, showing that r(x) 6= r(y). Conversely, suppose the latter.
Then either x 6∈ R! and y ∈ R1 , or vice versa, for some R1 ∈ R. Assuming
the former we get by (15): x ∨ z1 = 1, for some z1 ∈ R1 ; and y ∨ z 6= 1, for
all z ∈ R1 . Thus {x}⊥ 6= {y}⊥ .
Call L join-complemented if for any x 6= 0 there is an y 6= 1 such that
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Boguslaw Wolniewicz
x ∨ y = 1. And call the zero in L separated if r(x) 6= R, for any x 6= 0.
(Clearly, if L = {0, 1}, then L is join-complemented, and it has a separated
zero.)
(17) L is join-complemented iff L has a separated zero.
Indeed observe that L is join-complemented iff for any x 6= 0 there is
an y =
6 1 such that y ∈ {x}⊥ . And that we have:
L has a separated zero
iff
iff
iff
iff
r(x) 6= r(0), for any x 6= 0;
{x}⊥ 6= {0}⊥ , for any x 6= 0;
⊥
{x}
6= {1}, for any x 6= 0;
W
⊥
y6=1 y ∈ {x} , for any x 6= 0.
A lattice with zero is sectionally complemented if for any x, z ∈ L such
that x ≤ z there is an y ∈ L such that x ∨ y = z, and x ∧ y = 0.
(18) Let L be both finitely and uniquely atomistic. Then L is sectionally
complemented iff S has a separated zero. In fact, for any z 6= 1 and x ≤ z
the sectional complement of x is then x0 = sup(At(z) − At(x)).
By assumption, L is bounded; and by hypothesis it is a lattice in view
of (5). Hence clearly if L is sectionally complemented, then it is also joincomplemented. Thus L has a separated zero by (17). Conversely, let us
consider the cases z 6= 1 and z = 1 separately.
Supposing z 6= 1, take any x ≤ z. Then A0 = At(z) − At(x) is finite
by hypothesis, and x0 = supA0 exists. It is easily verified that x ∨ x0 = z
and x ∧ x0 = 0.
Supposing z = 1 observe that if either x = 1 or x = 0, then x0 = 0
and x0 = 1, respectively, are the sectional complements of x. So take an
x 6= 0, 1. Having a separated zero by hypothesis, L is join-complemented by
(17). Hence, with x 6= 0, there is at least one y 6= 1 such that x ∨ y = 1 = z.
Setting B 0 = At(y) − At(x), and y 0 = supB 0 , we get: x ∨ y 0 = supAt(x) ∨
supB 0 = sup(At(x) ∪ B 0 ) = sup(At(x) ∪ At(y)) = supAt(x) ∨ supAt(y) =
x ∨ y = 1. And x ∧ y 0 = supAt(x) ∧ supB 0 = 0, as At(x) ∩ B 0 = ∅ and
the conditions of (8) are satisfied (x 6= 1 by assumption, and y 0 6= 1 as
B 0 ⊂ At(y) and y 6= 1).
Clearly, in the latter case the sectional complement is not unique.
On Atomic Join-Semilattices
111
References
[1] G. Grätzer, General Lattice Theory, Berlin 1978.
[2] B. Wolniewicz, An Algebra of Subsets for Join-Semilattices, Bulletin of the Section of Logic 13:1 (1984).
[3] B. Wolniewicz, Entailments and Independence in Join-Semilattices,
Bulletin of the Section of Logic 18:1 (1989).
Institute of Philosophy
Warsaw University
Poland