Acta Polytechnica Vol. 51 No. 4/2011

Two Remarks to Bifullness of Centers of Archimedean Atomic
Lattice Effect Algebras

M. Kalina

Abstract

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect
algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is
the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that
simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect
algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of
lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find
a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic
lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E)
is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular
lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be
a bifull sublattice of E.

Keywords: lattice effect algebra, orthomodular lattice, center, atom, bifullness.

1 Preliminaries
Effect algebras, introduced by D. J. Foulis and
M. K. Bennett [3], have their importance in the
investigation of uncertainty. Lattice ordered effect
algebras generalize orthomodular lattices and MV-
algebras. Thus they may include non-compatible
pairs of elements as well as unsharp elements.

Definition 1 (Foulis and Bennett [3]) An effect al-
gebra is a system (E; ⊕,0,1) consisting of a set E
with two different elements 0 and 1, called zero and
unit, respectively and ⊕ is a partially defined binary
operation satisfying the following conditions for all
p, q, r ∈ E:
(E1) If p ⊕ q is defined, then q ⊕ p is defined and

p ⊕ q = q ⊕ p.
(E2) If q ⊕ r is defined and p ⊕ (q ⊕ r) is defined,

then p ⊕ q and (p ⊕ q) ⊕ r are defined and
p ⊕ (q ⊕ r) = (p ⊕ q) ⊕ r.

(E3) For every p ∈ E there exists a unique q ∈ E
such that p ⊕ q is defined and p ⊕ q = 1.

(E4) If p ⊕1 is defined then p = 0.

The element q in (E3) will be called the supplement
of p, and will be denoted as p′.

In the whole paper, for an effect algebra
(E, ⊕,0,1), writing a ⊕ b for arbitrary a, b ∈ E will
mean that a ⊕ b exists. On an effect algebra E we
may define another partial binary operation � by

a � b = c ⇔ b ⊕ c = a.

The operation � induces a partial order on E.

Namely, for a, b ∈ E b ≤ a if there exists a c ∈ E
such that a � b = c. If E with respect to ≤ is lattice
ordered, we say that E is a lattice effect algebra. For
the sake of brevity we will write just LEA. Further,
in this article we often briefly write ‘an effect algebra
E’ skipping the operations.

S. P. Gudder ( [5, 6]) introduced the notion of
sharp elements and sharply dominating lattice effect
algebras. Recall that an element x of the LEA E is
called sharp if x ∧ x′ = 0. Jenča and Riečanová in [7]
proved that in every lattice effect algebra E the set
S(E) = {x ∈ E; x ∧ x′ = 0} of sharp elements is an
orthomodular lattice which is a sub-effect algebra of
E, meaning that if among x, y, z ∈ E with x ⊕ y = z
at least two elements are in S(E) then x, y, z ∈ S(E).
Moreover S(E) is a full sublattice of E, hence a supre-
mum of any set of sharp elements, which exists in E,
is again a sharp element. Further, each maximal sub-
set M of pairwise compatible elements of E, called
a block of E, is a sub-effect algebra and a full sub-
lattice of E and E =

⋃
{M ⊆ E; M is a block of E}

(see [16, 17]). Central elements and centers of effect
algebras were defined in [4]. In [14, 15] it was proved
that in every lattice effect algebra E the center

C(E) = {x ∈ E; (∀y ∈ E)y = (y ∧ x) ∨ (y ∧ x′)}
= S(E) ∩ B(E), (1)

where B(E) =
⋂

{M ⊆ E; M is a block of E}.
Since S(E) is an orthomodular lattice and B(E) is an

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Acta Polytechnica Vol. 51 No. 4/2011

MV-effect algebra, we obtain that C(E) is a Boolean
algebra. Note that E is an orthomodular lattice if
and only if E = S(E) and E is an MV-effect alge-
bra if and only if E = B(E). Thus E is a Boolean
algebra if and only if E = S(E) = B(E) = C(E).

Recall that an element p of an effect algebra E
is called an atom if and only if p is a minimal non-
zero element of E and E is atomic if for each x ∈ E,
x �= 0, there exists an atom p ≤ x.

Definition 2 Let (E, ⊕, 0) be an effect algebra. To
each a ∈ E we define its isotropic index, notation
ord(a), as the maximal positive integer n such that

na := a ⊕ . . . ⊕ a︸ ︷︷ ︸
n-times

exists. We set ord(a) = ∞ if na exists for each posi-
tive integer n. We say that E is Archimedean, if for
each a ∈ E, a �= 0, ord(a) is finite.

An element u ∈ E is called finite, if there exists
a finite system of atoms a1, . . . , an (which are not
necessarily distinct) such that u = a1 ⊕ . . . ⊕ an. An
element v ∈ E is called cofinite, if there exists a finite
element u ∈ E such that v = u′.

We say that for a finite system F = (xj )
k
j=1

of not necessarily different elements of an effect al-
gebra (E, ⊕,0,1) is ⊕-orthogonal if for all n ≤ k
x1 ⊕ x2⊕· · ·⊕ xn = (x1 ⊕ x2 ⊕· · ·⊕ xn−1) ⊕ xn exists

in E (briefly we will write
n⊕

j=1

xj ). We define also

⊕∅ = 0.

Definition 3 For a lattice (L, ∧, ∨) and a subset
D ⊆ L we say that D is a bifull sublattice of L, if
and only if for any X ⊆ D,

∨
L

X exists if and only if∨
D

X exists and
∧
L

X exists if and only if
∧
D

X exists,

in which case
∨
L

X =
∨
D

X and
∧
L

X =
∧
D

X.

It is known that if E is a distributive effect al-
gebra (i. e., the effect algebra E is a distributive
lattice — e. g., if E is an MV-effect algebra) then
C(E) = S(E). If moreover E is Archimedean and
atomic then the set of atoms of C(E) = S(E) is the
set {naa; a ∈ E is an atom of E}, where na = ord(a)
(see [20]). Since S(E) is a bifull sublattice of E if E
is an Archimedean atomic LEA (see [13]), we obtain
that

1 =
∨

C(E)

{p ∈ C(E); p is an atom of C(E)}

=
∨
E

{p ∈ C(E); p is an atom of C(E)}

for every Archimedean atomic distributive lattice ef-
fect algebra E. In [8] it was shown that there exists
an LEA E for which this property fails to be true.
Important properties of Archimedean atomic lattice
effect algebras with an atomic center were proven by
Riečanová in [21].

Theorem 1 (Riečanová [21]) Let E be an Archime-
dean atomic lattice effect algebra with an atomic cen-
ter C(E). Let AE be the set of all atoms of E and
AC(E) the set of all atoms of C(E). The following
conditions are equivalent:

1.
∨
E

AC(E) = 1.

2. For every atom a ∈ AE there exists an atom
pa ∈ AC(E) such that a ≤ pa.

3. For every z ∈ C(E) it holds

z =
∨

C(E)

{p ∈ AC(E); p ≤ z}

=
∨
E

{p ∈ AC(E); p ≤ z}.

4. C(E) is a bifull sub-lattice of E.
In this case E is isomorphic to a subdirect product
of Archimedean atomic irreducible lattice effect alge-
bras.

Theorem 2 (Paseka, Riečanová [13]) Let E be an
atomic Archimedean lattice effect algebra. Then the
set S(E) of all sharp elements of E is a bifull sublat-
tice of E.

We will deal only with atomic Archimedean lattice
effect algebras E. We have C(E) ⊂ S(E) ⊂ E. Be-
cause of this inclusion and Theorem 2, considering
the bifullness of the center C(E) in E is equivalent
to considering the bifullness of C(E) in S(E). And
S(E) is an orthomodular lattice. For this reason, in
the rest of the paper we will restrict our attention
to atomic orthomodular lattices L and their centers
C(L). For the sake of completeness, we give the def-
inition of an orthomodular lattice.

Definition 4 Let L be a bounded lattice with a
unary operation ′ (called complementation) satisfy-
ing the following conditions

1. for all a ∈ L (a′)′ = a,
2. for all a, b ∈ L if a ≤ b then b′ ≤ a′,
3. for all a, b ∈ L if a ≤ b then a ∨ (a′ ∧ b) = b.

Then L is said to be an orthomodular lattice (OML
for brevity).

Remark 1 Though in OML’s we have just lattice-
theoretical operations ∨ and ∧, we will use also ef-
fect algebraic operations ⊕ and � with the meaning
a ⊕ b = a ∨ b iff a ≤ b′ and a � b = c iff b ⊕ c = a.

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Acta Polytechnica Vol. 51 No. 4/2011

2 Orthomodular lattice L
whose center is not a bifull
sublattice

Let us have the following sequences of atoms (sets):

a0 = {(x, y) ∈ R2; 0 ≤ x ≤ 1, y ∈ R},
al = {(x, y) ∈ R2; l < x ≤ l + 1, y ∈ R},

for l = 1, 2, . . .,

b0 = {(x, y) ∈ R2; −1 ≤ x < 0, y ∈ R},
bl = {(x, y) ∈ R2; −l − 1 ≤ x < −l, y ∈ R},

for l = 1, 2, . . ., (2)

cj = {(x, y) ∈ R2; −j ≤ x ≤ j, y ≤ j · x},
for j = 1, 2, . . .,

dj = {(x, y) ∈ R2; −j ≤ x ≤ j, y > j · x},
for j = 1, 2, . . .,

pj = {j}, for j = 1, 2, . . ..

For such a choice of atoms, q1 �= q2 are compatible if
and only if q1∩q2 = ∅. Fig. 1 shows the compatibility
among atoms. For their non-compatibility (denoted
by �↔) the following rules hold

cj �↔ ai, cj �↔ bi for all j = 1, 2, . . .
and i = 0, . . . , j − 1,

dj �↔ ai, dj �↔ bi for all j = 1, 2, . . .
and i = 0, . . . , j − 1,

cj �↔ di for all i, j = 1, 2, . . .
such that i �= j,

cj �↔ ci, dj �↔ di for all i, j = 1, 2, . . .
such that i �= j.

� � �� � � � � � � � � � �

p1 p2 p3 p4 p5 p6 pn pn+1

� � �� � � � � � � � � � �
a0 b0 a1 b1 a2 b2 an bn

�
�

�
�

�
�

� � �

� � �

�
�

�

�

c1 c2 c3 cn−1

d1 d2 d3 dn−1

Fig. 1: Greechie diagram of sets of atoms

For non-compatible atoms the following equalities
hold

cj ⊕ dj =
j−1⊕
i=0

(ai ⊕ bi)

= ck ∨ cj = dk ∨ dj
= ck ∨ dj = dk ∨ cj
= cj ∨ al = cj ∨ bl
= dj ∨ al = dj ∨ bl

for 1 ≤ k < j and 0 ≤ l < j.

Denote B̂0, B̂j (for j = 1, 2, . . .) complete atomic
Boolean algebras with the corresponding sets of
atoms A0, Aj (j = 1, 2, . . .), given by

A0 =
∞⋃

i=0

{ai} ∪
∞⋃

i=0

{bi} ∪
∞⋃

j=1

{pj}, (3)

Aj =
∞⋃
i=j

{ai} ∪
∞⋃
i=j

{bi} ∪
∞⋃

j=1

{pj}

∪{cj , dj}. (4)

Disjointness occurring among some atoms of the sys-
tem (2) is equivalent to the fact that A0 and Aj
(j = 1, 2, . . .) are unique maximal sets of pairwise
compatible atoms.

Theorem 3 (Kalina [9]) Let L̂ =
∞⋃

i=0

B̂i. Let L1

be the complete OML generated by sets of atoms
∞⋃

i=0

{ai, bi} ∪
∞⋃

j=1

{cj , dj} and N the complete Boolean

algebra generated by the set of atoms
∞⋃

j=1

{pj}. Then

(L̂, ∨, ∧,0,1) is a complete OML and L̂ ∼= L1 ×N.

An element u ∈ B̂l is finite if and only if u =
q1⊕ q2⊕ . . .⊕ qn for an n ∈ N and q1, q2, . . . , qn ∈ Al.
Set Ql = {u ∈ Bl; u is finite}, l = 0, 1, 2, . . .. Then Ql
is a generalized Boolean algebra, since Bl = Ql ∪̇ Q∗l
is a Boolean algebra, where Q∗l = {u

∗; u∗ = 1l � u
and u ∈ Ql} (see [22], or [2, pp. 18-19]). This means
that Bl is a Boolean subalgebra of finite and cofinite
elements of B̂l (l = 0, 1, 2, . . .).

Theorem 4 (Kalina [8]) Denote L =
∞⋃
l=0

Bl. Then

(L, ∨, ∧,0,1) is a compactly generated orthomodular
lattice with the family (Bl)

∞
l=0 of atomic blocks of L.

The center of L, C(L), is not a bifull sublattice of L.

3 Completion of the center
of L

We are going to show that it is possible to extend the
orthomodular lattice L from Theorem 4 to L̄, whose
center, C(L̄), is a complete Boolean algebra which is
not a bifull sublattice of L̄.

Denote F a fixed non-trivial ultrafilter on N (the
index set of atoms pj ). Then F has the following
properties which will be important for our construc-
tion:
• Let F ⊂ N. Then either F ∈ F or N \ F ∈ F.
• Let F ⊂ N be a finite set. Then F /∈ F.
• If F1 ∈ F and F2 ∈ F then F1 ∩ F2 ∈ F.
• If F1 ∈ F and F2 ⊃ F1 then F2 ∈ F.

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Acta Polytechnica Vol. 51 No. 4/2011

Let QL1 denote the set of all finite elements of L1.
Further set

PF =

{⊕
i∈F

pi; F /∈ F
}

(5)

and

G = {f ⊕ g; g ∈ QL1, f ∈ PF},
G⊥ = {h′ ∈ L̂; h ∈ G}.

Theorem 5 Let L̃ = G ∪̇ G⊥. Then the system(
L̃, ∨, ∧,0,1

)
is an orthomodular lattice.

The center

C(L̃) = {f ∈ L̃; f ∈ PF or f ′ ∈ PF},

and C(L̃) is a complete Boolean algebra which is not
bifull in L̃.

Proof. First we show that L̃ is a bounded lattice.
Consider elements h1, h2 ∈ G. Then there exist el-
ements g1, g2 ∈ QL1 and elements f1, f2 ∈ PF such
that

h1 = f1 ⊕ g1, h2 = f2 ⊕ g2. (6)
By the properties of the non-trivial ultrafilter F we
get that f1 ∨ f2 ∈ PF and f1 ∧ f2 ∈ PF . Since g1, g2
are finite elements of L1, we get that g1 ∨ g2 ∈ QL1
and also g1 ∧ g2 ∈ QL1. Since L1 is generated by

the sets of atoms
∞⋃

i=0

{ai, bi} and
∞⋃

j=1

{cj, dj}, each

g ∈ QL1 is ⊕-orthogonal to each f ∈ PF . This im-
plies that G is closed under ∨ and ∧. Because G⊥
consists of complements of elements of G, we have
that also G⊥ is closed under ∨ and ∧. Now assume
that h1 ∈ G and h2 ∈ G⊥. Then h′2 ∈ G and we can
write

h1 = f1 ⊕ g1, h′2 = f2 ⊕ g2
with the same meaning of f1, f2, g1, g2 as in formula
(6). This means that h2 = f

′
2 � g2. Then, be-

cause of the monotonicity of the ultrafilter F, we have
(f1 ∨ f ′2)

′ ∈ PF and hence f1 ∨ f ′2 ∈ G
⊥. Moreover,

g2 ∈ QL1 is orthogonal to f1 which implies

(f1 ∨ f ′2) � g2 = f1 ∨ (f
′
2 � g2) ∈ G

⊥.

Since G is a monotone system (meaning that with an
arbitrary element δ1 ∈ G it contains also all elements
δ2 ∈ L̂ such that δ2 ≤ δ1), we get from the duality
between G and G⊥ that

(f1 ∨ g1) ∨ (f ′2 � g2) = h1 ∨ h2 ∈ G
⊥

Dually we get that h1 ∧ h2 ∈ G. This implies that
L̃ = G ∪̇ G⊥ is a lattice. Obviously it is a bounded
and orthocomplemented lattice. Showing that it is
an OML is a matter of routine. We will omit the
detailed proof.

Let us consider an element f ∈ L̃ such that
f ∈ PF or f ′ ∈ PF . Then f is a central element.
If f is such that neither f ∈ PF nor f ′ ∈ PF , then

there exist atoms α1, α2 ∈
∞⋃

i=0

{ai, bi} ∪
∞⋃

j=1

{cj , dj}

fulfilling α1 �↔ α2 and α1 ≤ f , α2 �≤ f . Then f is
not a central element. This proves that

C(L̃) = {f ∈ L̃; f ∈ PF or f ′ ∈ PF}.

Due to the fact that F is a non-trivial ultrafilter,
C(L̃) is a complete Boolean algebra.

The only central element that is greater than all
atoms pj for j = 1, 2, . . ., is 1, hence we have that∨

C(L̃)

{pj ; j = 1, 2, . . .} = 1.

On the other hand, let us take an arbitrary atom α ∈
∞⋃

i=0

{ai, bi} ∪
∞⋃

j=1

{cj , dj} and assume that
∨
L̃

{pj ; j =

1, 2, . . .} does exist. Since α is orthogonal to all atoms

from the set
∞⋃

j=1

{pj}, we have that α is orthogonal to∨
L̃

{pj ; j = 1, 2, . . .} and hence

∨
L̃

{pj ; j = 1, 2, . . .} �= 1.

It can be shown (see [8]) that
∨
L̃

{pj ; j = 1, 2, . . .}

does not exist. This implies that C(L̃) is not a bifull
sublattice of L̃. �

4 σ-complete orthomodular
lattice L̃σ whose center is
not a bifull sublattice

Let I denote the set of all ordinal numbers less than
Ω (the first uncountable ordinal number). Further,
denote E the set of all limit ordinal numbers up to Ω
and J = I \ E.

Assume sets of elements {pi; i ∈ I}, {ai; i ∈ I},
{bi; i ∈ I}, {ci; i ∈ I}, {di; i ∈ I}, where the cor-
responding elements for i ∈ J will act as atoms.
We will have a partial relation �↔ modelling non-
compatibility. This partial relation will have the fol-
lowing form among atoms

cj �↔ ai, cj �↔ bi for all j ∈ J and i ≤ j,
dj �↔ ai, dj �↔ bi for all j ∈ J and i ≤ j,

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Acta Polytechnica Vol. 51 No. 4/2011

cj �↔ di for all i, j ∈ J
such that i �= j,

cj �↔ ci, dj �↔ di for all i, j ∈ J
such that i �= j.

Sets of elements {pi; i ∈ I}, {ai; i ∈ I}, {bi; i ∈ I},
{ci; i ∈ I}, {di; i ∈ I} will present atoms for i ∈ J
and for κ ∈ E we will have

pκ =
∨
i<κ

pi, (7)

aκ =
∨
i<κ

ai, (8)

bκ =
∨
i<κ

bi, (9)

cκ =
∨
i<κ

ci =
∨
i<κ

di = dκ = aκ ⊕ bκ. (10)

As a possible model for the just presented sets of
elements fulfilling the non-compatibility relation we
may have the following:
Let us choose a good order of positive real numbers of
type Ω, i.e., positive real numbers will be enumerated
by ordinal numbers from J . For i ∈ J and r > 0,
r ∈ R, we denote ri the i-th number in the chosen
good order. Then we identify the set {pi; i ∈ J }
with the set of all positive real numbers, i.e., pi = ri.
Further we put for i, j ∈ J

ai = {(ri, y) ∈ R2; y ∈ R},
bi = {(−ri, y) ∈ R2; y ∈ R},
ci = {(rj , y) ∈ R2; j ≤ i, y ≤ ri · rj}

∪ {(−rj , y) ∈ R2; j ≤ i, y ≤ −ri · rj},
di = {(rj , y) ∈ R2; j ≤ i, y > ri · rj}

∪ {(−rj , y) ∈ R2; j ≤ i, y > −ri · rj}.

For κ ∈ E we define the corresponding elements
pκ, aκ, bκ, cκ, dκ by equalities 7, 8, 9, 10, respectively.
Compatibility among different atoms is given by dis-
jointness of the corresponding sets. This implies that
the uniquely given maximal sets of pairwise compat-
ible atoms are

Ã0 =
⋃
i∈J

{ai, bi, pi},

Ãj =
⋃

i ∈ J
i > j

{ai, bi} ∪
⋃
i∈J

{pi} ∪ {cj, dj}

for j ∈ J . Sets of atoms Ã0 and Ãj for j ∈ J , gener-
ate complete Boolean algebras B̃0 and B̃j for j ∈ J ,
respectively. For κ ∈ E we get complete atomic

Boolean algebras B̃κ generated by sets of atoms

Ãκ =
⋃
i∈J

{pi} ∪ {aκ, bκ} ∪
⋃

i ∈ J
i > κ

{ai, bi}.

This means that for κ ∈ E B̃κ ⊂ B̃0. The union of all
complete atomic Boolean algebras, L̃ = B̃0 ∪

⋃
i∈I

B̃i,

is a complete OML. An element f ∈ L̃ will be called
countable if there exists an at most countable set
of atoms (an at most countable set of indices K)
{qk}k∈K ⊂ Ã0 or {qk}k∈K ⊂ Ãi for i ∈ J , such
that

f =
⊕
k∈K

qk.

By definition of elements pi, ai, bi, ci, di for i ∈ I we
get that each of these elements is countable.

Let K denote the set of all countable elements of
L̃ and K⊥ = {f ∈ L̃; f ′ ∈ K}. Further, let P de-
note the set of all countable elements generated by
{pi, i ∈ J }, and P⊥ = {f ∈ L̃; f ′ ∈ P}.
Theorem 6 Let L̃σ = K∪̇K⊥. Then

(
L̃σ, ∨, ∧,0,1

)
is a σ-complete OML. The center C(L̃σ ) = P∪̇P⊥
and it is not a bifull sublattice of L̃σ.

Proof. Each of the atoms pi, ai, bi, ci, di for i ∈ J
(and hence also each of the elements pi, ai, bi, ci, di
for i ∈ I) is countable. This implies that L̃σ is an
OML. Since it is by definition closed under countable
meets and joins, it is σ-complete.

Elements pi for i ∈ I are central because each of
the elements pi is compatible with all atoms of L̃σ.
This implies that P∪̇P⊥ ⊂ C(L̃σ ). On the other
hand, let f be a countable element, f /∈ P. Then
there exists ci such that ci �≤ f for i ∈ J and an
atom out of e ∈ {aj, bj , cj, dj} for j < i, e ≤ f . Then
ci �↔ e and hence cj �↔ f . Similarly, if f ∈ K⊥, there
exists ci ≤ f and an atom out of e ∈ {aj, bj , cj, dj}
for j < i such that e �≤ f . In this case e �↔ ci and
hence also e �↔ f . We conclude that C(L̃σ ) = P∪̇P⊥.

We show that C(L̃σ ) is not a bifull sublattice of
L̃σ. Obviously ∨

C(L̃σ)

{pi, i ∈ I} = 1.

Assume that
∨
L̃σ

{pi, i ∈ I} does exist. Then all el-

ements e ∈
⋃
i∈I

{ai, bi, ci, di} are orthogonal with all

elements from the set
⋃
I
{pj} and consequently also

with
∨
L̃σ

{pi, i ∈ I}. This implies
∨
L̃σ

{pi, i ∈ I} �= 1.

This means that C(L̃σ ) is not a bifull sublattice of
L̃σ. �

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Acta Polytechnica Vol. 51 No. 4/2011

Acknowledgement

Support from the Science and Technology Assistance
Agency under contract No. APVV-0073-10, and from
the VEGA grant agency, grant number 1/0297/11, is
gratefully acknowledged.

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Martin Kalina
E-mail: kalina@math.sk
Dept. of Mathematics
Faculty of Civil Engineering
Slovak Univ. of Technology
Radlinského 11, Sk-813 68 Bratislava, Slovakia

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