ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

Sharply Orthocomplete Effect Algebras

M. Kalina, J. Paseka, Z. Riečanová

Abstract

Special types of effect algebras E called sharply dominating and S-dominating were introduced by S. Gudder in [7, 8].
We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of E.
Namely we prove that in every sharply orthocomplete S-dominating effect algebra E the set of sharp elements and the
center of E are complete lattices bifull in E. If an Archimedean atomic lattice effect algebra E is sharply orthocomplete
then it is complete.

Keywords: effect algebra, sharp element, central element, block, sharply dominating, S-dominating, sharply orthocom-
plete.

1 Introduction
Analgebraic structure calledan effect algebrawas in-
troduced by D. J. Foulis and M. K. Bennett (1994).
The advantage of effect algebras is that they pro-
vide a mechanism for studying quantum effects, or
more generally, in non-classical probability theory
their elements represent events that may be unsharp
or pairwise non-compatible. Lattice effect algebras
are in some sense a nearest common generalization
of orthomodular lattices [13] that may include non-
compatible pairs of elements, and MV-algebras [3]
that may include unsharp elements. More precisely,
a lattice effect algebra E is an orthomodular lattice
iff every element of E is sharp (i.e., x and “non x”
are disjoint) and it is an MV-effect algebra iff every
pair of elements of E is compatible. Moreover, in ev-
ery lattice effect algebra E the set of sharp elements
is an orthomodular lattice ([10]), and E is a union of
its blocks (i.e., maximal subsets of pairwise compat-
ible elements that are MV-effect algebras (see [21])).
Thus a lattice effect algebra E is a Boolean algebra
iff every pair of elements is compatible and every el-
ement of E is sharp.
However, non-lattice ordered effect algebra E is

so general that its set S(E) of sharp elements may
form neither an orthomodular lattice nor any reg-
ular algebraic structure. S. Gudder (see [7, 8]) in-
troduced special types of effect algebras E called
sharply dominating effect algebras, whose set S(E)
of sharp elements forms an orthoalgebra and also so-
called S-dominating effect algebras, whose set S(E)
of sharp elements forms an orthomodular lattice.
In [7], S. Gudder showed that a standard Hilbert
space effect algebra E(H) of bounded operators on
a Hilbert space H between zero and identity opera-
tors (with partially defined usual operation+) is S-

dominating. Hence S-dominating effect algebrasmay
be useful abstractmodels for sets of quantum effects
in physical systems.
We study these two special kinds of effect alge-

bras. We show properties of some remarkable sub-
effect algebras of such effect algebras E satisfying the
condition that E is sharply orthocomplete. Namely
properties of their blocks, sets of sharp elements and
their centers. It is worth noting that it was proved
in [11] that there are evenArchimedean atomic MV-
effect algebras which are not sharply dominating,
hence they are not S-dominating.

2 Basic definitions and some
known facts

Definition 1 ([4]) A partial algebra (E;⊕,0,1) is
called an effect algebra if 0, 1 are two distinct ele-
ments and ⊕ is a partially defined binary operation
on E which satisfy the following conditions for any
x, y, z ∈ E:
(Ei) x ⊕ y = y ⊕ x if x ⊕ y is defined,
(Eii) (x ⊕ y)⊕ z = x ⊕(y ⊕ z) if one side is defined,
(Eiii) for every x ∈ E there exists a unique y ∈ E

such that x ⊕ y =1 (we put x′ = y),
(Eiv) if 1 ⊕ x is defined then x =0.
We often denote the effect algebra (E;⊕,0,1)

briefly by E. On every effect algebra E the par-
tial order ≤ and a partial binary operation ! can be
introduced as follows:

x ≤ y and y !x = z iff x⊕z is defined and x⊕z = y.
If E with the defined partial order is a lattice (a

complete lattice) then (E;⊕,0,1) is called a lattice
effect algebra (a complete lattice effect algebra).

This paper is a contribution to the Proceedings of the 6-th Microconference “Analytic and Algebraic Methods VI”.

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Acta Polytechnica Vol. 50 No. 5/2010

Definition 2 Let E be an effect algebra. Then Q ⊆
E is called a sub-effect algebra of E if

(i) 1 ∈ Q,
(ii) if out of elements x, y, z ∈ E with x ⊕ y = z
two are in Q, then x, y, z ∈ Q.

If E is a lattice effect algebra and Q is a sub-lattice
and a sub-effect algebra of E then Q is called a sub-
lattice effect algebra of E.

Note that a sub-effect algebra Q (sub-lattice ef-
fect algebra Q) of an effect algebra E (of a lattice
effect algebra E) with inherited operation ⊕ is an
effect algebra (lattice effect algebra) in its own right.
For an element x of an effect algebra E we write

ord(x) = ∞ if nx = x ⊕ x ⊕ . . . ⊕ x (n-times) exists
for everypositive integer n andwewrite ord(x)= nx
if nx is the greatest positive integer such that nxx
exists in E. An effect algebra E is Archimedean if
ord(x) < ∞ for all x ∈ E.
A minimal nonzero element of an effect algebra

E is called an atom and E is called atomic if under
every nonzero element of E there is an atom.
For a poset P and its subposet Q ⊆ P we denote,

for all X ⊆ Q, by
∨
Q

X the join of the subset X in

the poset Q whenever it exists.
We say that a finite system F = (xk)

n
k=1 of

not necessarily different elements of an effect alge-
bra (E;⊕,0,1) is orthogonal if x1 ⊕ x2 ⊕ . . . ⊕ xn

(written
n⊕

k=1

xk or
⊕

F) exists in E. Here we define

x1⊕x2⊕. . .⊕xn =(x1⊕x2⊕. . .⊕xn−1)⊕xn suppos-

ing that
n−1⊕
k=1

xk is defined and
n−1⊕
k=1

xk ≤ x′n. We also

define
⊕

" =0. An arbitrary system G =(xκ)κ∈H
of not necessarily different elements of E is called
orthogonal if

⊕
K exists for every finite K ⊆ G.

We say that for an orthogonal system G = (xκ)κ∈H
the element

⊕
G (more precisely

⊕
E

G) exists iff∨
{
⊕

K | K ⊆ G is finite} exists in E and then we
put

⊕
G =

∨
{
⊕

K | K ⊆ G is finite}. (Here
we write G1 ⊆ G iff there is H1 ⊆ H such that
G1 =(xκ)κ∈H1).
We call an effect algebra E orthocomplete [9] if

every orthogonal system G =(xκ)κ∈H of elements of

E has the sum
⊕

G. It is known that every ortho-
complete Archimedean lattice effect algebra E is a
complete lattice (see [22, Theorem 2.6]).
Recall that elements x, y of a lattice effect al-

gebra E are called compatible (written x ↔ y) iff
x ∨ y = x ⊕ (y ! (x ∧ y)) (see [15]). P ⊆ E is a
set of pairwise compatible elements if x ↔ y for all
x, y ∈ P . M ⊆ E is called a block of E iff M is a

maximal subset of pairwise compatible elements. Ev-
ery block of a lattice effect algebra E is a sub-effect
algebra and a sub-lattice of E and E is a union of
its blocks (see [21]). A lattice effect algebra with a
unique block is called an MV-effect algebra. Every
block of a lattice effect algebra is an MV-effect alge-
bra in its own right.
Anelement w ofaneffect algebra E is called sharp

(see [7, 8]) if w ∧ w′ =0.

Definition 3 ([7, 8]) An effect algebra E is called
sharply dominating if for every x ∈ E there exists
x̂ ∈ S(E) such that

x̂ =
∧
E

{w ∈ S(E) | x ≤ w} =
∧

S(E)

{w ∈ S(E) | x ≤ w}.

Note that clearly E is sharply dominating iff for
every x ∈ E there exists x̃ ∈ S(E) such that

x̃ =
∨
E

{w ∈ S(E) | x ≥ w} =
∨

S(E)

{w ∈ S(E) | x ≥ w}.

A sharply dominating effect algebra E is called
S-dominating [8] if x ∧ w exists for every x ∈ E,
w ∈ S(E).

It is awell known fact that in every S-dominating
effect algebra E the subset S(E)= {w ∈ E | w∧w′ =
0} of sharp elements of E is a sub-effect algebra of
E being an orthomodular lattice (see [8, Theorem

2.6]). Moreover if for D ⊆ S(E) the element
∨
E

D

exists then
∨
E

D ∈ S(E) hence
∨

S(E)

D =
∨
E

D. We

say that S(E) is a full sublattice of E (see [10]).
Let G be a sub-effect algebra of an effect algebra

E. We say that G is bifull in E, if, for any D ⊆ G
the element

∨
G

D exists iff the element
∨
E

D exists

and they are equal. Clearly, any bifull sub-effect al-
gebra of E is full but not conversely (see [12]).
The notion of a central element of an ef-

fect algebra E was introduced by Greechie-Foulis-
Pulmannová [6]. An element c ∈ E is called central
(see [18]) iff for every x ∈ E there exist x ∧ c and
x ∧ c′ and x =(x ∧ c) ∨ (x ∧ c′). The center C(E) of
E is the set of all central elements of E. Moreover,
C(E) is a Boolean algebra, see [6]. If E is a lattice
effect algebra then z ∈ E is central iff z ∧ z′ = 0
and z ↔ x for all x ∈ E, see [19]. Thus in a lat-
tice effect algebra E, C(E) = B(E) ∩ S(E), where
B(E) =

⋂
{M ⊆ E | M is a block of E} is called

the compatibility center of E.

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Acta Polytechnica Vol. 50 No. 5/2010

Aneffect algebra E is calledcentrally dominat-
ing (see also [5] for the notion central cover) if for
every x ∈ E there exists cx ∈ C(E) such that

cx =
∧
E

{c ∈ C(E) | x ≤ c} =
∧

C(E)

{c ∈ C(E) | x ≤ c}.

An element a of a lattice L is called compact iff,
for any D ⊆ L, a ≤

∨
D implies a ≤

∨
F for some

finite F ⊆ D. A lattice L is called compactly gen-
erated iff every element of L is a join of compact
elements.

3 Sharply orthocomplete
effect algebras

In an effect algebra E the set S(E) = {x ∈ E |
x ∧ x′ = 0} of sharp elements plays an important
role. In some sense we can say that an effect alge-
bra E is a “smeared set S(E)” of its sharp elements,
while unsharp effects are important in studies of un-
sharpmeasurements [4, 2]. S.Gudderproved(see [8])
that, in standard Hilbert space effect algebra E(H)
of bounded operators A on a Hilbert space H be-
tween null operator and identity operator, which are
endowed with usual + defined iff A + B is in E(H),
the set S(E(H)) of sharp elements forms an ortho-
modular lattice of projection operators on H. Fur-
ther in [8, Theorem 2.2] it was shown that in every
sharply dominating effect algebra the set S(E) is a
sub-effect algebra of E. Moreover, in [7, Theorem
2.6] it is proved that in every S-dominating effect al-
gebra E the set S(E) is an orthomodular lattice. We
are going to show that in this case S(E) is bifull in
E.
Theorem 1 Let E be an S-dominating effect alge-
bra. Then S(E) is bifull in E.

Proof. Let S ⊆ S(E).
(1) Assume that z =

∨
S(E)

S ∈ S(E) exists. Let

us show that z is the least upper bound of S in E.
Let y ∈ E be an upper bound of S. Then y ∧ z exists
and it is an upper bound of S as well. Hence, for
any s ∈ S, s ≤ y ∧ z. As E is sharply dominating,
there exists a greatest sharp element ỹ ∧ z ≤ y ∧ z
This yields that s ≤ ỹ ∧ z ≤ y ∧ z, for all s ∈ S,
ỹ ∧ z ∈ S(E). Hence z ≤ ỹ ∧ z ≤ y ∧ z ≤ z. Then
z = y ∧ z ≤ y i.e., z is really the least upper bound
of S in E.
(2) Conversely, let z =

∨
E

S ∈ E exist. Let

y ∈ S(E) be an upper bound of S in S(E). Then
y ∧ z exists and it is again an upper bound of S. As

in (1) we have that ỹ ∧ z is the greatest sharp ele-
ment under y ∧ z and hence s ≤ ỹ ∧ z ≤ y ∧ z ≤ z, for
all s ∈ S. This gives that z = ỹ ∧ z ∈ S(E). Thus
z =

∨
S(E)

S ∈ S(E).

Corollary 1 If E is a sharply dominating lattice ef-
fect algebra then S(E) is bifull in E.

Definition 4 An effect algebra E is called shar-
ply orthocomplete (centrally orthocomplete
(see [5])) if for any system (xκ)κ∈H of elements
of E such that there exists an orthogonal system
(wκ)κ∈H , wκ ∈ S(E) with xκ ≤ wκ, κ ∈ H (an or-
thogonal system (cκ)κ∈H , cκ ∈ C(E) with xκ ≤ cκ,
κ ∈ H) there exists⊕

{xκ | κ ∈ H} =∨
E

{
⊕
E

{xκ | κ ∈ F } | F ⊆ H, F finite}.

Theorem 2 Let E be a sharply orthocomplete S-
dominating effect algebra. Then
(i) S(E) is a complete orthomodular lattice bifull in

E.
(ii) C(E) is a complete Boolean algebra bifull in E.
(iii) E is centrally dominating and centrally ortho-

complete.
(iv) If C(E) is atomic then

∨
E

{p ∈ C(E) | p atom of

C(E)} =1.

Proof. (i): From [8, Theorem 2.6] we know that
S(E) is an orthomodular lattice and a sub-lattice ef-
fect algebra of E.
Let us show that S(E) is orthocomplete. Let

S ⊆ S(E), S orthogonal. Then for everyfinite F ⊆ S
we have that

⊕
E

F =
∨
E

F =
∨

S(E)

F ∈ S(E). More-

over, for any s ∈ S, s ≤ s. Since S(E) is bifull
in E by Theorem 1 and E is sharply orthocomplete
we have that

⊕
E

S =
∨
E

S =
∨

S(E)

S ∈ S(E) exists.

Since S(E) is an Archimedean lattice effect algebra
we have from [22, Theorem 2.6] that S(E) is com-
plete.
(ii): As C(E) = {x ∈ E | y = (y ∧ x) ∨ (y ∧ x′) for
every y ∈ E}, we obtain that 1 = x ∨ x′ for every
x ∈ C(E) and by the de Morgan Laws 0 = x ∧ x′
for every x ∈ C(E). Hence C(E) ⊆ S(E). It
follows by (i) that, for any Q ⊆ C(E), there ex-
ists

∨
S(E)

Q =
∨
E

Q ∈ C(E) because C(E) is full

in E, hence
∨

C(E)

Q =
∨
E

Q. By the de Morgan

Laws there exists
∧
E

Q = (
∨
E

Q′)′, where evidently

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Acta Polytechnica Vol. 50 No. 5/2010

Q′ = {q′ ∈ E | q ∈ Q} ⊆ C(E). Hence
∧
E

Q ∈ C(E)

which gives
∧

C(E)

Q =
∧
E

Q (see also [5]).

(iii): Let x ∈ E. Using (ii) let us put cx =
∧

C(E)

{c ∈

C(E) | x ≤ c} ∈ C(E). Since C(E) is bifull in E we
have that cx =

∧
E

{c ∈ C(E) | x ≤ c} (see again [5]).

Since C(E) ⊆ S(E) we immediately obtain that E is
centrally orthocomplete.
(iv): Since C(E) is an atomic Boolean algebra we

have
∨

C(E)

{p ∈ C(E) | p atom of C(E)} =1. As C(E)

is bifull in E, we have that
∨
E

{p ∈ C(E) | p atom of

C(E)} =
∨

C(E)

{p ∈ C(E) | p atom of C(E)} =1.

4 Sharply orthocomplete
lattice effect algebras

M. Kalina in [12] has shown that even in an
Archimedean atomic lattice effect algebra E with
atomic center C(E) the join of atoms of C(E) com-
puted in E need not be equal to 1. Next examples
and theorems showconnections between sharportho-
completeness, sharp dominancy and completeness of
an effect algebra E aswell as bifullness of S(E), C(E)
and atomic blocks in a lattice effect algebra E. It is
worth noting that if S(E) = {0,1} then evidently E
is S-dominating and sharply orthocomplete.
Example 1 Example of a compactly generated
sharply orthocomplete MV-effect algebra that is not
complete.
It is enough to take the Chang MV-effect alge-

bra E = {0, a,2a,3a, . . . ,(3a)′,(2a)′, a′,1} that isnot
Archimedean (hence it is not complete). It is com-
pactly generated (every x ∈ E is compact) and ob-
viously sharply orthocomplete (the center C(E) =
S(E) is trivial) and hence sharply dominating.

Example 2 Example of a sharply dominatingArchi-
medean atomic latticeMV-effect algebra E with com-
plete and bifull S(E) that is not sharply orthocom-
plete.
Let E =

∏
{{0n, an,1n} | n =1,2, . . .} and let

E0 = {(xn)∞n=1 ∈ E | xk = ak
for at most finitely many k ∈ {1,2, . . .}}.

Then E0 is a sub-lattice effect algebra of E (hence
it is an MV-effect algebra), evidently sharply domi-
nating and it is not sharply orthocomplete (since it
is not complete).

S(E0) =
∏

{{0n,1n} | n = 1,2, . . .} is a com-
plete Boolean algebra and S(E0)= C(E0) is a bifull
sub-lattice of E0.

Lemma 1 Let E be a sharply orthocomplete Archi-
medean atomic MV-effect algebra. Then E is com-
plete.

Proof. Let A ⊆ E be a set of all atoms of E.
Then 1 =

∨
E

{naa|a ∈ A} =
⊕
E

{naa|a ∈ A},

naa ∈ C(E) = S(E) are atoms of C(E) for all
a ∈ A. By [23, Theorem 3.1] we have that E
is isomorphic to a subdirect product of the family
{[0, naa] | a ∈ A}. The corresponding lattice effect
algebra embedding ϕ:E →

∏
{[0, naa] | a ∈ A} is

given by ϕ(x)= (x ∧ naa)a∈A.
Let us check that E is isomorphic to

∏
{[0, naa] |

a ∈ A}. It is enough to check that ϕ is onto. Let
(xa)a∈A ∈

∏
{[0, naa] | a ∈ A}. Then (na)a∈A is an

orthogonal system and xa = kaa ≤ naa ∈ S(E) for
all a ∈ A. Hence x =

⊕
E

{xa | a ∈ A} =
∨
E

{kaa |

a ∈ A} ∈ E exists. Evidently, ϕ(x) = (x∧naa)a∈A =
(kaa)a∈A =(xa)a∈A.

Example 3 Example of a sharply orthocomplete
Archimedean MV-effect algebra that is not complete.
If we omit in Lemma 1 the assumption of atom-

icity in E it is enough to take the MV-effect alge-
bra E = {f: [0,1] → [0,1] | f continuous function},
which is a sub-lattice effect algebra of a direct prod-
uct of copies of the standard MV-effect algebra of
real numbers [0,1] that is Archimedean, sharply or-
thocomplete (the center C(E) = S(E) = {0,1} is
trivial) and hence sharply dominating. Moreover, E
is not complete.

It iswell knownthatanArchimedean lattice effect
algebra E is complete if and only if every block of E
is complete (see [22, Theorem 2.7]). If moreover E is
atomic then E mayhaveatomicaswell asnon-atomic
blocks [1]. K.Mosná [16, Theorem8] has proved that

in this case E =
⋃

{M ⊆ E | M atomic block of E}.
Hence every non-atomic block of E is covered

by atomic blocks. Moreover, many properties of
Archimedean atomic lattice effect algebras as well as
their non-atomic blocksdepend onproperties of their
atomic blocks.
Namely, the center C(E), the compatibility cen-

ter B(E) and the set S(E) of sharp elements of
Archimedean atomic lattice effect algebras E can
be expressed by set-theoretical operations on their
atomic blocks. As follows, B(E) =

⋂
{M ⊆ E |

M atomic block of E}, S(E) =
⋃

{C(M) | M ⊆
E, M atomic block of E} and C(E) = B(E) ∩ S(E)
(see [16]).

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Acta Polytechnica Vol. 50 No. 5/2010

For instance, an Archimedean atomic lattice ef-
fect algebra E is sharply dominating iff every atomic
block of E is sharply dominating (see [11]). More-
over, we can prove the following:

Theorem 3 Let E be an Archimedean atomic lat-
tice effect algebra. Then the following conditions are
equivalent:
(i) E is complete.
(ii) Every atomic block of E is complete.
In this case every block of E is complete.

Proof. (i)=⇒ (ii): This is trivial, as every block M
of E is a full sub-lattice effect algebra of E.
(ii) =⇒ (i): It is enough to show that E is ortho-
complete. From [22, Theorem 2.6] we then get that
E is complete.
Let G ⊆ E be a

⊕
-orthogonal system. Then,

for every x ∈ G, there is a set Ax of atoms of E and
positive integers ka, a ∈ Ax such that x =

⊕
E

{kaa |

a ∈ Ax}. Moreover, for any F ⊆ G finite we have
that

⋃
{Ax | x ∈ F } is an orthogonal set of atoms.

Hence AG =
⋃

{Ax | x ∈ G} is an orthogonal set of
atoms of E and there is a maximal orthogonal set A
of atoms of E such that AG ⊆ A. Therefore there is
an atomic block M of E with A ⊆ M. By assump-
tion

⊕
M

G exists and
⊕
M

G =
⊕
E

G, as M is bifull in

E because E is Archimedean and atomic (see [17]).

Theorem 4 Let E be a sharply orthocomplete lat-
tice effect algebra. Then
(i) S(E) is a complete orthomodular lattice bifull
in E.

(ii) C(E) is a complete Boolean algebra bifull in E.
(iii) E is sharply dominating, centrally dominating

and S-dominating.
(iv) If moreover E is Archimedean and atomic then

E is a complete lattice effect algebra.

Proof. (i),(iii): Let S ⊆ S(E), S be orthogonal.
Then, for any s ∈ S, s ≤ s. Hence (since S(E)
is full in E)

⊕
E

S =
∨
E

S =
∨

S(E)

S ∈ S(E) exists.

Since S(E) is an Archimedean lattice effect algebra
we have from [22, Theorem 2.6] that S(E) is com-
plete. Moreover, let x ∈ E and let G = (wκ)κ∈H,
wκ ∈ S(E), κ ∈ H be a maximal orthogonal sys-
tem of mutually different elements such that wx =⊕
E

{wκ | κ ∈ H} ≤ x. Let us show that y ∈ S(E),

y ≤ x =⇒ y ≤ wx ∈ S(E). Clearly, wx ∈ S(E).
Assume that y 
≤ wx. Then wx < y ∨ wx ≤ x. Hence
z = (y ∨ wx) ! wx 
= 0 and G ∪ {z} is an orthogo-
nal system of mutually different elements such that

y ∨ wx = wx ⊕ z =
⊕
E

{wκ | κ ∈ H} ⊕ z ≤ x,

a contradiction with the maximality of G. There-
fore y ≤ wx and E is sharply dominating, hence S-
dominating and from Theorem 2 we get that E is
centrally dominating. From Theorem 1, we get that
S(E) is bifull in E.
(ii): It follows from (i),(iii) and Theorem 2.
(iv): Assume now that E is a sharply orthocomplete
Archimedean atomic lattice effect algebra. Then ev-
ery atomic block M of E is a sharply orthocomplete
Archimedean atomic MV-effect algebra and hence it
is a complete MV-effect algebra by Lemma 1. By
Theorem 3, E is a complete lattice effect algebra.

Theorem 5 Let E be an atomic lattice effect alge-
bra. Then the following conditions are equivalent:
(i) E is complete.
(ii) E is Archimedean and sharply orthocomplete.

Proof. (i) =⇒ (ii): By [20, Theorem 3.3] we
have that any complete lattice effect algebra is
Archimedean. Evidently, any complete lattice effect
algebra is sharply orthocomplete.
(ii)=⇒ (i): It follows from Theorem 4, (iv).

Acknowledgement

The work of the first author was supported by the
SlovakResearchandDevelopmentAgencyunder con-
tract No. APVV–0375–06 and by the VEGA grant
agency, grant number 1/0373/08. The second au-
thor gratefully acknowledges financial support from
the Ministry of Education of the CzechRepublic un-
der project MSM0021622409. The third author was
supported by the Slovak Research and Development
Agency under contract No. APVV–0071–06.
The authors also thank the referee for reading

very thoroughly and for improving the presentation
of the paper.

References

[1] Beltrametti, E. G., Cassinelli, G.: The Logic of
Quantum Mechanics. Addison-Wesley, Reading,
MA, 1981.

[2] Busch, P., Lahti, P. J., Mittelstaedt, P.: The
quantum theory of measurement, Lecture Notes
in Physics, New Series m: Monographs, Vol. 2,
Springer-Verlag, Berlin, 1991.

[3] Chang, C. C.: Algebraic analysis of many val-
ued logics, Trans. Amer. Math. Soc. 88 (1958),
467–490.

[4] Foulis, D. J., Bennett, M. K.: Effect algebras
and unsharp quantum logics, Found. Phys. 24
(1994), 1331–1352.

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[5] Foulis, D. J., Pulmannová, S.: Type-Decompo-
sition of an Effect Algebra, Found. Phys.

[6] Greechie, R. J., Foulis, D. J., Pulmannová, S.:
The center of an effect algebra,Order 12 (1995),
91–106.

[7] Gudder, S. P.: Sharply dominating effect alge-
bras, Tatra Mt. Math. Publ. 15 (1998), 23–30.

[8] Gudder, S. P.: S-dominating effect algebras, In-
ternat. J. Theoret. Phys. 37 (1998), 915–923.

[9] Jenča, G., Pulmannová, S.: Orthocomplete ef-
fect algebras, Proc. Amer. Math. Soc. 131
(2003), 2663–2671.

[10] Jenča, G., Riečanová, Z.: On sharp elements
in lattice ordered effect algebras, BUSEFAL 80
(1999), 24–29.

[11] Kalina, M., Olejček, V., Paseka, J., Riečano-
vá, Z.: Sharply Dominating MV-Effect Alge-
bras, Internat. J. Theoret. Phys.,
DOI: 10.1007/s10773-010-0338-x.

[12] Kalina, M.: On central atoms of Archimedean
atomic lattice effect algebras, Kybernetika, ac-
cepted.

[13] Kalmbach, G.: Orthomodular lattices, Mathe-
matics and its Applications, Vol. 453, Kluwer
Academic Publishers, Dordrecht, 1998, 1998.

[14] Kôpka, F.: Compatibility in D-posets, Internat.
J. Theoret. Phys. 34 (1995), 1525–1531.

[15] Kôpka, F., Chovanec, F.: Boolean D-posets, In-
ternat. J. Theoret. Phys.34 (1995), 1297–1302.

[16] Mosná, K.: Atomic lattice effect algebras and
their sub-lattice effect algebras,J. Electrical En-
gineering 58 (2007), 7/S, 3–6.

[17] Paseka, J., Riečanová, Z.: The inheritance of
BDE-property in sharply dominating lattice ef-
fect algebras and (o)-continuous states, Soft
Comput., DOI: 10.1007/s00500-010-0561-7.

[18] Riečanová, Z.: Compatibility and central ele-
ments in effect algebras, Tatra Mt. Math. Publ.
16 (1999), 151–158.

[19] Riečanová, Z.: Subalgebras, intervals and cen-
tral elements of generalized effect algebras, In-
ternat. J. Theoret. Phys.38 (1999), 3209–3220.

[20] Riečanová, Z.: Archimedean and block-finite
lattice effect algebras, Demonstratio Mathemat-
ica 33 (2000), 443–452.

[21] Riečanová, Z.: Generalization of blocks for D-
lattices and lattice-ordered effect algebras, In-
ternat. J. Theoret. Phys. 39 (2000), 231–237.

[22] Riečanová, Z.: Orthogonal sets in effect alge-
bras, Demonstratio Math. 34 (2001), 525–532.

[23] Riečanová, Z.: Subdirect decompositions of lat-
tice effect algebras, Internat. J. Theoret. Phys.
42 (2003), 1415–1433.

Doc. RNDr. Martin Kalina, Ph.D.
E-mail: kalina@math.sk
Department of Mathematics
Faculty of Civil Engineering
Slovak University of Technology
Radlinského 11, SK-813 68 Bratislava, Slovakia

Doc. RNDr. Jan Paseka, CSc.
E-mail: paseka@math.muni.cz
Department of Mathematics and Statistics
Faculty of Science
Masaryk University
Kotlářská 2, CZ-611 37 Brno, Czech Republic

Prof. RNDr. Zdenka Riečanová, Ph.D.
E-mail: zdena.riecanova@gmail.com
Department of Mathematics
Faculty of Electrical Engineering and Information
Technology
Slovak University of Technology
Ilkovičova 3, SK-812 19 Bratislava, Slovak Republic

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