Acta Polytechnica


Acta Polytechnica 53(3):314–316, 2013 © Czech Technical University in Prague, 2013
available online at http://ctn.cvut.cz/ap/

INTERVALS IN GENERALIZED EFFECT ALGEBRAS AND THEIR
SUB-GENERALIZED EFFECT ALGEBRAS

Zdenka Riečanová∗, Michal Zajac

Department of Mathematics, Faculty of Electrical Engineering and Information Technology STU, Ilkovičova 3,
SK-81219 Bratislava

∗ corresponding author: zdenka.riecanova@stuba.sk

Abstract. We consider subsets G of a generalized effect algebra E with 0 ∈ G and such that every
interval [0, q]G = [0, q]E ∩ G of G (q ∈ G, q 6= 0) is a sub-effect algebra of the effect algebra [0, q]E. We
give a condition on E and G under which every such G is a sub-generalized effect algebra of E.

Keywords: generalized effect algebra, effect algebra, Hilbert space, densely defined linear operators,
embedding, positive operators valued state.

1. Introduction and some basic
definitions and facts

The Hilbert space effect algebra E(H) on a Hilbert
space H is the set of positive operators dominated by
the identity operator I. In the quantum mechanical
framework the elements of an effect algebra represent
quantum effects and these are important for quantum
statistics and for quantum mechanical theory (see
[2, 3]).

One may think of quantum effects as elementary yes-
no measurements that may be unsharp or imprecise.
Effect algebras were introduced by D. Foulis and

M.K. Bennet in 1994 [1]. The prototype for the
abstract definition of an effect algebra was the set
E(H) (Hilbert space effects) of all selfadjoint opera-
tors between null and identity operators in a complex
Hilbert space H. If a quantum mechanical system is
represented in the usual way by a complex Hilbert
space H then self-adjoint operators from E(H) rep-
resent yes-no measurements that may be unsharp.
Recently several examples and properties of opera-
tor (generalized) effect algebras were studied in pa-
pers Polakovič, Riečanová [9], Polakovič [10], Paseka,
Riečanová [8], Riečanová, Zajac, Pulmannová [12],
Pulmannová, Riečanová, Zajac [11], Riečanová, Zajac
[13] and Riečanová [14]
The abstract definition of an effect algebra follows

the properties of the usual sum of operators in the
interval [0, I] (i.e. between null and identity operators
in H) and it is the following.

Definition 1.1 (Foulis, Bennet [1]). A partial al-
gebra (E;⊕, 0, 1) is called an effect algebra if 0,1 are
two distinguished elements and ⊕ is a partially de-
fined binary operation on E which satisfy the following
conditions for any x, y, z ∈ E:
(E1) x ⊕ y = y ⊕ x if x ⊕ y is defined,
(E2) (x⊕y)⊕z = x⊕(y ⊕z) if one side is defined,
(E3) for every x ∈ E there exists a unique y ∈ E

such that x ⊕ y = 1 (we put x′ = y),

(E4) if 1 ⊕ x is defined then x = 0.

Immediately in 1994 the study of generalizations of
effect algebras (without the top element 1) was started
by several authors (Foulis and Bennet [1], Kalmbach
and Riečanová [4], Hedlíková and Pulmannová [5],
Kôpka and Chovanec [6]). It was found out that
all these generalizations coincide and their common
definition is the following:

Definition 1.2. A generalized effect algebra (E;⊕, 0)
is a set E with element 0 ∈ E and partial binary
operation ⊕ satisfying for any x, y, z ∈ E conditions
(GE1) x ⊕ y = y ⊕ x if one side is defined,
(GE2) (x⊕y)⊕z = x⊕(y⊕z) if one side is defined,
(GE3) if x ⊕ y = x ⊕ z then y = z,
(GE4) if x ⊕ y = 0 then x = y = 0,
(GE5) x ⊕ 0 = x for all x ∈ E.

In every (generalized) effect algebra E a partial
order ≤ and a binary operation 	 can be introduced
as follows: for any a, b ∈ E, a ≤ b and b 	 a = c iff
a ⊕ c is defined and a ⊕ c = b.
Throughout the paper we assume that H is an

infinite-dimensional complex Hilbert space. For no-
tions and results on Hilbert space operators we refer
the reader to [7]. We will assume that the domains
D(A) of all considered linear operators A are dense
linear subspaces of H (in the metric topology induced
by the inner product). We say that operators A are
densely defined in H. The set of all densely defined
linear operators on H will be denoted by L(H).

Recall that A : D(A) →H is a bounded operator if
there exists a real constant C > 0 such that ‖Ax‖≤
C‖x‖ for all x ∈ D(A). If A is not bounded then it is
called unbounded.
Recall that if (E;⊕, 0, 1) is an effect algebra

((E;⊕, 0) is a generalized effect algebra) then a subset
G 6= ∅ such that 1 ∈ G (0 ∈ G respectively) is a
sub-effect algebra (sub-generalized effect algebra) of E
iff

314

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vol. 53 no. 3/2013 Intervals in Generalized Effect Algebras and their Sub-generalized Effect Algebras

(S) for any a, b, c ∈ E with a ⊕ b = c in E the fact
that two out of elements a, b, c are in G implies
that all a, b, c ∈ G.

Moreover, as we can easily check, every sub-
generalized effect algebra is a generalized effect algebra
in its own right.

2. Sub-generalized effect
algebras of generalized effect
algebras

A significant property of a generalized effect algebra
(E;⊕, 0) is the fact that for any q ∈ E, q 6= 0, the
interval [0, q]E = {c ∈ E | there exists d ∈ E such
that c ⊕ d = q} is an effect algebra with top element
q and the partial binary operation ⊕q defined for
a, b ∈ [0, q]E iff a ⊕ b ≤ q. Then we set a ⊕q b = a ⊕ b
(we write ⊕q = ⊕|[0,q]E ). Thus if a set G ⊆ E with
0 ∈ G is a sub-generalized effect algebra of E, then
the same is true for all [0, q]E ∩ G and [0, q]E, q ∈ E.
More precisely,

Theorem 2.1. Let E be a generalized effect algebra
and 0 ∈ G ⊆ E. Then the following assertions are
equivalent:
(1.) G is a sub-generalized effect algebra of E.
(2.) For all nonzero q ∈ E the set G ∩ [0, q]E is a
sub-generalized effect algebra of [0, q]E considered
as a generalized effect algebra.

Proof. (1.) ⇒ (2.) This implication is obvious.
(2.) ⇒ (1.) Let a, b, c ∈ E, a ⊕ b = c. Substituting

c = q into (2) we obtain that G ∩ [0, c]E is a sub-
generalized effect algebra of [0, c]E. Hence a, b, c ∈
[0, c]E satisfy the property (S), i.e., G is a sub-
generalized effect algebra of E.

The following example shows that the condition (2.)
in Theorem 2.1 cannot be replaced by a stronger one:
(2’.) For all nonzero q ∈ E the set G ∩ [0, q]E is a
sub-effect algebra of the effect algebra [0, q]E.

Example 2.2. Let E = R+ and G = Q+ be the
sets of all non-negative real and rational numbers,
respectively and let + denote the usual sum of real
numbers. Then (2) obviously holds but for nonrational
q > 0, e.g. for q =

√
2, we have G ∩ [0, q]E is not a

sub-effect algebra of [0, q]E.

It is easy to see that if G is a sub-generalized effect
algebra of a generalized effect algebra E, then for all
q ∈ G, q 6= 0, the intersection [0, q]G = [0, q]E ∩G is a
sub-generalized effect algebra of [0, q]E considered as a
generalized effect algebra. Our goal, roughly speaking,
is to investigate under what conditions the converse
holds.

Theorem 2.3. Let G be a subset of a generalized
effect algebra (E;⊕, 0) such that 0 ∈ E and for every
c ∈ E there exists g ∈ G with c ≤ g. Then the
following conditions are equivalent:

(1.) G is a sub-generalized effect algebra of (E;⊕, 0).
(2.) For any q ∈ G, q 6= 0 the interval [0, q]G =

[0, q]E ∩ G in G is a sub-effect algebra of the effect
algebra [0, q]E.

Proof. (1.) ⇒ (2.) This is obvious since (1.) implies
that G satisfies the condition (S), hence [0, q]E is
an effect algebra for every nonzero q ∈ G. Thus
the intersection of two generalized effect algebras
[0, q]E ∩ G = [0, q]G is an effect algebra, as q ∈ G.

(2.) ⇒ (1.) Let a, b ∈ G with a ⊕ b = c ∈ E.
There exists g ∈ G with c ≤ g and hence a ⊕ b ∈
[0, g]E ∩ G = [0, g]G which, by (2.), gives a ⊕ b ∈ G.
If a, c ∈ G, b ∈ E and a ⊕ b = c, then by (2.) we
have b ∈ G. This proves that G is a sub-generalized
effect algebra of E.

Example 2.4. Assume that H is an infinite-
dimensional complex Hilbert space. Further, let
B+(H) be the set of all bonded positive linear op-
erators with domain H. In [12] it was proved that for
any dense linear subspace D ⊆ H the set GD(H) =
B+(H) ∪{A : D →H| A ≥ 0, unbounded linear op-
erator with D(A) = D} is a generalized effect algebra
with the operation ⊕D which for any A, B ∈GD(H)
coincides with the usual sum of linear operators, i.e.
A ⊕D B = A + B.

It is easy to show that B+(H) is a sub-generalized
effect algebra of GD (H). For every Q ∈ B+(H), Q 6= 0,
the intervals under Q in B+(H) and GD (H) coincide,
i.e. [0, Q]B+(H) = [0, Q]GD (H) ∩ B

+(H) = [0, Q]GD (H)
and they also coincide as effect algebras. This shows
that conditions (1.) and (2.) from Theorem 2.3 hold.

Open Problem 2.5. Example 2.4 shows that, in
Theorem 2.3, the condition “to every q ∈ E there
exists c ∈ G with q ≤ c” is only sufficient but not nec-
essary for the equivalence of conditions (1.) and (2.).
Thus the open problem remains to find a necessary
and sufficient condition for the equivalence of (1.) and
(2.) in Theorem 2.3.

In fact, for every dense subspace D of H, the gen-
eralized effect algebra

GD (H) = B+(H) ∪U+D (H),

where U+D (H) is the set of all unbounded positive
linear operators with domain D and the null operator
0. Clearly, B+(H) ∩U+D (H) = {0}. On the other
hand, while B+(H) is a sub-generalized effect algebra
of GD(H), the same is not true for U+D (H). This
shows that the union of {0} with the difference of two
sub-generalized effect algebras of the generalized effect
algebra E need not be again a sub-generalized effect
algebra of E.

Example 2.6. Let U+D (H) =
(
GD (H)\B+(H)

)
∪{0}

be the set of all positive unbounded operators in H
with domain D and the null operator 0.

315



Z. Riečanová, M. Zajac Acta Polytechnica

Then U+D (H) is not a sub-generalized effect algebra
of GD (H) because for A ∈ B+(H) and U, V ∈U+D (H)
such that V = U + A we have A /∈ U+D (H). It fol-
lows that there are Q ∈ U+D (H), Q 6= 0, such that
[0, Q]U +

D
(H) = [0, Q]GD (H) ∩U

+
D (H) is not a sub-effect

algebra in [0, Q]GD (H).

Acknowledgements
This work was supported by grants VEGA 1/0297/11
and VEGA 1/0426/12 of the Ministry of Education of
the Slovak Republic and by grant APVV-0178-11 of the
Slovak Research and Development Agency.

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	Acta Polytechnica 53(3):314–316, 2013
	1 Introduction and some basic definitions and facts
	2 Sub-generalized effect algebras of generalized effect algebras
	Acknowledgements
	References