Algebra Univers. 78 (2017) 67–91
Published online May 23, 2017
© Springer International Publishing 2017
When the lexicographic product of two po-groups has
the Riesz decomposition property
censkij and Omid Zahiri
Abstract. We study conditions when a certain type of the Riesz Decomposition
Property (RDP for short) holds in the lexicographic product of two po-groups. Deﬁn-
ing two important properties of po-groups, we extend known situations showing that
the lexicographic product satisﬁes RDP or even RDP
, a stronger type of RDP. We
recall that a very strong type of RDP, RDP
, entails that the group is lattice ordered.
RDP’s of the lexicographic products are important for the study of lexicographic
pseudo eﬀect algebras, or perfect types of pseudo MV-algebras and pseudo eﬀect al-
gebras, where inﬁnitesimal elements play an important role both for algebras as well
as for the ﬁrst order logic of valid but not provable formulas.
In the last decades, we observe that there is a growing interest in the study
of some algebraic structures using lattice ordered groups or po-groups both for
Abelian and non-Abelian groups. A prototypical situation is due to Mundici,
see [22, 1], where any MV-algebra is represented as an interval in a unital
Abelian -group. This result was extended in , where it was proved that
pseudo MV-algebras, a non-commutative generalization of MV-algebras, see
[18, 23], can be represented by intervals in unital -groups that are not neces-
For mathematical foundations of quantum mechanics, Foulis and Bennett
introduced in  eﬀect algebras, which are partial algebras with a partially
deﬁned operation +, where a + b means disjunction of two mutually excluded
events a and b. These eﬀect algebras are in many cases also intervals in Abelian
po-groups (= partially ordered groups). A suﬃcient condition for such a po-
group representation is the Riesz Decomposition property, RDP, of the eﬀect
algebra and of the po-group, as follows from . RDP means roughly speak-
ing a possibility to perform a joint reﬁnement of any two decompositions of
the same element, and po-groups with RDP are intensively studied in the
literature, see e.g., [17, 20]. Recently, eﬀect algebras have been extended to
non-commutative algebras, called pseudo eﬀect algebras in [13, 14]. Also, if
such a pseudo eﬀect algebra satisﬁes a stronger form of RDP, namely RDP
Presented by S. Pulmannova.
Received April 26, 2016; accepted in ﬁnal form November 13, 2016.
2010 Mathematics Subject Classiﬁcation: Primary: 06D35; Secondary: 03G12.
Key words and phrases: po-group, lexicographic product, unital po-group, antilattice
po-group, Riesz Decomposition Property, pseudo eﬀect algebra.
This work was supported by the grant VEGA No. 2/0069/16 SAV, and GA