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On the Schröder-Bernstein problem for Carathéodory vector lattices

Ján Jakubík — 2009

Czechoslovak Mathematical Journal

In this note we prove that there exists a Carathéodory vector lattice V such that V V 3 and V V 2 . This yields that V is a solution of the Schröder-Bernstein problem for Carathéodory vector lattices. We also show that no Carathéodory Banach lattice is a solution of the Schröder-Bernstein problem.

Projectability and weak homogeneity of pseudo effect algebras

Ján Jakubík — 2009

Czechoslovak Mathematical Journal

In this paper we deal with a pseudo effect algebra 𝒜 possessing a certain interpolation property. According to a result of Dvurečenskij and Vettterlein, 𝒜 can be represented as an interval of a unital partially ordered group G . We prove that 𝒜 is projectable (strongly projectable) if and only if G is projectable (strongly projectable). An analogous result concerning weak homogeneity of 𝒜 and of G is shown to be valid.

On the distributive radical of an Archimedean lattice-ordered group

Ján Jakubík — 2009

Czechoslovak Mathematical Journal

Let G be an Archimedean -group. We denote by G d and R D ( G ) the divisible hull of G and the distributive radical of G , respectively. In the present note we prove the relation ( R D ( G ) ) d = R D ( G d ) . As an application, we show that if G is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.

Direct product decompositions of bounded commutative residuated -monoids

Ján Jakubík — 2008

Czechoslovak Mathematical Journal

The notion of bounded commutative residuated -monoid ( B C R -monoid, in short) generalizes both the notions of M V -algebra and of B L -algebra. Let A ̧ be a B C R -monoid; we denote by ( A ̧ ) the underlying lattice of A ̧ . In the present paper we show that each direct product decomposition of ( A ̧ ) determines a direct product decomposition of A ̧ . This yields that any two direct product decompositions of A ̧ have isomorphic refinements. We consider also the relations between direct product...

On some types of radical classes

Ján Jakubík — 2008

Czechoslovak Mathematical Journal

Let 𝔪 be an infinite cardinal. We denote by C 𝔪 the collection of all 𝔪 -representable Boolean algebras. Further, let C 𝔪 0 be the collection of all generalized Boolean algebras B such that for each b B , the interval [ 0 , b ] of B belongs to C 𝔪 . In this paper we prove that C 𝔪 0 is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized M V -algebras.

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