On algebraic operations of a lattice-ordered group
This paper contains a result of Cantor-Bernstein type concerning archimedean lattice ordered groups.
By dealing with absolute retracts of l-groups we use a definition analogous to that applied by Halmos for the case of Boolean algebras. The main results of the present paper concern absolute convex retracts in the class of all archimedean l-groups and in the class of all complete l-groups.
In this paper we investigate sufficient conditions for the validity of certain implications concerning direct products of lattice-ordered groups.
In this paper it is proved that the lattice of additive hereditary properties of finite graphs is completely distributive and that it does not satisfy the Jordan-Dedekind condition for infinite chains.
In this note we prove that there exists a Carathéodory vector lattice such that and . This yields that 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.
In this paper we deal with a pseudo effect algebra possessing a certain interpolation property. According to a result of Dvurečenskij and Vettterlein, can be represented as an interval of a unital partially ordered group . We prove that is projectable (strongly projectable) if and only if is projectable (strongly projectable). An analogous result concerning weak homogeneity of and of is shown to be valid.
Let be an Archimedean -group. We denote by and the divisible hull of and the distributive radical of , respectively. In the present note we prove the relation . As an application, we show that if is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.
The notion of bounded commutative residuated -monoid ( -monoid, in short) generalizes both the notions of -algebra and of -algebra. Let be a -monoid; we denote by the underlying lattice of . In the present paper we show that each direct product decomposition of determines a direct product decomposition of . This yields that any two direct product decompositions of have isomorphic refinements. We consider also the relations between direct product...
Let and be a nonzero abelian linearly ordered group or a nonzero abelian lattice ordered group, respectively. In this paper we prove that the wreath product of and fails to be affine complete.
Let be an infinite cardinal. We denote by the collection of all -representable Boolean algebras. Further, let be the collection of all generalized Boolean algebras such that for each , the interval of belongs to . In this paper we prove that is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized -algebras.
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