On algebraic operations of a lattice-ordered group
Projectable kernel of a lattice ordered group
Convex isomorphisms of Archimedean lattice ordered groups.
This paper contains a result of Cantor-Bernstein type concerning archimedean lattice ordered groups.
Projectability and weak homogeneity of pseudo effect algebras
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.
On the distributive radical of an Archimedean lattice-ordered group
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.
Direct product decompositions of bounded commutative residuated -monoids
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...
On the Schröder-Bernstein problem for Carathéodory vector lattices
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.
On absolute retracts and absolute convex retracts in some classes of l-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.
Isomorphisms of direct products of lattice-ordered groups
In this paper we investigate sufficient conditions for the validity of certain implications concerning direct products of lattice-ordered groups.
On the lattice of additive hereditary properties of finite graphs
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.
Distinguished completion of a direct product of lattice ordered groups
The distinguished completion of a lattice ordered group was investigated by Ball [1], [2], [3]. An analogous notion for -algebras was dealt with by the author [7]. In the present paper we prove that if a lattice ordered group is a direct product of lattice ordered groups , then is a direct product of the lattice ordered groups . From this we obtain a generalization of a result of Ball [3].
On free -algebras
In the present paper we show that free -algebras can be constructed by applying free abelian lattice ordered groups.
Graph automorphisms of a finite modular lattice
Complete generators and maximal completions of -algebras
On half cyclically ordered groups
In this paper we introduce and investigate the notion of half cyclically ordered group generalizing the notion of half partially ordered group whose study was begun by Giraudet and Lucas.
Lateral and Dedekind completions of strongly projectable lattice ordered groups
State-homomorphisms on -algebras
Riečan [12] and Chovanec [1] investigated states in -algebras. Earlier, Riečan [11] had dealt with analogous ideas in -posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on -algebras. We remark that a different definition of a state in an -algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity...
On archimedean -algebras
On monotone permutations of -cyclically ordered sets
For an -cyclically ordered set with the -cyclic order let be the set of all monotone permutations on . We define a ternary relation on the set . Further, we define in a natural way a group operation (denoted by ) on . We prove that if the -cyclic order is complete and , then is a half cyclically ordered group.
On orthogonally -complete lattice ordered groups
In this paper we prove a theorem of Cantor-Bernstein type for orthogonally -complete lattice ordered groups.
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