### On algebraic operations of a lattice-ordered group

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This paper contains a result of Cantor-Bernstein type concerning archimedean lattice ordered groups.

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

Let $G$ be an Archimedean $\ell $-group. We denote by ${G}^{d}$ and ${R}_{D}\left(G\right)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation ${\left({R}_{D}\left(G\right)\right)}^{d}={R}_{D}\left({G}^{d}\right)$. 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.

The notion of bounded commutative residuated $\ell $-monoid ($BCR$ $\ell $-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $A\u0327$ be a $BCR$ $\ell $-monoid; we denote by $\ell \left(A\u0327\right)$ the underlying lattice of $A\u0327$. In the present paper we show that each direct product decomposition of $\ell \left(A\u0327\right)$ determines a direct product decomposition of $A\u0327$. This yields that any two direct product decompositions of $A\u0327$ have isomorphic refinements. We consider also the relations between direct product...

In this note we prove that there exists a Carathéodory vector lattice $V$ such that $V\cong {V}^{3}$ and $V\ncong {V}^{2}$. 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.

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.

The distinguished completion $E\left(G\right)$ of a lattice ordered group $G$ was investigated by Ball [1], [2], [3]. An analogous notion for $MV$-algebras was dealt with by the author [7]. In the present paper we prove that if a lattice ordered group $G$ is a direct product of lattice ordered groups ${G}_{i}$ $(i\in I)$, then $E\left(G\right)$ is a direct product of the lattice ordered groups $E\left({G}_{i}\right)$. From this we obtain a generalization of a result of Ball [3].

In the present paper we show that free $MV$-algebras can be constructed by applying free abelian lattice 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.

Riečan [12] and Chovanec [1] investigated states in $MV$-algebras. Earlier, Riečan [11] had dealt with analogous ideas in $D$-posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on $MV$-algebras. We remark that a different definition of a state in an $MV$-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...

For an $\ell $-cyclically ordered set $M$ with the $\ell $-cyclic order $C$ let $P\left(M\right)$ be the set of all monotone permutations on $M$. We define a ternary relation $\overline{C}$ on the set $P\left(M\right)$. Further, we define in a natural way a group operation (denoted by $\xb7$) on $P\left(M\right)$. We prove that if the $\ell $-cyclic order $C$ is complete and $\overline{C}\ne \varnothing $, then $(P\left(M\right),\xb7,\overline{C})$ is a half cyclically ordered group.

In this paper we prove a theorem of Cantor-Bernstein type for orthogonally $\sigma $-complete lattice ordered groups.

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