In this paper we investigate convergence structures on a generalized Boolean algebra and their relations to convergence structures on abelian lattice ordered groups.
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 was assumed). Below we work with the definition from [13]; but, in order to avoid terminological problems we use the term “state-homomorphism” (instead of “state”). The author is indebted to the referee for his suggestion concerning terminology. Let $\mathcal A$ be an $MV$-algebra which is defined on a set $A$ with $\mathop {\mathrm card}A>1$. In the present paper we show that there exists a one-to-one correspondence between the system of all state-homomorphisms on $\mathcal A$ and the system of all $\sigma $-closed maximal ideals of $\mathcal A$. For $MV$-algebras we apply the notation and the definitions as in Gluschankof [3]. The relations between $MV$-algebras and abelian lattice ordered groups (cf. Mundici [8]) are substantially used in the present paper.
In this paper we prove that the lateral completion of a projectable lattice ordered group is strongly projectable. Further, we deal with some properties of Specker lattice ordered groups which are related to lateral completeness and strong projectability.
The extension of a lattice ordered group $A$ by a generalized Boolean algebra $B$ will be denoted by $A_B$. In this paper we apply subdirect decompositions of $A_B$ for dealing with a question proposed by Conrad and Darnel. Further, in the case when $A$ is linearly ordered we investigate (i) the completely subdirect decompositions of $A_B$ and those of $B$, and (ii) the values of elements of $A_B$ and the radical $R(A_B)$.
In this paper we investigate the relations between torsion classes of Specker lattice ordered groups and torsion classes of generalized Boolean algebras.
In this paper we prove a theorem on weak homogeneity of $MV$-algebras which generalizes a known result on weak homogeneity of Boolean algebras. Further, we consider a homogeneity condition for $MV$-algebras which is defined by means of an increasing cardinal property.
In this paper we deal with weakly homogeneous direct factors of lattice ordered groups. The main result concerns the case when the lattice ordered groups under consideration are archimedean, projectable and conditionally orthogonally complete.
In this paper we prove that the collection of all weakly distributive lattice ordered groups is a radical class and that it fails to be a torsion class.