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 $b\in B$, 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 $MV$-algebras.

In the present note we characterize finite lattices which are isomorphic to the congruence lattice of an abelian lattice ordered group.

We develop elementary methods of computing the monoid $𝒱\left(R\right)$ for a directly-finite regular ring $R$. We construct a class of directly finite non-cancellative refinement monoids and realize them by regular algebras over an arbitrary field.

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.

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.

In this paper we deal with the vector lattice $C\left(B\right)$ of all elementary Carathéodory functions corresponding to a generalized Boolean algebra $B$.

It is proved by an order theoretical and purely algebraic method that any order bounded orthosymmetric bilinear operator $b:E\times E\to F$ where $E$ and $F$ are Archimedean vector lattices is symmetric. This leads to a new and short proof of the commutativity of Archimedean almost $f$-algebras.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

We show that the Bruschlinsky group with the winding order is a homomorphism invariant for a class of one-dimensional inverse limit spaces. In particular we show that if a presentation of an inverse limit space satisfies the Simplicity Condition, then the Bruschlinsky group with the winding order of the inverse limit space is a dimension group and is a quotient of the dimension group with the standard order of the adjacency matrices associated with the presentation.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.

On a non-trivial partially ordered real vector space (V,≤) the orthogonality relation is defined by incomparability and ζ(V,⊥) is a complete lattice of double orthoclosed sets. We say that A ⊆ V is an orthogonal set when for all a,b ∈ A with a ≠ b, we have a ⊥ b. In our earlier papers we defined an integrally open ordered vector space and two closure operations A → D(A) and $A\to A^{\perp \perp }$. It was proved that V is integrally open iff $D\left(A\right)=A^{\perp \perp }$ for every orthogonal set A ⊆ V. In this paper we generalize this result. We...