In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation $𝒟$ on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain...

We give some necessary and sufficient conditions for transitive $l$-permutation groups to be $2$-transitive. We also discuss primitive components and give necessary and sufficient conditions for transitive $l$-permutation groups to be normal-valued.

Let $A$ and $B$ be two Archimedean vector lattices and let ${\left(A^{\text{'}}\right)}_n^{\text{'}}$ and ${\left(B^{\text{'}}\right)}_n^{\text{'}}$ be their order continuous order biduals. If $\Psi :A\times A\to B$ is a positive orthosymmetric bimorphism, then the triadjoint ${\Psi }^{***}:{\left(A^{\text{'}}\right)}_n^{\text{'}}\times {\left(A^{\text{'}}\right)}_n^{\text{'}}\to {\left(B^{\text{'}}\right)}_n^{\text{'}}$ of $\Psi$ is inevitably orthosymmetric. This leads to a new and short proof of the commutativity of almost $f$-algebras.

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{b_n\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda }_n\right\}$ of positive reals, and $b\in B$, with ${\lambda }_n{b}_n\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma$” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma$ iff the cardinal $\left|X\right|<𝔟$, Rothberger’s bounding number. Consequences and...

It is shown that any set-open topology on the automorphism group A(X) of a chain X coincides with the pointwise topology and that A(X) is a topological group with respect to this topology. Topological properties of connectedness and compactness in A(X) are investigated. In particular, it is shown that the automorphism group of a doubly homogeneous chain is generated by any neighborhood of the identity element.

For an order embedding $G\stackrel{h}{\to }\to \Gamma$ of a partly ordered group $G$ into an $l$-group $\Gamma$ a topology ${𝒯}_{\widehat{W}}$ is introduced on $\Gamma$ which is defined by a family of valuations $W$ on $G$. Some density properties of sets $h\left(G\right)$, $h\left(X_t\right)$ and $(h\left(X_t\right)\setminus \{h\left(g_1\right),\cdots ,h\left(g_n\right)\})$ ($X_t$ being $t$-ideals in $G$) in the topological space $(\Gamma ,{𝒯}_{\widehat{W}})$ are then investigated, each of them being equivalent to the statement that $h$ is a strong theory of quasi-divisors.