Let $L_c\left(X\right)=\{f\in C\left(X\right):\overline{C_f}=X\}$, where $C_f$ is the union of all open subsets $U\subseteq X$ such that $\left|f\left(U\right)\right|\le {\aleph }_0$. In this paper, we present a pointfree topology version of $L_c\left(X\right)$, named ${ℛ}_{\ell c}\left(L\right)$. We observe that ${ℛ}_{\ell c}\left(L\right)$ enjoys most of the important properties shared by $ℛ\left(L\right)$ and ${ℛ}_c\left(L\right)$, where ${ℛ}_c\left(L\right)$ is the pointfree version of all continuous functions of $C\left(X\right)$ with countable image. The interrelation between $ℛ\left(L\right)$, ${ℛ}_{\ell c}\left(L\right)$, and ${ℛ}_c\left(L\right)$ is examined. We show that $L_c\left(X\right)\cong {ℛ}_{\ell c}\left(𝔒\left(X\right)\right)$ for any space $X$. Frames $L$ for which ${ℛ}_{\ell c}\left(L\right)=ℛ\left(L\right)$ are characterized.

In analogy with effect algebras, we introduce the test spaces and $MV$-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between $MV$-algebras and $MV$-test spaces.

If element $z$ of a lattice effect algebra $(E,\oplus ,\mathbf{0},\mathbf{1})$ is central, then the interval $[\mathbf{0},z]$ is a lattice effect algebra with the new top element $z$ and with inherited partial binary operation $\oplus$. It is a known fact that if the set $C\left(E\right)$ of central elements of $E$ is an atomic Boolean algebra and the supremum of all atoms of $C\left(E\right)$ in $E$ equals to the top element of $E$, then $E$ is isomorphic to a subdirect product of irreducible effect algebras ([18]). This means that if there exists a MacNeille completion $\widehat{E}$ of $E$ which is its extension...

Probability on collections of fuzzy sets can be developed as a generalization of the classical probability on $\sigma$-algebras of sets. A Łukasiewicz tribe is a collection of fuzzy sets which is closed under the standard fuzzy complementation and under the pointwise application of the Łukasiewicz t-norm to countably many fuzzy sets. An observable is a fuzzy set-valued mapping defined on a $\sigma$-algebra of sets and satisfying some additional properties; formally, the role of an observable is in a sense analogous...

In the present paper we investigate the relations between maximal completions of lattice ordered groups and maximal completions of pseudo $MV$-algebras.

In this paper we define maximal MV-algebras, a concept similar to the maximal rings and maximal distributive lattices. We prove that any maximal MV-algebra is semilocal, then we characterize a maximal MV-algebra as finite direct product of local maximal MV-algebras.

Coherent ideals, strongly coherent ideals, and $\tau$-closed ideals are introduced in pseudo-complemented distributive lattices and their characterization theorems are derived. A set of equivalent conditions is derived for every ideal of a pseudo-complemented distributive lattice to become a coherent ideal. The notion of median prime ideals is introduced and some equivalent conditions are derived for every maximal ideal of a pseudo-complemented distributive lattice to become a median prime ideal which...

This paper is an erratum of H. Mühle: Distributive lattices have the intersection property, Math. Bohem. (2021). Meet-distributive lattices form an intriguing class of lattices, because they are precisely the lattices obtainable from a closure operator with the so-called anti-exchange property. Moreover, meet-distributive lattices are join semidistributive. Therefore, they admit two natural secondary structures: the core label order is an alternative order on the lattice elements and the canonical...

A locallic version of Hager’s metric-fine spaces is presented. A general definition of $𝒜$-fineness is given and various special cases are considered, notably $𝒜=$ all metric frames, $𝒜=$ complete metric frames. Their interactions with each other, quotients, separability, completion and other topological properties are discussed.

The purpose of this paper is to study the topological properties of the interval topology on a completely distributive lattice. The main result is that a metrizable completely distributive lattice is an ANR if and only if it contains at most finite completely compact elements.