The aim of this paper is to define an abstract quantale framework for extending some properties of the zip rings (studied by Faith, Zelmanowitz, etc.) and the weak zip rings (defined by Ouyang). By taking as prototype the quantale of ideals of a zip ring (resp. a weak zip ring) we introduce the notion of zipped quantale (resp. weakly zipped quantale). The zipped quantales also generalize the zipped frames, defined by Dube and Blose in a recent paper. We define the zip (bounded distributive) lattices and we prove that a coherent quantale A is weakly zipped iff the reticulation L(A) of A is a zip lattice. From this result we obtain the following corollary: the coherent quantale A is weakly zipped iff the frame R(A) of the radical elements of A is zipped. Such theorems allow us to extend to quantale framework a lot of results obtained by Dube and Blose for the zipped frames and for the weak zip rings.

The commutator theory, developed by Fresee and McKenzie in the framework of a congruence-modular variety $\mathcal{V}$, allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993).The reticulation of an algebra $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (endowed with the topology defined by Agliano). In a recent paper, C. Mure\c{s}an and the author defined the reticulation for the algebras $A$ in a semidegenerate congruence-modular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ is closed under commutators. This theory does not cover the Belluce reticulation for non-commutative rings. In this paper we shall introduce the quasi-commutative algebras in a semidegenerate congruence-modular variety $\mathcal{V}$ as a generalization of the Belluce quasi-commutative rings. We define and study a notion of reticulation for the quasi-commutative algebras such that the Belluce reticulation for the quasi-commutative rings can be obtained as a particular case. We prove a characterization theorem for the quasi-commutative algebras and some transfer properties by means of the reticulation.

Based on the notion of Q-sup-lattices (a fuzzy counterpart of complete join-semilattices valuated in a commutative quantale), we present the concept of Q-sup-algebras–Q-sup-lattices endowed with a collection of fini-tary operations compatible with the fuzzy joins. Similarly to the crisp case investigated in [30], we characterize their subalgebras and quotients, and following [20], we show that the category of Q-sup-algebras is isomorphic to a certain subcategory of a category of Q-modules.

We introduce a quantale-valued generalization of approach spaces in terms of quantale-valued gauges. The resulting category is shown to be topological and to possess an initially dense object. Moreover we show that the category of quantale-valued approach spaces defined recently in terms of quantale-valued closures is a coreflective subcategory of our category and, for certain choices of the quantale, is even isomorphic to our category. Finally, the category of quantale-valued metric spaces is shown to be coreflectively embedded in our category.

The aim of this paper is to extend the truth value table of lattice-valued convergence spaces to a more general case and then to use it to introduce and study the quantale-valued fuzzy Scott topology in fuzzy domain theory. Let (L; ∗ ; ") be a commutative unital quantale and let ⊗ be a binary operation on L which is distributive over nonempty subsets. The quadruple (L; ∗ ; ⊗ ; ") is called a generalized GL-monoid if (L; ∗ ; ") is a commutative unital quantale and the operation ∗ is ⊗-semi-distributive. For generalized GL-monoid L as the truth value table, we systematically propose the stratified L-generalized convergence spaces based on stratified L-filters, which makes various existing lattice-valued convergence spaces as special cases. For L being a commutative unital quantale, we define a fuzzy Scott convergence structure on L-fuzzy dcpos and use it to induce a stratified L-topology. This is the inducing way to the definition of quantale-valued fuzzy Scott topology, which seems an appropriate way by some results.

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantalevalued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactness and show that the indices of compactness of the quantale-valued metric space and of the hyperspaces equipped with the quantale-valued Hausdorff metric and with the quantale-valued Wijsman structure coincide.

Let L be an integral and commutative quantale. In this paper, by fuzzifying the notion of generalized neighborhood systems, the notion of L-fuzzy generalized neighborhood system is introduced and then a pair of lower and upper approximation operators based on it are defined and discussed. It is proved that these approximation operators include generalized neighborhood system-based approximation operators, L-fuzzy relation-based approximation operators and L-fuzzy covering-based approximation operators as their special circumstances. Therefore, the research on L-fuzzy generalized neighborhood system-based approximation operators has more general significance. In addition, when the L-fuzzy generalized neighborhood system is serial, re exive, unary and transitive, then the corresponding approximation operators are discussed and characterized, respectively.

In this paper, we characterize local T0 and T1 quantale-valued gauge spaces, show how these concepts are related to each other and apply them to L-approach distance spaces and L-approach system spaces. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.

In this work, we describe an adjunction between the comma category of Set-based monads under the V -powerset monad and the category of associative lax extensions of Set-based monads to the category of V -relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V -relations.

The purpose of this paper is to construct a weak hyper semi-quantale as a generalization of the concept of semi-quantale and used it as an appropriate hyperlattice-theoretic basis to formulate new lattice-valued topological theories. Based on such weak hyper semi-quantale, we aim to construct the notion of a weak hypervalued-topology as a generalized form of the so-called lattice-valued topology (or many-valued topology). Some properties of weak hyper semi-quantales and weak hypervalued-topologies will be studied. An adjunction between the category of weak hyper semi-quantales and the category of weak hypervalued quasi-topological spaces will be established.