This paper introduces a new approach to TOPOLOGY, based on category theory and universal algebra, and called CATEGORICALLY-ALGEBRAIC (catalg) TOPOLOGY. It incorporates the most important settings of lattice-valued TOPOLOGY, including poslat TOPOLOGY of S. E. Rodabaugh, (L; M)-fuzzy TOPOLOGY of T. Kubiak and A. Sostak, andM-fuzzy TOPOLOGY on L-fuzzy sets of C. Guido.Moreover, its respective categories of topological structures are topological over their ground categories. The theory also extends the notion of topological system of S. Vickers (and its numerous many-valued modifications of J. T. Denniston, A. Melton and S. E. Rodabaugh), and shows that the categories of catalg topological structures are isomorphic to coreective subcategories of the categories of catalg topological systems. This extension initiates a new approach to soft TOPOLOGY, induced by the concept of soft set of D. Molodtsov, and currently pursued by various researchers.

The paper continues the study of the authors on relationships between topological systems of S. Vickers and attachments of C. Guido. We extend topological systems to ALGEBRAICally-topological systems. A particular instance of the latter, called attachment system, incorporates the notion of attachment, thus, making it CATEGORICALLY redundant in mathematics. We show that attachment systems are equipped with an internal TOPOLOGY, which is similar to the TOPOLOGY induced by locales. In particular, we provide an attachment system analogue of the well-known categorical equivalence between sober topological spaces and spatial locales.

This article aims to provide a method for defining L-fuzzy ALGEBRAIC substructures on general algebras. Concretely, the properties of L-fuzzy sets are first reviewed, and their representations are then provided. Next, ALGEBRAIC substructures are generalised as the closure systems on the power set of the algebra, and the properties of the prime and maximal elements in the above closure system are investigated. Based on these facts, L-fuzzy ALGEBRAIC substructures with respect to the closure system are defined and studied. Two equivalence characterisations of the sup property of the ordered set L are provided using L-fuzzy substructures. Similarly, some properties of L-fuzzy prime and maximal substructures with respect to the closure system are discussed. Finally, to demonstrate the broad applicability of the theory of L-fuzzy ALGEBRAIC substructures, the theory is applied to some specific ALGEBRAIC structures, such as groups and pseudo MV-algebras.

The theory of ALGEBRAIC frames for a Hilbert space H is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to de, ne the dual of ALGEBRAIC frames with densely de, ned unbounded analysis operators. It is shown that every ALGEBRAIC frame has an ALGEBRAIC dual frame, and if an ALGEBRAIC frame has a nonzero redundancy, then it is not Riesz-type. An example of an ALGEBRAIC frame with , nite redundancy is constructed which is not a Riesz-type ALGEBRAIC frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its ALGEBRAIC dual are studied and shown to be interrelated.