Formal Concept Analysis (FCA) has been proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notion of contractive mappings over formal contexts is proposed, which can be viewed as a generalization of interior operators on sets into the framework of FCA. Then, by considering subset-selections consistent with contractive mappings, the notions of attribute continuous formal contexts and continuous concepts are introduced. It is shown that the set of continuous concepts of an attribute continuous formal context forms a continuous domain, and every continuous domain can be restructured in this way. Moreover, the notion of F-morphisms is identified to produce a category equivalent to that of continuous domains with Scott continuous functions. The paper also investigates the representations of various subclasses of continuous domains including algebraic domains and stably continuous semilattices.
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Motivated by the recent study of several researchers on extended-order algebras introduced by C.Guido and P.Toto as a possible common framework for the majority of algebraic structures used in many valued mathematics,the paper focuses on the properties of homomorphisms of the new structures,considering extended order algebras as a generalization of partially ordered sets.The manuscript also introduces the notion of extended-relation algebra providing a new framework for developing the theory of rough sets.
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