In this paper we discuss how biresiduations provide a unifying paradigm for fuzzy formal concept analysis and mathematical morphology. In particular we provide constructions of morphological operators such as dilation and erosion within the framework of convolution algebras of action networks (also known as small covariant categories) over join complete semirings. This unifies and generalizes previous contributions in mathematical morphology, such as the work of Isabelle Bloch. Further we show how decomposition, factorization, and hedges naturally align themselves in the framework of fuzzy formal concept analysis.
In this paper we discuss how biresiduations provide a unifying paradigm for fuzzy formal concept analysis and mathematical morphology. In particular we provide constructions of morphological operators such as dilation and erosion within the framework of convolution algebras of action networks (also known as small covariant categories) over join complete semirings. This unifies and generalizes previous contributions in mathematical morphology, such as the work of Isabelle Bloch. Further we show how decomposition, factorization, and hedges naturally align themselves in the framework of fuzzy formal concept analysis.
We promote biresiduation as a fundamental unifying principle in Formal Concept Analysis, including fuzzification and factor analysis. In particular, we show that maximal formal rectangles are exactly formal concepts within the presented framework of biresiduated maps on ordered sets. Macroscopic implications yield the particular derivation operators in specific settings such as Fuzzy Formal Concept Analysis, Factor Analysis, and degree of containment (i.e. degree of being a subset).
We exhibit an intimate connection between the concept of an annotated ordered set and that of a pattern structure. This enables an exchange of ideas and techniques between both domains.
We study the connection between certain many-valued contexts and general geometric structures. The known one-to-one correspondence between attribute-complete many-valued contexts and complete affine ordered sets is used to extend the investigation to π -lattices, class geometries, and lattices with classification systems. π -lattices are identified as a subclass of complete affine ordered sets, which exhibit an intimate relation to concept lattices closely tied to the corresponding context. Class geometries can be related to complete affine ordered sets using residuated mappings and the notion of a weak parallelism. Lattices with specific sets of classification systems allow for some sort of “reverse conceptual scaling”.
We show how the concept of an annotated ordered set can be used to model large taxonomically structured ontologies such as the Gene Ontology. By constructing a formal context consistent with a given annotated ordered set, their concept lattice representations are derived. We develop the fundamental mathematical relations present in this formulation, in particular deriving a conceptual pre-ordering of the taxonomy, and constructing a correspondence between the annotations of an ordered set and the closure systems of its filter lattice. We study an example from the Gene Ontology to demonstrate how the introduced technique can be utilized for taxonomy review.
The notion of an affine ordered set is specialized to that of a complete affine ordered set, which can be linked to attribute-complete many-valued contexts and is categorically equivalent to the notion of a closed system of equivalence relations (SER). This specialization step enables us to give conditions under which the complete affine ordered set can be interpreted as the set of congruence classes labeled with the congruence relation they stem from yielding a coordinatization theorem for affine ordered sets.
We show how the concept of an annotated ordered set can be used to model large taxonomically structured ontologies such as the Gene Ontology. By constructing a formal context consistent with a given annotated ordered set, their concept lattice representations are derived. We develop the fundamental mathematical relations present in this formulation, in particular deriving a conceptual pre-ordering of the taxonomy, and constructing a correspondence between the annotations of an ordered set and the closure systems of its filter lattice. We study an example from the Gene Ontology to demonstrate how the introduced technique can be utilized for ontology review.
We present a possibility for coordinatizing many-valued contexts and their concept lattices, i.e. we investigate when an algebra (in the sense of universal algebra) can be assigned to the object set of a many-valued context such that the extents can be described by the congruence classes of the algebra. Since congruence class spaces have a natural geometric nature the outlined approach can be interpreted as a geometric representation of concept lattices.
The development of the theory of Formal Concept Analysis has been accompanied from its beginning by applications of the theory to real-world problems. Those applications gave rise to the implementation of the software Toscana and the creation of Toscana-systems. In this paper, we provide a mathematical model for these systems. This model – called Conceptual Data System – enables us to describe Toscana-systems and to discuss possible extensions in mathematical terminology.
We introduce the notions of systems of equivalence relations, affine ordered sets and projective ordered sets to draw connections between many-valued contexts, geometry, and order-theory. This gives rise to applications in data analysis and data visualization.
