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What we propose here is to reduce the size of Galois lattices still conserving their formal structure and exhaustivity. For that purpose we use a preliminary partition of the instance set, representing the association of a “type” to each instance. By redefining the notion of of a term in order to cope, to a certain degree (denoted as ), with this partition, we define a particular family of Galois lattices denoted as . We also discuss the related implication rules defined as inclusion of such -extents and show that Iceberg concept lattices are Alpha Galois lattices where the partition is reduced to one single class.

About ten years ago, triadic contexts were presented by Lehmann and Wille as an extension of Formal Concept Analysis. However, they have rarely been used up to now, which may be due to the rather complex structure of the resulting diagrams. In this paper, we go one step back and discuss how traditional line diagrams of standard (dyadic) concept lattices can be used for exploring and navigating triadic data.

Our approach is inspired by the paradigm of On-Line-Analytical Processing (OLAP).We recall the basic ideas of OLAP, and showhowthey may be transferred to triadic contexts. For modeling the navigation patterns a user might follow, we use the formalisms of finite state machines. In order to present the benefits of our model, we show how it can be used for navigating the IT Baseline Protection Manual of the German Federal Office for Information Security.

In order to define a negation on formal concepts in Formal Concept Analysis, the more general notions of semiconcepts and protoconcepts were introduced. The theory of the resulting protoconcept and semiconcept algebras is developed in Boolean Concept Logic as a part of Contextual Logic. In this paper it is shown that each complete subalgebra of a semiconcept algebra is itself the semiconcept algebra of an appropriate context. An analogous result holds for the complete subalgebras of protoconcept algebras. These contexts can be obtained from the original context through partitions of the object and the attribute set satisfying certain conditions. Characterizations of the complete subalgebras of semiconcept and protoconcept algebras in terms of contexts, in terms of subsets, and through closed subrelations are given.

For representing different views and their connections, networks of formal contexts are considered which are coded by so-called . The coincidences between the network contexts of a multicontext give rise to a . It is the aim of this paper to state and to prove the as an extension of the Basic Theorem on Concept Lattices.

The purpose of this paper is to investigate the connection between the theory of computation and Temporal Concept Analysis, the temporal branch of Formal Concept Analysis.

The main idea is to represent for each possible input of a given algorithm the uniquely determined sequence of computation steps as a life track of an object in some conceptually described state space. For that purpose we introduce for a given Turing machine a Conceptual Time System with Actual Objects and a Time Relation (CTSOT) which yields the state automaton of a Turing machine as well as its configuration automaton. The conceptual role of the instructions of a Turing machine is understood as a set of background implications of the derived context of a Turing CTSOT.

The development of a mathematical model for judgments understood as compositions of concepts and relations has been an important branch of research in recent years. It led to the definitions of and which are based on information contained in a , where incidence relations between objects (or tuples of objects) and attributes are stored.

A theory of the information those graphs represent (called ) has been developed for concept graphs in [PW99] and [Wi03]. In [HK04], an extension of this theory to protoconcept graphs not considering object implications (as it is done for concept graphs) has been established. The first part of this paper concentrates on the investigation of the protoconceptual content of protoconcept graphs respecting both protoconceptual and object implications.

The second part compares the different structures of conceptual and protoconceptual contents of a given power context family, showing how more background information (using object implications and concepts instead of protoconcepts) reduces the number of possible contents.

The third and final part analyzes how the different approaches can be generalized. Here we will concentrate on the (generalized) conceptual content of a formal context.

In each part an will be defined, which provides an accessible representation of the lattice of (proto-)conceptual closures.

Popular lattice drawing algorithms do not take planarity into account and find plane diagrams mainly heuristically. We present a characterization of planar lattices based on a theorem of Dushnik and Miller [4] and the “left”-relation introduced by Kelly and Rival [6]. In particular, our work is helpful for drawing plane attribute additive diagrams.

In [8], Vogt used so-called contexts to represent the lattice () of all sublattices of a finite distributive lattice as the substructure lattice of an appropriately defined finite (universal) algebra, based on Rival’s description (see [4] and [5]) by means of deleting suitable intervals from . We show how to extend Vogt’s context in order to obtain a conceptually simpler description of () – the lattice of all 0-1-preserving sublattices of – by means of quasiorders and an associated total binary operation on (), the set of all pairs of non-zero join-irreducibles of . Our approach is based on Birkhoff- resp. Priestley-duality, a standard reference is [1].

We give a contextual characterization of pseudocomplementation by means of the arrow relations.

AMS Subject Classification: 06D15