Editing Formal concept analysis (section)

== Extensions of the theory ==
* '''Triadic concept analysis''' replaces the binary incidence relation between objects and attributes by a ternary relation between objects, attributes, and conditions. An incidence {{tmath|(g,m,c)}} then expresses that ''the object {{mvar|g}} has the attribute {{mvar|m}} under the condition {{mvar|c}}''. Although ''triadic concepts'' can be defined in analogy to the formal concepts above, the theory of the ''trilattices'' formed by them is much less developed than that of concept lattices, and seems to be difficult.<ref>{{cite journal |first=R. |last=Wille |title=The basic theorem of triadic concept analysis".  |journal=Order |volume=12 |issue= 2|pages=149–158 |date=1995 |doi=10.1007/BF01108624 |s2cid=122657534 |url=}}</ref> Voutsadakis has studied the ''n''-ary case.<ref name="Voutsadakis">{{cite journal |first=G. |last=Voutsadakis |title=Polyadic Concept Analysis |journal=Order |volume=19 |issue=3 |pages=295–304 |date=2002 |doi=10.1023/A:1021252203599 |s2cid=17738011 |url=https://www.voutsadakis.com/RESEARCH/PUBLISHED/polyadic.pdf }}</ref>
* '''Fuzzy concept analysis''': Extensive work has been done on a fuzzy version of formal concept analysis.<ref>{{Cite web |url=http://www.glc.us.es/cla2010/slides/tutorialI_Belohlavek.pdf |title=Formal Concept Analysis and Fuzzy Logic |access-date=2017-12-08 |archive-url=https://web.archive.org/web/20171209043947/http://www.glc.us.es/cla2010/slides/tutorialI_Belohlavek.pdf |archive-date=2017-12-09 |url-status=dead }}</ref>
* <span id="concept algebra">'''Concept algebras'''</span>: Modelling negation of formal concepts is somewhat problematic because the complement {{math|1=(''G'' \ ''A'', ''M'' \ ''B'')}} of a formal concept (''A'', ''B'') is in general not a concept. However, since the concept lattice is complete one can consider the join (''A'', ''B'')<sup>Δ</sup> of all concepts (''C'', ''D'') that satisfy {{math|1=''C'' ⊆ ''G'' \ ''A''}}; or dually the meet (''A'', ''B'')<sup>𝛁</sup> of all concepts satisfying {{math|1=''D'' ⊆ ''M'' \ ''B''}}. These two operations are known as ''weak negation'' and ''weak opposition'', respectively. This can be expressed in terms of the ''derivation operators''. Weak negation can be written as {{math|1=(''A'', ''B'')<sup>Δ</sup> = ((''G'' \ ''A''){{pprime}}, (''G'' \ ''A'')')}}, and weak opposition can be written as {{math|1=(''A'', ''B'')<sup>𝛁</sup> = ((''M'' \ ''B'')', (''M'' \ ''B''){{pprime}})}}. The concept lattice equipped with the two additional operations Δ and 𝛁 is known as the ''concept algebra'' of a context. Concept algebras generalize [[power set]]s. Weak negation on a concept lattice ''L'' is a ''weak complementation'', i.e. an [[order-reversing]] map {{math|1=Δ: ''L'' → ''L''}} which satisfies the axioms {{math|1=''x''<sup>ΔΔ</sup> ≤ ''x'' and (''x''⋀''y'') ⋁ (''x''⋀''y''<sup>Δ</sup>) = ''x''}}. Weak opposition is a dual weak complementation. A (bounded) lattice such as a concept algebra, which is equipped with a weak complementation and a dual weak complementation, is called a ''weakly dicomplemented lattice''. Weakly dicomplemented lattices generalize distributive [[orthocomplemented lattice]]s, i.e. [[Boolean algebra (structure)|Boolean algebras]].<ref>{{Citation |last=Wille |first=Rudolf |year=2000 |contribution=Boolean Concept Logic |editor1-last=Ganter |editor1-first=B. |editor2-last=Mineau |editor2-first=G. W. |title=ICCS 2000 Conceptual Structures: Logical, Linguistic and Computational Issues |publisher=Springer |pages=317–331 |isbn=978-3-540-67859-5 |series=LNAI 1867}}.</ref><ref name=kwuida2004>{{Citation |last=Kwuida |first=Léonard |title=Dicomplemented Lattices. A contextual generalization of Boolean algebras |year=2004 |publisher=[[Shaker Verlag]] |isbn=978-3-8322-3350-1 |url=http://hsss.slub-dresden.de/documents/1101148726640-2926/1101148726640-2926.pdf }}</ref>

=== Temporal concept analysis ===
Temporal concept analysis (TCA) is an extension of Formal Concept Analysis (FCA) aiming at a conceptual description of temporal phenomena. It provides animations in concept lattices obtained from data about changing objects. It offers a general way of understanding change of concrete or abstract objects in continuous, discrete or hybrid space and time. TCA applies conceptual scaling to temporal data bases.<ref>
{{Citation |last=Wolff |first=Karl Erich |year=2010 |contribution=Temporal Relational Semantic Systems |editor1-last=Croitoru |editor1-first= Madalina |editor2-last=Ferré |editor2-first=Sébastien |editor3-last=Lukose |editor3-first=Dickson |title=Conceptual Structures: From Information to Intelligence. ICCS 2010. LNAI 6208 |publisher=Springer |pages=165–180 |isbn=978-3-642-14196-6 |doi=10.1007/978-3-642-14197-3 |series=Lecture Notes in Artificial Intelligence|volume=6208 |url=https://basepub.dauphine.fr/handle/123456789/12138 }}.</ref>

In the simplest case TCA considers objects that change in time like a particle in physics, which, at each time, is at exactly one place. That happens in those temporal data where the attributes 'temporal object' and 'time' together form a key of the data base. Then the state (of a temporal object at a time in a view) is formalized as a certain object concept of the formal context describing the chosen view. In this simple case, a typical visualization of a temporal system is a line diagram of the concept lattice of the view into which trajectories of temporal objects are embedded.
<ref>{{Citation |last=Wolff |first=Karl Erich |year=2019 |contribution=Temporal Concept Analysis with SIENA |url=http://ceur-ws.org/Vol-2378/shortAT12.pdf |editor1-last=Cristea |editor1-first=Diana |editor2-last=Le Ber |editor2-first=Florence |editor3-last=Missaoui |editor3-first=Rokia |editor4-last=Kwuida |editor4-first=Léonard |editor5-last=Sertkaya |editor5-first=Bariş |title=Supplementary Proceedings of ICFCA 2019, Conference and Workshops |publisher=Springer |pages=94–99 }}.</ref>

TCA generalizes the above mentioned case by considering temporal data bases with an arbitrary key. That leads to the notion of distributed objects which are at any given time at possibly many places, as for example, a high pressure zone on a weather map. The notions of 'temporal objects', 'time' and 'place' are represented as formal concepts in scales. A state is formalized as a set of object concepts.
That leads to a conceptual interpretation of the ideas of particles and waves in physics.<ref>{{Citation |last=Wolff |first=Karl Erich |year=2004 |contribution='Particles' and 'Waves' as Understood by Temporal Concept Analysis. |editor1-last=Wolff |editor1-first=Karl Erich |editor2-last=Pfeiffer |editor2-first=Heather D. |editor3-last=Delugach |editor3-first=Harry S. |title=Conceptual Structures at Work. 12th International Conference on Conceptual Structures, ICCS 2004. Huntsville, AL, USA, July 2004, LNAI 3127. Proceedings |publisher=Springer |pages=126–141 |isbn=978-3-540-22392-4 |series=Lecture Notes in Artificial Intelligence|volume=3127 |doi=10.1007/978-3-540-27769-9_8 }}.</ref>