In universal algebra and in lattice theory the notion of varieties is very prominent, since varieties describe the classes of all algebras (or of all lattices) modeling a given set of equations. While a comprehensive translation of that notion to a similar notion of varieties of complete lattices – and thus to Formal Concept Analysis – has not yet been accomplished, some characterizations of the doubly founded complete lattices of some special varieties (e.g. the variety of modular or that of distributive lattices) have been discovered. In this paper we use the well-known arrow relations to give a characterization of the formal contexts of doubly founded concept lattices in the variety that is generated by M 3 – the smallest modular, non-distributive lattice variety.

@inproceedings{doerfel2011contextbased, author = {Doerfel, Stephan}, title = {A Context-Based Description of the Doubly Founded Concept Lattices in the Variety Generated by M_3}, editor = {Valtchev, Petko and Jäschke, Robert}, booktitle = {Formal Concept Analysis}, series = {Lecture Notes in Computer Science}, publisher = {Springer}, address = {Berlin / Heidelberg}, year = {2011}, volume = {6628}, pages = {93-106}, url = {http://dx.doi.org/10.1007/978-3-642-20514-9_9}, doi = {10.1007/978-3-642-20514-9_9}, keywords = {2011, characterization, context, da, itegpub, modular, myown, variety}, abstract = {In universal algebra and in lattice theory the notion of varieties is very prominent, since varieties describe the classes of all algebras (or of all lattices) modeling a given set of equations. While a comprehensive translation of that notion to a similar notion of varieties of complete lattices – and thus to Formal Concept Analysis – has not yet been accomplished, some characterizations of the doubly founded complete lattices of some special varieties (e.g. the variety of modular or that of distributive lattices) have been discovered. In this paper we use the well-known arrow relations to give a characterization of the formal contexts of doubly founded concept lattices in the variety that is generated by M 3 – the smallest modular, non-distributive lattice variety.} }

%0 = inproceedings %A = Doerfel, Stephan %B = Formal Concept Analysis %C = Berlin / Heidelberg %D = 2011 %I = Springer %T = A Context-Based Description of the Doubly Founded Concept Lattices in the Variety Generated by M_3 %U = http://dx.doi.org/10.1007/978-3-642-20514-9_9

@book{JipsenRose, author = {Jipsen, Peter and Rose, Henry}, title = {Varieties of Lattices}, publisher = {Springer-Verlag}, year = {1992}, keywords = {cited, da, diplomarbeit, lattice, variety} }

%0 = book %A = Jipsen, Peter and Rose, Henry %D = 1992 %I = Springer-Verlag %T = Varieties of Lattices

We prove that there is no free object over a countable set in the category of complete distributive lattices with homomorphisms preserving binary meets and arbitrary joins.

@article{octavio1985nonexistence, author = {Garcia, Octavio and Nelson, Evelyn}, title = {On the nonexistence of free complete distributive lattices}, journal = {Order}, year = {1985}, volume = {1}, number = {4}, pages = {399--403}, url = {http://dx.doi.org/10.1007/BF00582745}, keywords = {distributive, fca, free, lattice, nonexistence, variety}, abstract = {We prove that there is no free object over a countable set in the category of complete distributive lattices with homomorphisms preserving binary meets and arbitrary joins.} }

%0 = article %A = Garcia, Octavio and Nelson, Evelyn %D = 1985 %T = On the nonexistence of free complete distributive lattices %U = http://dx.doi.org/10.1007/BF00582745

@article{Jonsson68, author = {Jónsson, Bjarni}, title = {Equational Classes of Lattices}, journal = {Math. Scand.}, year = {1968}, volume = {22}, pages = {187--196}, keywords = {cited, da, diplomarbeit, jonsson, lattice, variety} }

%0 = article %A = Jónsson, Bjarni %D = 1968 %T = Equational Classes of Lattices

@article{Jonsson, author = {Jónsson, Bjarni}, title = {Algebras whose Congruence Lattices are Distributive}, journal = {Math. Scand.}, year = {1967}, volume = {21}, pages = {110--121}, keywords = {cited, da, diplomarbeit, distributive, jonsson, lattice, variety} }

%0 = article %A = Jónsson, Bjarni %D = 1967 %T = Algebras whose Congruence Lattices are Distributive