In the last few years my interest turned to  higher dimensional (not planar) semimodular lattices.

(This research is inspired by some papers of George Grtzer and E. Knapp on planar semimodular lattices,

published in the Acta Sci. Math. Szeged around 2007)  pdf

The new results can be considered as a geometric approach. I published another 7 papers with Gbor Czdli, the first of these is

G. Czdli and E. T. Schmidt, How can we derive semilmoduar lattices from distributive lattices?

Acta Math. Acad. Sci. Hungar. 121(2008), 277-282 ( pdf )

 is for arbitrary  semimodular lattices the another joint papers consider planar lattices.

The main structure theorem of 2D semimodular lattices is in the following paper:

 

G. Czdli and E. T. Schmidt, Slim semimodular lattices. II.

A description by patchwork systems, Order, 30 (2013), 689-721. ( pdf)

My conjecture: every semimodular lattice is the patchwork of patch lattices.

 

 

(New version: January 2016)  A structure theorem of semimodular lattices and the Rubik's cube, submitted t AU

 pdf    figures and tex

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Not published results,  (sketches):

 124.  A characterisation of the sources in semimodular lattic ( 2012, renewed: January 2016)   pdf

 125.  A new look at the semimodular lattices, a geometric approach;    (2011, under rewriting 2016)  pdf

 126.   Diamond-free  3-dimensional semimodular lattices,  (2011, 2012)  pdf  

 127.   Rectangular hulls of semimodular lattices,  (2011)   pdf

 128.    An extension theorem for finite  semimodular lattices, (2014,  renewed: January 2016)

         pdf   

 129.    Semimodular lattices,  Slide show (2012)   pdf

 132.     Rectangular lattices as geometric shapes, (2013) pdf

 133.     Congruences of 2-dimensional semimodular lattices ,  (2014), 

          pdf

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Slide show: Semimodular lattices,  ppt

A 2-dimensional patchwork,  htm,  gif

A 3-dimensional patchwork,  htm,  gif

The patch lattice (pigeonhole) gif

The zip gif

OP lattices

 

 

Patchwork in the everyday life: pieces of cloth of various colors and shape sewn together

28  January  2016