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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The geometric structure of skew lattices


Author: Jonathan Leech
Journal: Trans. Amer. Math. Soc. 335 (1993), 823-842
MSC: Primary 20M10; Secondary 06A06, 20M25
MathSciNet review: 1080169
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Abstract: A skew lattice is a noncommutative associative analogue of a lattice and as such may be viewed both as an algebraic object and as a geometric object. Whereas recent papers on skew lattices primarily treated algebraic aspects of skew lattices, this article investigates their intrinsic geometry. This geometry is obtained by considering how the coset geometries of the maximal primitive subalgebras combine to form a global geometry on the skew lattice. While this geometry is derived from the algebraic operations, it can be given a description that is independent of these operations, but which in turn induces them. Various aspects of this geometry are investigated including: its general properties; algebraic and numerical consequences of these properties; connectedness; the geometry of skew lattices in rings; connections between primitive skew lattices and completely simple semigroups; and finally, this geometry is used to help classify symmetric skew lattices on two generators.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1080169-X
PII: S 0002-9947(1993)1080169-X
Keywords: Band, (partial) skew lattice, primitive skew lattice, partial ordering, quasi-ordering, coset, connected subalgebra
Article copyright: © Copyright 1993 American Mathematical Society