ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.05
https://doi.org/10.26493/2590-9770.1305.f45
(Also available at http://adam-journal.eu)
On the coset structure of distributive skew
lattices
João Pita Costa ∗
Jožef Stefan Institute, Ljubljana, Slovenia
Jonathan Leech
Westmont College, Santa Barbara, CA, USA
Received 14 July 2019, accepted 19 September 2019, published online 29 December 2019
Abstract
In the latest developments in the theory of skew lattices, the study of distributivity has
been one of the main topics. The largest classes of examples of skew lattices thus far en-
countered are distributive. In this paper, we will discuss several aspects of distributivity in
the absence of commutativity, and review recent related results in the context of the coset
structure of skew lattices. We show that the coset perspective is essential to fully under-
stand the nature of skew lattices and distributivity in particular. We will also discuss the
combinatorial implications of these results and their impact in the study of skew lattices.
Keywords: Skew lattices, distributive structures, noncommutative structures, ordered structures, bands
of semigroups.
Math. Subj. Class.: 06A75, 06B20, 06B75
∗The author thanks the support of the Croatian Science Foundation’s funding of the project EVOSOFT, with
the reference UIP-2014-09-7945.
E-mail addresses: joao.pitacosta@ijs.si (João Pita Costa), leech@westmont.edu (Jonathan Leech)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 2590-9770
The Art of Discrete and Applied Mathematics 2 (2019) #P2.05
https://doi.org/10.26493/2590-9770.1305.f45
(Dostopno tudi na http://adam-journal.eu)
O odsekovni strukturi distributivnih
poševnih mrež
João Pita Costa ∗
Jožef Stefan Institute, Ljubljana, Slovenia
Jonathan Leech
Westmont College, Santa Barbara, CA, USA
Prejeto 14. julija 2019, sprejeto 19. septembra 2019, objavljeno na spletu 29. decembra 2019
Povzetek
V najnovejših raziskavah v teoriji poševnih mrež je študij distributivnosti ena glavnih
smeri. Med doslej obravnavanimi primeri poševnih mrež je največ distributivnih. V pris-
pevku obravnavamo različne aspekte distributivnosti, ki nastopijo v odsotnosti komuta-
tivnosti, in povzamemo pred kratkim dobljene rezultate v zvezi z odsekovno strukturo
poševne mreže. Pokažemo, da je prav odsekovna struktura odločilna pri razumevanju
poševnih mrež in še posebej njihove distributivnosti. Obravnavamo tudi kombinatorne
posledice teh rezultatov in njihove ustrezne vplive pri študiju poševnih mrež.
Ključne besede: Poševne mreže, distributivne strukture, nekomutativne strukture, urejene strukture,
pasovi polgrup.
Math. Subj. Class.: 06A75, 06B20, 06B75
∗Avtor se zahvaljuje za podporo s strani Croatian Science Foundation pri financiranju projekta EVOSOFT,
referenčna oznaka UIP-2014-09-7945.
E-poštna naslova: joao.pitacosta@ijs.si (João Pita Costa), leech@westmont.edu (Jonathan Leech)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/