ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.03
https://doi.org/10.26493/1855-3974.2894.b07
(Also available at http://amc-journal.eu)
A non-associative incidence near-ring with a
generalized Möbius function*
John Johnson †, Max Wakefield ‡
US Naval Academy, 572-C Holloway Rd, Annapolis MD, 21402 USA
This paper is dedicated to the memory of John Johnson.
Received 1 June 2022, accepted 27 February 2023, published online 20 September 2023
Abstract
There is a convolution product on 3-variable partial flag functions of a locally finite
poset that produces a generalized Möbius function. Under the product this generalized
Möbius function is a one sided inverse of the zeta function and satisfies many generaliza-
tions of classical results. In particular we prove analogues of Phillip Hall’s Theorem on
the Möbius function as an alternating sum of chain counts, Weisner’s Theorem, and Rota’s
Crosscut Theorem. A key ingredient to these results is that this function is an overlapping
product of classical Möbius functions. Using this generalized Möbius function we define
analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We
compute these polynomials for certain families of matroids and prove that this generalized
Möbius polynomial has -1 as root if the matroid is modular. Using results from Ardila and
Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.
Keywords: Incidence algebra, matroid, Möbius function, valuation.
Math. Subj. Class. (2020): 37K15, 42A99, 60E05, 05A17
*The authors are very thankful for detailed comments by the reviewer. The reviewers suggestions have signif-
icantly improved the article. The authors are thankful for discussions with Carolyn Chun, Joel Lewis, and Will
Traves. The authors are also thankful to George Andrews for help on Lemma 6.14. Frederico Ardila and Mario
Sanchez significantly helped with the material on valuations for which the authors are very thankful. Also, Jose
Bastidas made multiple excellent comments for which the authors are very thankful. The authors would like to
thank the US Naval Academy trident program for support during this project.
†Supported by the US Naval Academy as a Trident Scholar.
‡Corresponding author.
E-mail addresses: m213162@usna.edu (John Johnson), wakefiel@usna.edu (Max Wakefield)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.03
https://doi.org/10.26493/1855-3974.2894.b07
(Dostopno tudi na http://amc-journal.eu)
Neasociativni incidenčni približni kolobar s
posplošeno Möbiusovo funkcijo*
John Johnson †, Max Wakefield ‡
US Naval Academy, 572-C Holloway Rd, Annapolis MD, 21402 USA
Ta članek je posvečen spominu Johna Johnsona.
Prejeto 1. junija 2022, sprejeto 27. februarja 2023, objavljeno na spletu 20. septembra 2023
Povzetek
Obstaja konvolucijski produkt na parcialnih funkcijah (s 3 spremenljivkami) praporov
lokalno končne delno urejene množice, ki da posplošeno Möbiusovo funkcijo. Za ta pro-
dukt je ta posplošena Möbiusova funkcija enostranski inverz zeta funkcije in zadošča mno-
gim posplošitvam klasičnih rezultatov. Dokažemo analogije Phillip Hallovega izreka o
Möbiusovi funkciji kot izmenični vsoti verižnih štetij, Weisnerjevega izreka in Rotajevega
prečnega izreka. Ključna sestavina teh rezultatov je, da je ta funkcija prekrivajoči se pro-
dukt klasičnih Möbiusovih funkcij. Z uporabo te posplošene Möbiusove funkcije defini-
ramo analogne pojme karakterističnega polinoma in Möbiusovih polinomov za rangirane
mreže. Te polinome izračunamo za določene družine matroidov in dokažemo, da je -1 ničla
tega posplošenega Möbiusovega polinoma, če je matroid modularen. Z uporabo rezultatov
Ardile in Sancheza dokažemo, da je ta posplošeni karakteristični polinom matroidna valu-
acija.
Ključne besede: Incidenčna algebra, matroid, Möbiusova funkcija, valuacija.
Math. Subj. Class. (2020): 37K15, 42A99, 60E05, 05A17
*Avtorja se zahvaljujeta recenzentu za podrobne pripombe. Recenzentovi predlogi so znatno izboljšali članek.
Avtorja sta hvaležna za razprave s Carolyn Chun, Joel Lewisom in Willom Travesom. Avtorja se zahvaljujeta tudi
Georgeu Andrewsu za pomoč pri lemi 6.14. Frederico Ardila in Mario Sanchez sta znatno pomagala pri gradivu
o valuacijah, za kar sta jima avtorja zelo hvaležna. Jose Bastidas je podal številne odlične pripombe, za katere se
avtorja prav tako zahvaljujeta. Avtorja se zahvaljujeta tudi US Naval Academy trident programu za podporo med
tem projektom.
†Podprt s strani US Naval Academy kot Trident Scholar.
‡Kontaktni avtor.
E-poštna naslova: m213162@usna.edu (John Johnson), wakefiel@usna.edu (Max Wakefield)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/