DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

Article in volume


Authors:

L. Oluoch

Lilian Oluoch

Technical University of Kenya
Haile Selassie Avenue, P.O Box 52428 - 00200, Nairobi, Kenya

email: lilianoluoch@tukenya.ac.ke

A. Al-Najafi

Amenah Al-Najafi

University of Szeged, Bolyai Institute
Szeged, Aradi vértanúk tere 1, HUNGARY 6720

email: amenah@math.u-szeged.hu

Title:

Lower bound for the number of 4-element generating sets of direct products of two neighboring partition lattices

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Source:

Discussiones Mathematicae - General Algebra and Applications 42(2) (2022) 327-338

Received: 2020-12-21 , Revised: 2022-03-04 , Accepted: 2022-05-09 , Available online: 2022-10-05 , https://doi.org/10.7151/dmgaa.1393

Abstract:

H. Strietz proved in 1975 that the minimum size of a generating set of the partition lattice $\textrm{Part}(n)$ on the $n$-element set $(n \geq 4)$ equals $4$. This classical result forms the foundation for this study. Strietz's results have been echoed by L. Zádori (1983), who gave a new elegant proof confirming the outcome. Several studies have indeed emerged henceforth concerning four-element generating sets of partition lattices. More recently more studies have presented the approach for the lower bounds on the number $\lambda(n)$ of the four-element generating sets of $\textrm{Part}(n)$ and statistical approach to $\lambda(n)$ for small values of $ n $. Also, G. Czédli and the present author have recently proved that certain direct products of partition lattices are also 4-generated. In a recent paper, G. Czédli has shown that this result has connection with information theory. On this basis, here we give a lower bound on the number $\nu(n) $ of 4-element generating sets of the direct product $\textrm{Part}(n)× \textrm{Part}(n+1)$ for $n\geq 7 $ using the results from previous studies. For $n=1,\dots,5$, we use a computer-aided approach; it gives exact values for $n=1,2,3,4$ but we need a statistical method for $n=5$.

Keywords:

partition lattice, four-element generating set, sublattice, statistics, computer program, direct product of lattices, generating, partition lattices

References:

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