Article in volume
Authors:
Title:
Lower bound for the number of 4-element generating sets of direct products of two neighboring partition lattices
PDFSource:
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:
- G. Czédli, Four-generated large equivalence lattices, Acta Sci. Math. (Szeged) 62 (1996) 47–69.
- G. Czédli, Lattice generation of small equivalences of a countable set, Order 13 (1996) 11–16.
https://doi.org/10.1007/BF00383964 - G. Czédli, (1+1+2)-generated equivalence lattices, J. Algebra 221 (1999) 439–462.
https://doi.org/10.1006/jabr.1999.8003 - G. Czédli and L. Oluoch, Four-element generating sets of partition lattices and their direct products, Acta Sci. Math. (Szeged) 86 (2020) 405–448.
https://doi.org/10.14232/actasm-020-126-7 - G. Czédli, Four-generated direct powers of partition lattices and authentication, Publicationes Mathematicae (Debrecen) (to appear).
- J.L. Hodges Jr. and E.L. Lehmann, Basic concepts of probability and statistics (Society for Industrial and Applied Mathematics, 2005).
https://doi.org/10.1137/1.9780898719123 - M. Lefebvre, Applied probability and statistics (Springer Science & Business Media, 2007).
https://doi.org/10.1007/0-387-28505-9 - W. Mendenhall, R.J. Beaver and B.M. Beaver, Introduction to probability and statistics (Cengage Learning, 2012).
- H. Strietz, Finite partition lattices are four-generated, in: Proc. Lattice Theory Conf., Gudrun Kalmbach (Ed(s)), (Universität Ulm 1975) 257–259.
- H. Strietz, Über Erzeugendenmengen endlicher Partitionenverbände, Studia Sci. Math. Hungar. 12 (1977) 1–17.
- L. Zádori, Generation of finite partition lattices, in: Lectures in universal algebra (Szeged, 1983), L. Szabó and Á. Szendrei (Ed(s)), (Math. Soc. János Bolyai and North-Holland 1986) 573–586.
https://doi.org/10.1016/b978-0-444-87759-8.50038-9
Close