ACTA issues

Lattices embeddable in three-generated lattices

Gábor Czédli

Acta Sci. Math. (Szeged) 82:3-4(2016), 361-382

Abstract. We prove that every finite lattice $L$ can be embedded in a three-generated \emph{finite} lattice $K$. We also prove that every \emph{algebraic} lattice with accessible cardinality is a \emph{complete} sublattice of an appropriate \emph{algebraic} lattice $K$ such that $K$ is completely generated by three elements. Note that ZFC has a model in which all cardinal numbers are accessible. Our results strengthen P. Crawley and R. A. Dean's 1959 results by adding finiteness, algebraicity, and completeness.

DOI: 10.14232/actasm-015-586-2

AMS Subject Classification (1991): 06B99, 06B15

Keyword(s): three-generated lattice, equivalence lattice, partition lattice, complete lattice embedding, inaccessible cardinal

Received December 12, 2015, and in final form September 19, 2016. (Registered under 86/2015.)