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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.)
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