Four generators of an equivalence lattice with consecutive block counts

The block count of an equivalence $\mu \in$ Equ$(A)$ is the number blnum$(\mu )$ of blocks of (the partition corresponding to) $\mu$. We say that $X=\{{\mu }_1,{\mu }_2,{\mu }_3,{\mu }_4\}$ is a four-element generating set of Equ$(A)$ with consecutive block counts if $X$ generates Equ$(A)$ and blnum$({\mu }_{i+1})$ = blnum$({\mu }_1)+i$ for $i\in \{1,2,3\}$. We prove that if the number of elements of a finite set $A$ is six or at least eight, then Equ$(A)$ has a four-element generating set with consecutive block counts. Also, we present a historical remark on the connection between equivalence lattices and quasiorder lattices.