Given a finite lattice $L$, suppose $L$ is generated by a set $X$ such that the distributive law holds for all $a,b,c\in X$ i.e. $a\lor (b \wedge c) = (a\lor b)\wedge (a \lor c)$.

Is $L$ distributive?

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Given a finite lattice $L$, suppose $L$ is generated by a set $X$ such that the distributive law holds for all $a,b,c\in X$ i.e. $a\lor (b \wedge c) = (a\lor b)\wedge (a \lor c)$.

Is $L$ distributive?

Modularity and Distributivity of 3-Generated Lattices with Special Elements Among Generators, Algebra and Logic, 56(1), 1–12. doi:10.1007/s10469-017-9423-z disprove your conjecture, or am I getting something wrong about the meanings of the words involved? $\endgroup$ – darij grinberg Apr 15 at 19:54