On orthocomplemented lattices with Elkan's law(Algorithmic problems in algebra, languages and computation systems)
Michiro Kondo · Kyoto University Research Information Repository (Kyoto University) · 2006
In this short note we answer the problem left open in [3] thatAre there orthocomplemented lattices different $\mathrm{h}\mathrm{o}\mathrm{m}$ Boolean algebras $\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{i}\mathfrak{h}\mathrm{i}\mathrm{n}\mathrm{g}(\mathrm{E}\mathrm{L})$ : $(a\cdot b')'=b+d\cdot b'$ for all $a,$ $b$ ?That is, we prove that any orthocomplemented lattice satisfying $(a\cdot$ $b')'=b+a'\cdot b'$ is a Boolean algebra.Moreover we show the stronger result that every bounded lattice with the conditions (C1) $1'=0$ and $(\mathrm{E}\mathrm{L})$ is a Boolean algebra.