Negations and contrapositions of complete lattices Original Research Article

We introduce the negation CL of a complete lattice L as the concept lattice of the complementary context (TL, ML, ≰), formed by the join-irreducible elements as objects and the meet-irreducible elements as attributes. We show that the double negation CCL is always order-embeddable in L, and that for finite lattices, the sequence (CnL)nϵω runs into a ‘flip-flop’ (i.e., CnL ⋍ Cn + 2 L for some n). Using vertical sums, we provide constructions of lattices which are isomorphic or dually isomorphic to their own negation. The only finite distributive examples among such ‘self-negative’ or ‘self-contrapositive’ lattices are vertical sums of four-element Boolean lattices. Explicitly, we determine all self-negative and all self-contrapositive lattices with less than 11 points.