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The Erd\H{o}s--Ko--Rado theorem is extended to designs in semilattices withcertain conditions. As an application, we show the intersection theorems forthe Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmannschemes, signed sets, partial permutations and restricted signed sets.

A generalization of Erd\H{o}s--Ko--Rado theorem to $t$-designs in certain semilattices