Association schemes and t-designs in regular semilattices

A concept of regularity is introduced for finite semilattices with a length function. It is shown that the upper fiber of a regular semilattice carries an association scheme, when points are associated according to the length of their meet. In this framework, a natural definition of t-design is also proposed. The theory applies to triangular- and hypercubic-type association schemes, and to their q-analogs.