| Abstract: | An algebra with two binary operations · and + that are commutative, associative, and idempotent is called a bisemilattice. A bisemilattice that satisfies Birkhoff’s equation x · (x + y) = x + (x · y) is a Birkhoff system. Each bisemilattice determines, and is determined by, two semilattices, one for the operation + and one for the operation ·. A bisemilattice for which each of these semilattices is a chain is called a bichain. In this note, we characterize the finite bichains that are weakly projective in the variety of Birkhoff systems as those that do not contain a certain three-element bichain. As subdirectly irreducible weak projectives are splitting, this provides some insight into the fine structure of the lattice of subvarieties of Birkhoff systems. |
