Lattices of Quasivarieties of 3-Element Algebras

Abstract Shafaat showed that if L ( Q ( A )) is the lattice of subquasivarieties of the quasivariety Q ( A ) generated by an algebra A , then, for a 2-element algebra A , L ( Q ( A )) is a 2-element chain. It is shown that, for the 3-element Kleene algebra K , L ( Q ( K )) has cardinality 2 ℵ0 and that, for the 3-element algebra K ∘ obtained by adjoining a suitably defined binary operation ∘ to K , L ( Q ( K ∘ )) has cardinality ℵ 0 . The lattice of all clones containing the clone Clo K of all term functions on K is described. As a result, it will be shown that Clo K and Clo K ∘ are maximal with respect to the preceding property. In addition, whilst L ( Q ( K ∘ )) is a distributive lattice, L ( Q ( K )) will be seen to fail every non-trivial lattice identity.