Lattices of Quasivarieties of 3-Element Algebras

AbstractShafaat 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 2ℵ0 and that, for the 3-element algebra K∘ obtained by adjoining a suitably defined binary operation ∘ to K, L(Q(K∘)) has cardinality ℵ0. 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