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.

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