Lattices of Quasivarieties of 3-Element Algebras Original Research Article

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 2aleph, Hebrew0 and that, for the 3-element algebra Kring operator obtained by adjoining a suitably defined binary operation ring operator to K, L(Q(Kring operator)) has cardinality aleph, Hebrew0. 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 Kring operator are maximal with respect to the preceding property. In addition, whilst L(Q(Kring operator)) is a distributive lattice, L(Q(K)) will be seen to fail every non-trivial lattice identity.