Lattices of quasivarieties of 3-element algebras (Q1330084)

This well-written paper contains many interesting results from the structure of lattices of subquasivarieties. A quasivariety is any class of similar algebraic structures that is closed under isomorphisms, substructures, direct products, and ultraproducts. \(L(Q(A))\) denotes the lattice of subquasivarieties of the quasivariety \(Q(A)\), generated by the algebra \(A\). Let \(K\) be the three-element Kleene algebra \(K=\langle \{0,a,1\}; \vee,\wedge, \sim, 0,1\rangle\), where omitting the operation \(\sim\) gives a bounded chain, and \(\sim 0=1\), \(\sim 1=0\), \(\sim a=a\). The main results are the following: Theorem 1.1. The cardinality of \(L(Q(K))\) is continuum and contains the free lattice with countable many generators. In particular it does not satisfy non-trivial lattice identity. Theorem 1.2. Let \(K^ \circ\) denote the algebra obtained from \(K\) by adjoining a specific binary operation \(\circ\). Then the lattice \(L(Q(K^ \circ))\) is isomorphic to \(\omega+3\). In addition, these algebras are in some sense maximal: Let \(K^*\) be any algebra on the three-element set \(\{0,a,1\}\). If the clone of term functions of \(K^*\) properly contains that of \(K\) (resp. \(K^ \circ\)), then \(L(Q(K^*))\) is countable (resp. finite).