Joins of minimal quasivarieties
Let \({\mathcal D}_2\) denote the variety of algebras \((L;\wedge, \vee, 0, c_0, c_1,1)\) which are distributive \((0,1)\)-lattices with two distinguished elements \(c_0, c_1\in L\). It is known that the only subdirectly irreducible algebras in \({\mathcal D}_2\) are \(2_{ij}= (\{0, 1\};\wedge, \vee, 0, i,j, 1)\) with \(i,j\in \{0, 1\}\). Let, further, \(V_0\), \(V_1\), \(V_2\), \(V_3\) denote the quasivarieties generated by the algebras \(2_{00}\), \(2_{01}\), \(2_{10}\), \(2_{11}\), respectively. These quasivarieties are minimal elements in the lattice \(L({\mathcal D}_2)\) of quasivarieties contained in \({\mathcal D}_2\). It is proved that 1) the lattice \(L(V_1+ V_2)\) of subquasivarieties of the join \(V_1+ V_2\) is a 7-element lattice, 2) for distinct \(i\) and \(j\) with \(\{i, j\}\neq \{1, 2\}\), \(L(V_i+ V_j)\) is a countable infinite non-distributive lattice, 3) for distinct \(i\), \(j\), and \(k\), \(L(V_i+ V_j+ V_k)\) contains as a sublattice the ideal lattice of a free lattice with \(\omega\) free generators and is of cardinality \(2^\omega\).
scientific article; zbMATH DE number 769071
scientific article