Two-based duplicate-clones (Q2744619)

Let \(A=(A_1,A_2)\) be an ordered pair of disjoint sets each containing at least two elements. A subset of \(\{f:A_{i_1}\times\dots\times A_{i_n}\to A_{i_{n+1}}\mid n\geq 1\); \(i_1,\dots,i_{n+1}\in\{1,2\}\}\) is called a two-based clone on \(A\) if it both contains all projections and is closed under composition. A two-based clone on \(A\) is called a duplicate-clone if \(|A_1|=|A_2|\) and if it contains two mutually inverse bijections between \(A_1\) and \(A_2\). A two-based clone is called a \(2\)-Boolean clone if \(|A_1|=|A_2|=2\). If \(|A_1|=|A_2|\) then \(g:A_{i_1}\times\dots\times A_{i_n}\to A_{i_{n+1}}\) is called the conjugate of \(f:A_{3-i_1}\times\dots\times A_{3-i_n}\to A_{3-i_{n+1}}\) if \(g(x_1,\dots,x_n)=h(f(h(x_1),\dots, h(x_n)))\) for all \((x_1,\dots,x_n)\in A_{i_1}\times\dots\times A_{i_n}\) where \(h\) interchanges both the elements of \(A_1\) and those of \(A_2\). A \(2\)-Boolean clone is called double-dually closed if it is closed with respect to conjugation. The lattice consisting of all double-dually closed duplicate \(2\)-Boolean clones as well as of the minimal \(2\)-Boolean clone is described.