The point-termal complete clones of functions and the lattices of lattices of all subalgebras of algebras with fixed basic sets (Q1953985)

An \(n\)-ary function \(g\) on a set \(A\) is called point-termal for an algebra \(\mathcal A=(A,F)\) if for each \(n\)-tuple of elements of \(A\) there exists an \(n\)-ary term \(t\) of the algebra \(\mathcal A=(A,F)\) such that \(g\) coincides with \(t\) on this \(n\)-tuple. Denote by \(\mathit{PTr}(\mathcal A)\) the set of all point-termal functions for the algebra \(\mathcal A\). Let \(K\) be a clone of functions on a set \(A\) and \(A_K\) be an algebra whose operations are all functions from \(K\). Denote by \(K^P\) the so-called clone of point-termal functions \(\mathit{PTr}(A_K)\) for \(A_K\), and by \(\mathrm{PCL}_A\) the lattice of all point-termal clones on the set \(A\). The author presents some interesting properties of point-termal clones. Theorem 1: Every (finite) lattice is embeddable in \(\mathrm{PCL}_A\) for some (finite) set \(A\). This yields that \(\mathrm{PCL}_A\) cannot satisfy any non-trivial lattice identity. In the next theorem he determines the length and the automorphism group of \(\mathrm{PCL}_A\). It is proved that the elementary theory of \(\mathrm{PCL}_A\) is hereditary unsolvable. A connection to locally finite varieties generated by a given algebra is shown.