Superatomic treealgebras (Q1866831)

The tree algebra over a tree \(T\), denoted by \(B(T)\), is the subalgebra of the power set of \(T\) generated by \(\{ b_t;t\in T\}\), where \(b_t=\{ u\in T;t\leq u\}\) [see \textit{G. Brenner} and \textit{J. D. Monk}, ``Tree algebras and chains'', Lect. Notes Math. 1004, 54-66 (1983; Zbl 0527.06007)]. A Boolean algebra \(B\) is a canonically good algebra whenever there is a set \(E_B\) of representatives for \(B\) such that the sublattice of \(B\) generated by \(E_B\) is a well-founded lattice. In the paper, superatomic tree algebras are characterized and, for each \(\kappa \) different from \(\omega \), \(2^{\kappa }\) incomparable superatomic tree algebras are constructed with respect to the dense homomorphism relation. Also, the author shows that superatomic tree algebras are canonically good and therefore, in cardinality \(\kappa \), there are \(2^{\kappa }\) non-isomorphic canonically good algebras none of which is isomorphic to an interval algebra.