The automorphism group of the universal distributive lattice
The normal subgroup lattices of the automorphism groups of the countable universal homogeneous distributive lattice \(\mathbb D\) and of the countable atomless generalized Boolean lattice \(\mathbb B\) are described. It is known that there exists a unique (up to isomorphism) countable distributive lattice \(\mathbb D\) which is homogeneous, i.e. any isomorphism between two finite sublattices extends to an automorphism of \(\mathbb D\), and universal, i.e. it embeds every countable distributive lattice. A generalized Boolean lattice is a distributive lattice with a smallest element 0 (but not necessarily with 1) such that each interval is a Boolean algebra. There is also a unique countable universal homogeneous Boolean lattice \(\mathbb B\). For any lattice \(\Omega \), let \(A(\Omega)\) stand for the automorphism group of \(\Omega \), \(L(\Omega)=\{ g\in A(\Omega); \exists \;x\in \Omega , \forall \;y\in \Omega : x\leq y \Rightarrow \;g(y)=y\}\), \(R(\Omega)=\{ g\in A(\Omega); \exists \;x\in \Omega , \forall \;y\in \Omega : y\leq x \Rightarrow \;g(y)=y\}\) and \(B(\Omega)=L(\Omega)\cap R(\Omega).\) For \(\Omega =(\mathbb Q,\leq)\) it was known that these are the only non-trivial normal subgroups of \(A({\mathbb Q})\). In this paper the following theorem is proved: Theorem. (a) \(L({\mathbb B})\) is the only proper non-trivial normal subgroup of \(A({\mathbb B})\), and \(L({\mathbb B})\) is simple. (b) \(R({\mathbb D})\), \(L({\mathbb D})\) and \(B({\mathbb D})\) are the only proper non-trivial normal subgroups of \(A({\mathbb D})\), and \(L(\mathbb D)/B(\mathbb D)\cong R(\mathbb D)/B(\mathbb D)\) and \(B(\mathbb D)\) are simple. If \(G\) is a permutation group on \(X\), and \(Y\subseteq X,\) then \(G_{\{ Y\}}, G_{(Y)}\) denote respectively the setwise and pointwise stabilizers of \(Y\) in \(G\). A countably infinite first-order structure \(M\) is \(\omega \)-categorical if it is determined among all countable structures by its first-order theory. An \(\omega \)-categorical structure \(M\) has the strong small index property if, for any \(H\leq \text{Aut}(M)\) with \(|\text{Aut}(M):H|<2^{\aleph _0},\) there is a finite \(F\subset M\) such that \(\text{Aut} (M)_{(F)}\leq H\leq \text{Aut}(M)_{\{ F\}}.\) The second main theorem of the paper is: Theorem. The structures \(\mathbb D\) and \(\mathbb B\) have the strong small index property.
scientific article; zbMATH DE number 1899940