On Learning Families of Ideals in Lattices and Boolean Algebras

The paper studies learnability from positive data for families of ideals in lattices and Boolean algebras. We established some connections between learnability and algebraic properties of the underlying structures. We prove that for a computable lattice L, the family of all its ideals is \(\textbf{BC}\) -learnable (i.e., learnable w.r.t. semantical convergence) if and only if each ideal of L is principal. In addition, \(\textbf{BC}\) -learnability for the family of all ideals is equivalent to \(\textbf{Ex}\) -learnability (learnability w.r.t. syntactic convergence). In general, learnability depends on the choice of a computable isomorphic copy of L. Indeed, we show that for every infinite, computable atomic Boolean algebra B, there exist computable algebras A and C isomorphic to B such that the family of all computably enumerable ideals in A is \(\textbf{BC}\) -learnable, while the family of all computably enumerable ideals in C is not \(\textbf{BC}\) -learnable. Finally, we obtain a reverse-mathematical result about conservative \(\textbf{Ex}\) -learning for ideals.