Abstract
A complete solution is given to the problem of describing algebras with distinguished ideals, formulated by Peretyatkin. It is proven that such an algebra is isomorphic toℬ ωω× η, an interval algebra of the linear ordering ωω × η. I-algebras the elementary theory of each of which is axiomatizable by a single atom in some finite quotient with respect to the Frechet ideal of the Lindenbaum-Tarski algebra for the class of Boolean algebras with distinguished ideals are fully described in terms of direct summands.
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