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Algebras with hyperidentities of the variety of Boolean algebras. (English. Russian original) Zbl 0885.08002
Izv. Math. 60, No. 6, 1219-1260 (1996); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 60, No. 6, 127-168 (1996).
A hyperidentity is a formula of the form $\forall X_1,\dots, X_m,\;\forall x_1,\dots, x_n\;(w_1= w_2),$ where $$w_1$$ and $$w_2$$ are words over the functional letters $$X_1,\dots, X_m$$ and the object letters $$x_1,\dots, x_n$$. Such a formula is valid in an algebra $${\mathcal Q}= \langle Q,\Sigma\rangle$$ when $$w_1= w_2$$ in $${\mathcal Q}$$ for any assignment of elements of $$Q$$ to the object letters and of fundamental operations (elements of $$\Sigma$$) of the suitable arity to the functional letters.
For a class $$K$$ of algebras, let $$K^*$$ denote the set of all hyperidentities valid in every member of $$K$$; also for a set $$H$$ of hyperidentities, let $$H^*$$ denote the class of all algebras satisfying the hyperidentities of $$H$$. Hyperidentities have been studied since the 1960’s under the impulse of Mal’tsev, and the author has already characterized $$BA^*$$, where $$BA$$ is the variety of all Boolean algebras. This paper is a natural sequel. Two kinds of new results are obtained: 1) a characterization of $$K^*$$ when $$K$$ is the class of all lattices, or modular or distributive lattices; 2) a characterization of $$BA^{**}$$. In particular, it is shown that $$BA\neq BA^{**}$$ and every member of $$BA^{**}$$ is a Boolean sum of Boolean algebras.

##### MSC:
 08B10 Congruence modularity, congruence distributivity 06E05 Structure theory of Boolean algebras 06B20 Varieties of lattices
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