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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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