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On pseudo-Boolean algebras with a finite number of dense elements. (Russian) Zbl 0709.06006
Computable invariants in the theory of algebraic systems, Collect. Sci. Works, Novosibirsk, 35-45 (1987).
[For the entire collection see Zbl 0687.00004.]
The author studies pseudo-Boolean algebras, that is, relatively pseudo- complemented lattices with zero element. An element x of a pseudo-Boolean algebra D is said to be dense if \(-x=0,\) and regular if \(--x=x.\) The set \(\nabla (D)\) of all dense elements is a filter; the set A(D) of all regular elements forms a Boolean algebra (with respect to a new join operation: \(a\cup^*b=--(a\cup b)).\) For \(a\in A(D)\) let \(\nabla_ a(D)\) be \(\{\) \(b\in \nabla (D):\) \(a\leq b\}\); it is a filter.
Let A be a Boolean algebra, \(\nabla\) a relatively pseudo-complemented lattice, \(\{\nabla_ a:\) \(a\in A\}\) a family of filters in \(\nabla\). The author gives a criterium of existence of a pseudo-Boolean algebra D having A, \(\{\nabla_ a:\) \(a\in A\}\) as its A(D), \(\{\nabla_ a(D):\) \(a\in A(D)\}\). Using this result he reduces the problems of isomorphism, elementary equivalence, finite axiomatizability and decidability for pseudo-Boolean algebras with finitely many dense elements to the analogous ones for Boolean algebras with distinguished unary predicates.
Reviewer: O.V.Belegradek

06D15 Pseudocomplemented lattices
03C60 Model-theoretic algebra