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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.