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Universal recursively enumerable Boolean algebras. (English. Russian original) Zbl 0552.03026
Sib. Math. J. 24, 852-858 (1983); translation from Sib. Mat. Zh. 24, No. 6(142), 36-43 (1983).
Let $${\mathfrak K}$$ be a class of r.e. Boolean algebras. A r.e. Boolean algebra $${\mathfrak A}_{\nu}$$ is called universal in the class $${\mathfrak K}$$, if $${\mathfrak A}_{\nu}\in {\mathfrak K}$$ and for each r.e. Boolean algebra $${\mathfrak B}_{\mu}\in {\mathfrak K}$$ there exists a r.e. Boolean algebra $${\mathfrak C}_{\pi}$$ such that $${\mathfrak B}_{\mu}\times {\mathfrak C}_{\pi}$$ and $${\mathfrak A}_{\nu}$$ are recursively isomorphic. Let B denote the class of atomless r.e. Boolean algebras, A be the class of atomic r.e. Boolean algebras and $$A_ r$$ be the class of atomic recursive Boolean algebras. The author studies the question of existence of universal r.e. Boolean algebras for the classes B, A, $$A_ r$$ and proves the following theorems: Theorem 1. There is no universal r.e. Boolean algebra in the class B of atomless r.e. Boolean algebras. Theorem 2. For each atomic r.e. Boolean algebra ($${\mathfrak A},\mu)$$ there exists a recursive atomic Boolean algebra which is not a direct summand in ($${\mathfrak A},\mu)$$. Corollary 1. The class $$A_ r$$ has no universal recursive Boolean algebra. Corollary 2. The class A has no universal r.e. Boolean algebra.
##### MSC:
 03D45 Theory of numerations, effectively presented structures 06E99 Boolean algebras (Boolean rings)
##### Keywords:
r.e. Boolean algebras; recursive Boolean algebra
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##### References:
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