Computable Imbeddability Condition and Degrees
of Abelian Groups and Abelian Algebras

Asylkhan Khisamiev (hisamiev@math.nsc.ru)
Novosibirsk State University (Russia)

We will consider countable algebraic systems of finite signature. L. Richter has introduced in [1] the Recursive Embeddability Condition and has obtained the result: if a non-computable system A satisfies the recursive embeddability condition then A has no degree.

In the given work we show:

- Any countable Abelian p-group (Boolean algebra) A satisfies the Recursive Embeddability Condition.

- Let B be a countable Boolean algebra and A be an antisymmetric connected system S-definable in HF(B). Then A is computable.

From these results we obtain

-Any countable not computable Abelian p-group (Boolean algebra) has no degree.

- If L is a linear ordering S-definable in HF(B) over a countable Boolean algebra B then L is computable.



References

1. L.J. Richter, Degrees of structures, The Journal of Symbolic Logic, 1981, V.46, N.4, p. 723-731.