The join degree spectrum of successor and block relations on computable linear orders

A. Frolov (2012) proved that the degree spectrum of the successor relation of a non-$\eta$-like linear order is closed upwards in enumerable degrees. In the paper of R. Downey, S. Lempp, G. Wu (2012) the similar prove for $\eta$-like linear orders contains a fatal mistake. In this talk we consider the join spectrum of successor and block relations on computable $\eta$-like linear orders. We prove that such join spectrum is closed upwards in enumerable degrees. It is a join work with A. Frolov.