A Semilattice of Degrees of Computable Metrics
R. A. Kornev · Siberian Mathematical Journal · 2021
Under study is the ordering $ {\mathcal{CM}}_{c}({\mathbf{X}}) $ of $ c $ -degrees of computable metrics on a Polish space $ {\mathbf{X}} $ with a distinguished dense subset. We prove that this ordering forms a lower semilattice. If, for a computable metric $ \rho $ on $ {\mathbf{X}} $ , there is a computable limit point in $ (X,\rho) $ ; it is possible to construct a computable metric $ \rho^{\prime}<_{c}\rho $ . Under the same assumption, there exists a computable metric $ \widehat{\rho} $ such that $ \deg_{c}(\rho) $ and $ \deg_{c}(\widehat{\rho}) $ have no common upper bounds in $ {\mathcal{CM}}_{c}({\mathbf{X}}) $ ; thus, in this case $ {\mathcal{CM}}_{c}({\mathbf{X}}) $ is neither an updirected poset nor an upper semilattice.