Turing Computable Embeddings, Computable Infinitary Equivalence, and Linear Orders

Abstract

We study Turing computable embeddings for various classes of linear orders. The concept of a Turing computable embedding (or tc-embedding for short) was developed by Calvert, Cummins, Knight, and Miller as an effective counterpart for Borel embeddings. We are focused on tc-embeddings for classes equipped with computable infinitary \(\varSigma _{\alpha }\) equivalence, denoted by \(\sim ^c_{\alpha }\). In this paper, we isolate a natural subclass of linear orders, denoted by WMB, such that \((WMB,\cong )\) is not universal under tc-embeddings, but for any computable ordinal \(\alpha \ge 5\), \((WMB, \sim ^c_{\alpha })\) is universal under tc-embeddings. Informally speaking, WMB is not tc-universal, but it becomes tc-universal if one imposes some natural restrictions on the effective complexity of the syntax. We also give a complete syntactic characterization for classes \((K,\cong )\) that are Turing computably embeddable into some specific classes \((\mathcal {C},\cong )\) of well-orders. This extends the similar result of Knight, Miller, and Vanden Boom for the class of all finite linear orders \(\mathcal {C}_{fin}\).