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We give a systematic account of the current state of knowledge of an e↵ective analogue of the ultraproduct construction. We start with a product of a uniformly computable sequence of computable structures indexed by the set of natural numbers. The equality of elements and sat- isfaction of formulas are defined modulo a subset of the index set, which is cohesive, i.e., indecomposable with respect to computably enumerable sets. We present an analogue of Lo ́s’s theorem for e↵ective ultraprod- ucts and a number of results on definability and isomorphism types of the e↵ective ultrapowers of the field of rational numbers, when the com- plements of cohesive sets are computably enumerable. These e↵ective ultraproducts arose naturally in the study of the automorphisms of the lattice of computably enumerable vector spaces. Previously, a number of authors considered related constructions in the context of nonstandard models of fragments of arithmetic. 

Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let$$\omega $$,$$\zeta $$, and$$\eta $$denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of$$\omega $$. If$$\mathcal {L}$$is a computable copy of$$\omega $$that is computably isomorphic to the usual presentation of$$\omega $$, then every cohesive power of$$\mathcal {L}$$has order-type$$\omega + \zeta \eta $$. However, there are computable copies of$$\omega $$, necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to$$\omega + \zeta \eta $$. For example, we show that there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \eta $$. Our most general result is that if$$X \subseteq \mathbb {N} \setminus \{0\}$$is a Boolean combination of$$\Sigma _2$$sets, thought of as a set of finite order-types, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$, where$$\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$$denotes the shuffle of the order-types inXand the order-type$$\omega + \zeta \eta + \omega ^*$$. Furthermore, ifXis finite and non-empty, then there is a computable copy of$$\omega $$with a cohesive power of order-type$$\omega + \boldsymbol {\sigma }(X)$$.