If G is an Abelian lattice-ordered (l-) group, then G is existentially closed (e.c.) ist in case every finite system of l-group equations and inequations, involving elements of G, that is solvable in some Abelian l-group extending G is solvable already in G. If GC Hare Abelian l-groups, H is an e.c. prime-model extension of G just in case H is e.c. and is embeddable over G into every e.c. l-group containing, G. Though not every Abelian l-group has an e.c. prime-model extension, this paper establishes existence and uniqueness of such extensions for three special kinds of Abelian l-group. @2026 Elsevier B.V. All rights are reserved, including those for text and data mining, Al training, and similar technologies.
Using relativized ordinal analysis, we give a proof-theoretic characterization of the provably total set-recursive-from-w functions of KPl and related theories. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
We study local consequence relations in modal extensions of product logic over Kripke models with either valued (fuzzy) or crisp accessibility relations. In both settings, we consider semantics over the full class of product algebras as well as over the standard product algebra on [0, 1]. Our main result is a constructive reduction of these modal logics to propositional product logic. As consequences, we prove that all the resulting systems are decidable and standard complete, i.e., the local consequence relation over all product algebras coincides with the one induced by the standard product algebra. In the valued-accessibility case, our methods strengthen previous results on decidability by extending them from theoremhood to arbitrary local consequence relations, and covering standard completeness. In the crisp case, the techniques are substantially different and yield, to the best of our knowledge, the first decidability and standard completeness results for local modal product logics with crisp accessibility relations. (c) 2026 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In [2], we introduced a syntactically defined and highly general class of calculi known as semi-analytic. We then demonstrated that any sufficiently strong (modal) substructural logic with a semi-analytic calculus must satisfy the Craig interpolation property. In this paper, we show that if the calculus is also terminating in a certain formal sense, then its logic has the Uniform Interpolation Property (UIP). This result has significant applications. On the positive side, it provides a uniform and modular method for proving UIP for various logics, including FLe, FLew, CFLe, CFLew, and their K, D, and T-type modal extensions, as well as CPC, K, and KD. However, its more striking consequence lies in the negative direction. It extends the negative results of [2] to logics with CIP but without UIP. In particular, it shows that the modal logics K4 and S4 do not have a terminating semi-analytic calculus. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Learning from positive data models a scenario in which a learner receives longer and longer initial segments of an enumeration of an unknown set and repeatedly outputs a conjecture about it. Classical identification in the limit requires the learner to eventually stabilise to a hypothesis naming the target exactly, while anomalous learning relaxes this requirement by allowing finitely many errors in the final hypothesis. We develop a natural generalisation for computably enumerable (c.e.) sets in which correctness is determined by an arbitrary equivalence relation on their indices that respects equality of the sets they enumerate. We call the resulting framework tolerant learning. Exact learning and finite-anomaly learning arise as special cases. By comparing tolerance relations according to the families they make learnable, we obtain a new degree structure, the tolerant learning hierarchy. We extend the classical methods of locking sequences and tell-tales to this setting and derive a combinatorial characterisation of those tolerance relations that are universal (that is, under which every family of nonempty computably enumerable sets is learnable). We then show that the hierarchy is a bounded lower semilattice but not an upper semilattice, is atomless and coatomless, and contains chains and antichains of size continuum. We also analyse natural benchmark relations, including c.e.-index analogues of standard equivalence relations from descriptive set theory, and establish substantial incomparability among their learning powers. Finally, we separate non-effective from computable tolerant learning by exhibiting a universal relation that is not computably universal. These results show that semantic tolerance gives rise to a genuine degree theory at the interface of algorithmic learning and the theory of equivalence relations.
