On Extensions of Partial Isomorphisms

Construction of A Lattice on the completion space of an algebra and an isomorphism to its Caratheodory Extension

In this paper, we will show how the Caratheodory Extension process is intimately related to the metric completion process. In particular, it will be shown how one is able to construct a lattice on the completion and to obtain an isomorphism to its Caratheodory Extension.

Isomorphism property in nonstandard extensions of ZFC universe

We study models of HST, a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the language containing the membership and standardness predicates, and Saturation for well-orderable families of internal sets. This theory admits an adequate formulation of the isomorphism property IP: "any two elementarily equivalent internally presented structures of a wellorderable language are isomorphic." IP implies, for instance, that all infinite internal sets are equinumerous, and there exists a unique (modulo isomorphisms) internal elementary extension of th…

The Isomorphism Relation Between Tree-Automatic Structures

An $ω$-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for $ω$-tree-automatic structures. We prove first that the isomorphism relation for $ω$-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n >1) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for $ω$-tree-automatic boolean algebras (respectively, partial orders…

Classes of structures with no intermediate isomorphism problems

We say that a theory $T$ is intermediate under effective reducibility if the isomorphism problems among its computable models is neither hyperarithmetic nor on top under effective reducibility. We prove that if an infinitary sentence $T$ is uniformly effectively dense, a property we define in the paper, then no extension of it is intermediate, at least when relativized to every oracle on a cone. As an application we show that no infinitary sentence whose models are all linear orderings is intermediate under effective reducibility relative to every oracle on a cone.

Blok-Esakia定理在一元谓词逻辑环境中的失效

Failure of the Blok-Esakia Theorem in the monadic setting

研究证明Blok-Esakia同构无法扩展到仅含一个固定变量的一元谓词逻辑片段，即其不再是一元直觉逻辑扩张格与一元Grzegorczyk逻辑扩张格的同构，揭示该定理在一元谓词逻辑环境中失效。

Potential isomorphism and semi-proper trees

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notions of semi-proper and weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. We prove the existence of semi-proper trees under certain cardinal arithmetic assumptions. W…