In this paper we consider a general way of constructing profinite structures based on a given framework — a countable family of objects and a countable family of recognisers (e.g. formulas). The main theorem states: A subset of a family of recognisable sets is a lattice if and only if it is definable by a family of profinite equations. This result extends Theorem 5.2 from [GGEP08] expressed onl...

We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in אα for some (equivalently for every ordinal α > 0). Another central result here is, in this context: the number of models of a countable first order T of cardinality אα is either ≥ |α| for every אα or it has a small upper bound (close to i2). The author would like to ...

We show that the existence of a countable, first countable, zerodimensional, compact Hausdorff space which is not second countable, hence not metrizable, is consistent with ZF. 1. Notation and terminology Definition 1.1. Let (X,T ) be a topological space and B a base for T . (i) X is said to be compact if every open cover U of X has a finite subcover V . X is said to be compact with respect to ...

We find necessary and sufficient conditions for a complete local (Noetherian) ring containing the rationals to be completion of countable excellent domain. Furthermore, we noncatenary domain, as well it unique factorization

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.

We answer a question of P. Cameron’s by giving examples of 2א0 many non-isomorphic acyclic orientations of the infinite random graph with a topological ordering that do not have the pigeonhole property. Our examples also embed each countable linear ordering. A graph is n-existentially closed or n-e.c. if for each n-subset S of vertices, and each subset T of S (possibly empty), there is a vertex...

In this paper‎, ‎we mainly investigate how the generalized metrizability properties of the remainders affect the metrizability of rectifiable spaces‎, ‎and how the character of the remainders affects the character‎ ‎and the size of a rectifiable space‎. ‎Some results in [A. V‎. ‎Arhangel'skii and J‎. ‎Van Mill‎, ‎On topological groups with a first-countable remainder‎, ‎Topology Proc. 42 (2013...

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened ...

Region Connection Calculus (RCC) is the most widely studied formalism of Qualitative Spatial Reasoning. It has been known for some time that each connected regular topological space provides an RCC model. These ‘standard’ models are inevitable uncountable and regions there cannot be represented finitely. This paper, however, draws researchers’ attention to RCC models that can be constructed fro...

We do a quantitative analysis of modal logic. For example. for each Kripke structure M , we study the least ordinal ,u such that for each state of M , the beliefs up to level ,u characterize the agents’ beliefs (that is, there is only one way to extend these beliefs to higher levels). As another example, we show the equivalence of three conditions, that on the face of it look quite different, f...