We study some properties of smooth Borel set with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperaritmectic reals in the context of smooth sets. We also present some results about the σ-ideal of closed smooth sets.

We will investigate the relation of countable closed linear orderings with respect to continuous monotone embeddability and will show that there are exactly א1 many equivalence classes with respect to this embeddability relation. This is an extension of Laver’s result [Lav71], who considered (plain) embeddability, which yields coarser equivalence classes. Using this result we show that there ar...

We introduce a new equivalence relation on the set of all polygonal billiards. We say that two billiards (or polygons) are order equivalent if each of the billiards has an orbit whose footpoints are dense in the boundary and the two sequences of footpoints of these orbits have the same combinatorial order. We study this equivalence relation with additional regularity conditions on the orbit.

We propose a novel enhancement to the task of collision-testing a set of local paths. Our approach circumvents expensive collision-tests, yet it declares a continuum of paths collision-free by exploiting both the structure of paths and the outcome of previous tests. We define a homotopy-like equivalence relation among local paths and provide algorithms to (1) classify paths based on equivalence...

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 ...

We study some properties of smooth Borel sets with respect to a Borel equivalence relation, showing some analogies with the collection of countable sets from a descriptive set theoretic point of view. We found what can be seen as an analog of the hyperarithmetic points in the context of smooth sets. We generalize a theorem of Weiss from Z-actions to actions by arbitrary countable groups. We sho...

I discuss and illustrate the extent to which different relation measures and pattern similarity measures can be expected to generate different structural equivalence results. Measures of network relations and pattern similarity are reviewed to establish clear comparisons between structural equivalence measures. Using Monte Carlo sociometric choice data drawn from four strategically designed stu...

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will no...

This paper deals with rough set approach on lattice theory. We represent the lattices for rough sets determined by an equivalence relation. Without any loss of generality, we have defined the rough set as a pair of sets (lower approximation set, upper approximation set) and then we showed that the collection of all rough sets of an approximations by an equivalence relation form a lattice by som...

For simplicity we adopt the following convention: I, X denote sets, M denotes a many sorted set indexed by I, R1 denotes a binary relation on X, and E1, E2, E3 denote equivalence relations of X. We now state the proposition (1) (E1 ⊔ E2) ⊔ E3 = E1 ⊔ (E2 ⊔ E3). Let X be a set and let R be a binary relation on X. The functor EqCl(R) yielding an equivalence relation of X is defined as follows: (De...