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We examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of [0, ω]. In fact, for any n ∈ [0, ω], there is a degree d and weakly precomplete ceers E₁, . . . , En in d so that any ceer R in d is isomorphic to Ei ⊕ D for some i ≤ n and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes. We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is Π⁰₃-hard, though the definition is Σ⁰₄. We close the gap by showing that the index set is Σ⁰₄-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide. Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if (Ei ) is a uniform c.e. sequence of nonuniversal ceers, then {⊕ i≤j Ei | j ∈ } has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4]. Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.

Given a set S and an equivalence relation R on S, one can define an equivalence graph with vertex set S. Given a graph with vertex set V, we can define an equivalence relation on V using the concept of degree of a vertex as follows: two vertices a and b in V are related if and only if they are of same degree. The degree equivalence graph of a graph G is the equivalence graph with vertex set V with respect to the above equivalence relation. In this paper, we study some properties of degree equivalence graph of a graph. Keywords: Equivalence relation, graph, energy of a graph. AMS Subject Classification: 05Cxx, 05C07, 97E60.

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before. As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

Let $ \\le _c $ be computable the reducibility on computably enumerable equivalence relations (or ceers). We show that for every ceer R with infinitely many equivalence classes, the index sets $\\left\\{ {i:R_i \\le _c R} \\right\\}$ (with R nonuniversal), $\\left\\{ {i:R_i \\ge _c R} \\right\\}$ , and $\\left\\{ {i:R_i \\equiv _c R} \\right\\}$ are ${\\rm{\\Sigma }}_3^0$ complete, whereas in case R has only finitely many equivalence classes, we have that $\\left\\{ {i:R_i \\le _c R} \\right\\}$ is ${\\rm{\\Pi }}_2^0$ complete, and $\\left\\{ {i:R \\ge _c R} \\right\\}$ (with R having at least two distinct equivalence classes) is ${\\rm{\\Sigma }}_2^0$ complete. Next, solving an open problem from [1], we prove that the index set of the effectively inseparable ceers is ${\\rm{\\Pi }}_4^0$ complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a ${\\rm{\\Sigma }}_3^0$ complete preordering relation, a fact that is used to show that the preordering relation $ \\le _c $ on ceers is a ${\\rm{\\Sigma }}_3^0$ complete preordering relation.

Given an Lω1ω\\mathcal {L}_{\\omega _1\\omega }-elementary class C\\mathcal {C}, that is, the collection of the countable models of some Lω1ω\\mathcal {L}_{\\omega _1 \\omega }-sentence, denote by ≅C\\cong _{\\mathcal {C}} and ≡C\\equiv _{\\mathcal {C}} the analytic equivalence relations of, respectively, isomorphisms and bi-embeddability on C\\mathcal {C}. Generalizing some questions of A. Louveau and C. Rosendal, in a paper by S. Friedman and L. Motto  Ros they proposed the problem of determining which pairs of analytic equivalence relations (E,F)(E,F) can be realized (up to Borel bireducibility) as pairs of the form (≅C,≡C)(\\cong _{\\mathcal {C}}, \\equiv _{\\mathcal {C}}), C\\mathcal {C} some Lω1ω\\mathcal {L}_{\\omega _1\\omega }-elementary class (together with a partial answer for some specific cases). Here we will provide an almost complete solution to such a problem: under very mild conditions on EE and FF, it is always possible to find such an Lω1ω\\mathcal {L}_{\\omega _1\\omega }-elementary class C\\mathcal {C}.

Determining whether people in certain countries score differently in measurements of interest or whether concepts relate differently to each other across nations can indisputably assist in testing theories and advancing our sociological knowledge. However, meaningful comparisons of means or relationships between constructs within and across nations require equivalent measurements of these constructs. This is especially true for subjective attributes such as values, attitudes, opinions, or behavior. In this review, we first discuss the concept of cross-group measurement equivalence, look at possible sources of nonequivalence, and suggest ways to prevent it. Next, we examine the social science methodological literature for ways to empirically test for measurement equivalence. Finally, we consider what may be done when equivalence is not supported by the data and conclude with a review of recent developments that offer exciting directions and solutions for future research in cross-national measurement equivalence assessment.

We provide axiomatic characterizations of two natural families of rules for aggregating equivalence relations: the family of join aggregators and the family of meet aggregators. The central conditions in these characterizations are two separability axioms. Disjunctive separability, neutrality, and unanimity characterize the family of join aggregators. On the other hand, conjunctive separability and unanimity characterize the family of meet aggregators. We show another characterization of the family of meet aggregators using conjunctive separability and two Pareto axioms, Pareto + and Pareto - . If we drop Pareto - , then conjunctive separability and Pareto + characterize the family of meet aggregators along with a trivial aggregator.

We prove a theorem giving the asymptotic number of binary quartic forms having bounded invariants; this extends, to the quartic case, the classical results of Gauss and Davenport in the quadratic and cubic cases, respectively. Our techniques are quite general and may be applied to counting integral orbits in other representations of algebraic groups. We use these counting results to prove that the average rank of elliptic curves over ℚ, when ordered by their heights, is bounded. In particular, we show that when elliptic curves are ordered by height, the mean size of the 2-Selmer group is 3. This implies that the limsup of the average rank of elliptic curves is at most 1.5.

Weakly linear systems of fuzzy relation inequalities and equations have recently emerged from research in the theory of fuzzy automata. From the general aspect of the theory of fuzzy relation equations and inequalities homogeneous and heterogeneous weakly linear systems have been discussed in two recent papers. Here we give a brief overview of the main results from these two papers, as well as from a series of papers on applications of weakly linear systems in the state reduction of fuzzy automata, the study of simulation, bisimulation and equivalence of fuzzy automata, and in the social network analysis. Especially, we present algorithms for computing the greatest solutions to weakly linear systems.