Equivalence relations, invariants, and normal forms
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In this paper, we consider the family of pattern-replacement equivalence relations referred to as the "indices and values adjacent" case. Each such equivalence is determined by a partition $P$ of a subset of $S_c$ for some $c$. In 2010, Linton, Propp, Roby, and West posed a number of open problems in the area of pattern-replacement equivalences. Five, in particular, have remained unsolved until now, the enumeration of equivalence classes under the $\{123, 132\}$-equivalence, under the $\{123, 321\}$-equivalence, under the $\{123, 132, 213\}$ equivalence, and under the $\{123, 132, 213, 321\}$-equivalence. We find formulas for three of the five equivalences and systems of representatives for the equivalence classes of the other two. We generalize our results to hold for all replacement partitions of $S_3$, as well as for an infinite family of other replacement partitions. In addition, we characterize the equivalence classes in $S_n$ under the $S_c$-equivalence, finding a generalization of Stanley's results on the $\{12, 21\}$-equivalence. To do this, we introduce a notion of confluence that often allows one to find a representative element in each equivalence class under a given equivalence relation. Using an inclusion-exclusion argument, we are able to use this to count the equivalence classes under equivalence relations satisfying certain conditions.
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This paper is the third in a series of papers studying equivalence classes of fuzzy subgroups of a given group under a suitable equivalence relation. We introduce the notion of a pinned flag in order to study the operations sum, intersection and union, and their behavior with respect to the equivalence. Further, we investigate the extent to which a homomorphism preserves the equivalence. Whenever the equivalences are not preserved, we have provided suitable counterexamples.
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Introduction.There have been discussions from time to time of "abstract measures" the values of which need not be numerical (e.g.[2], [3], [4], [6], [7]).One of the purposes of this paper is to present arguments in favor of the use of cardinal algebras as values for these measures.Cardinal algebras were introduced and developed by A. Tarski in [8].They have many of the good properties of real numbers and arise naturally in situations like the following:A (pseudo) group G of one-one functions is given with domain and range in a a-ring of sets Jf.An equivalence relation between members of Jf is defined as follows :A^B iff there are A¡, Bi e Cti,fi e G for /
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We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations Eλ and id. In addition, we consider equivalence relations with finitely many nontrivial equivalence classes and those whose equivalence classes are all finite.
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This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence problem is to determine conditions on two differential operators such that there exists a fiber-preserving transformations mapping one to the other according to gauge equivalence.
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