Transfinite sequences of topologies, descriptive complexity, and approximating equivalence relations

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

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Isomorphism, canonization, and definability for graphs of bounded rank width

Communications of the ACM ◽

10.1145/3453943 ◽

2021 ◽

Vol 64 (5) ◽

pp. 98-105

Author(s):

Martin Grohe ◽

Daniel Neuen

Keyword(s):

Fixed Point ◽

Complexity Theory ◽

Polynomial Time ◽

Graph Isomorphism ◽

Isomorphism Problem ◽

Descriptive Complexity ◽

Graph Isomorphism Problem ◽

Structural Graph ◽

Graph Classes ◽

Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

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Query Optimization in the Databases Distributed by Means of Product Equivalence Relations

Fundamenta Informaticae ◽

10.3233/fi-1988-11303 ◽

1988 ◽

Vol 11 (3) ◽

pp. 241-265

Author(s):

W. Marek ◽

C. Rauszer

Keyword(s):

Query Optimization ◽

Selection Process ◽

Distributed Databases ◽

Optimization Techniques ◽

Equivalence Relations ◽

Database Operations ◽

By Products

In this paper, we address the problem of query optimization in distributed databases. We show that horizontal partitions of databases, generated by products of equivalence relations, induce optimization techniques for the basic database operations (i.e., the selection, projection, and join operators). In the case of selection, our method allows for restriction of the number of blocks to be searched in the selection process and subsequent simplification of the selection formula at each block. For the natural join operation, we propose an algorithm that reduces the computation of fragments. Proofs of the correctness of our algorithms are also included.

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Zeta functions of equivalence relations over finite fields

Finite Fields and Their Applications ◽

10.1016/j.ffa.2010.09.006 ◽

2011 ◽

Vol 17 (1) ◽

pp. 68-80 ◽

Cited By ~ 1

Author(s):

Tibor Beke

Keyword(s):

Finite Fields ◽

Zeta Functions ◽

Equivalence Relations

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L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002748 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 1169-1188 ◽

Cited By ~ 9

Author(s):

ROMAN SAUER

Keyword(s):

Probability Space ◽

Betti Numbers ◽

Free Action ◽

Countable Group ◽

Equivalence Relations ◽

Dimension Theory ◽

Discrete Groups ◽

Type Ii ◽

Orbit Equivalent ◽

Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

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