The isomorphism problem for uniserial modules over an arbitrary ring

AbstractAn important task in cheminformatics is to test whether two molecules are equivalent with respect to their 2D structure. Mathematically, this amounts to solving the graph isomorphism problem for labelled graphs. In this paper, we present an approach which exploits chemical properties and the local neighbourhood of atoms to define highly distinctive node labels. These characteristic labels are the key for clever partitioning molecules into molecule equivalence classes and an effective equivalence test. Based on extensive computational experiments, we show that our algorithm is significantly faster than existing implementations within , and . We provide our Java implementation as an easy-to-use, open-source package (via GitHub) which is compatible with . It fully supports the distinction of different isotopes and molecules with radicals.

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Nilpotency degree of the nilradical of a solvable Lie algebra on two generators and uniserial modules of free nilpotent Lie algebras

Journal of Algebra ◽

10.1016/j.jalgebra.2021.06.008 ◽

2021 ◽

Author(s):

Leandro Cagliero ◽

Fernando Levstein ◽

Fernando Szechtman

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

Nilpotent Lie Algebras ◽

Solvable Lie Algebra ◽

Uniserial Modules

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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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Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Fundamenta Informaticae ◽

10.3233/fi-2021-2002 ◽

2021 ◽

Vol 178 (3) ◽

pp. 173-185

Author(s):

Arthur Adinayev ◽

Itamar Stein

Keyword(s):

Isomorphism Problem ◽

Complete Characterization ◽

Hasse Diagram ◽

Subgraph Isomorphism ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting ◽

Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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Prime uniserial modules and rings

Journal of Algebra and Its Applications ◽

10.1142/s021949881950035x ◽

2019 ◽

Vol 18 (02) ◽

pp. 1950035 ◽

Cited By ~ 1

Author(s):

M. Behboodi ◽

Z. Fazelpour

Keyword(s):

Commutative Rings ◽

Finitely Generated ◽

Finitely Generated Module ◽

Direct Sum ◽

Dedekind Domains ◽

Prime Submodules ◽

Uniserial Modules ◽

Structure Formula

We define prime uniserial modules as a generalization of uniserial modules. We say that an [Formula: see text]-module [Formula: see text] is prime uniserial ([Formula: see text]-uniserial) if its prime submodules are linearly ordered by inclusion, and we say that [Formula: see text] is prime serial ([Formula: see text]-serial) if it is a direct sum of [Formula: see text]-uniserial modules. The goal of this paper is to study [Formula: see text]-serial modules over commutative rings. First, we study the structure [Formula: see text]-serial modules over almost perfect domains and then we determine the structure of [Formula: see text]-serial modules over Dedekind domains. Moreover, we discuss the following natural questions: “Which rings have the property that every module is [Formula: see text]-serial?” and “Which rings have the property that every finitely generated module is [Formula: see text]-serial?”.

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On commutative rings with uniserial dimension

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500085 ◽

2014 ◽

Vol 14 (01) ◽

pp. 1550008 ◽

Cited By ~ 2

Author(s):

A. Ghorbani ◽

Z. Nazemian

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Artinian Rings ◽

Uniform Dimension ◽

Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

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On the NP-Isomorphism Problem with Respect to Random Instances

Journal of Computer and System Sciences ◽

10.1006/jcss.1995.1014 ◽

1995 ◽

Vol 50 (1) ◽

pp. 151-164 ◽

Cited By ~ 16

Author(s):

J. Wang ◽

J. Belanger

Keyword(s):

Isomorphism Problem ◽

Random Instances

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On multiplicative systems defined by generators and relations

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100028772 ◽

1953 ◽

Vol 49 (4) ◽

pp. 579-589 ◽

Cited By ~ 11

Author(s):

Trevor Evans

Keyword(s):

Automorphism Group ◽

Finite Number ◽

Cyclic Group ◽

Complete Information ◽

Isomorphism Problem ◽

Infinite Cyclic Group ◽

Generators And Relations ◽

Decision Method ◽

Multiplicative Systems ◽

Finitely Related

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.

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