Directed Graphs and Combinatorial Properties of Semigroups
TL;DR: In this paper, a complete description of all commutative semigroups satisfying three other combinatorial properties defined in terms of directed graphs is given, by a graph we mean a directed graph without loops or multiple edges.
About: This article is published in Journal of Algebra.The article was published on 2002-05-01 and is currently open access. It has received 198 citations till now. The article focuses on the topics: Semigroup & Directed graph.
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TL;DR: In this article, it was shown that the multiplicative semigroup ℤn and its subgroup Un is complete if and only if n = 1,2,4,p or 2p, where p is a Fermat prime.
Abstract: The undirected power graph G(S) of a semigroup S is an undirected graph whose vertex set is S and two vertices a,b∈S are adjacent if and only if a≠b and am=b or bm=a for some positive integer m. In this paper we characterize the class of semigroups S for which G(S) is connected or complete. As a consequence we prove that G(G) is connected for any finite group G and G(G) is complete if and only if G is a cyclic group of order 1 or pm. Particular attention is given to the multiplicative semigroup ℤn and its subgroup Un, where G(Un) is a major component of G(ℤn). It is proved that G(Un) is complete if and only if n=1,2,4,p or 2p, where p is a Fermat prime. In general, we compute the number of edges of G(G) for a finite group G and apply this result to determine the values of n for which G(Un) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, G(G) is Hamiltonian and list some values of n for which G(Un) has no Hamiltonian cycle.
266 citations
TL;DR: It is conjectured that two finite groups with isomorphic power graphs have the same number of elements of each order, and that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.
225 citations
13 Nov 2013
TL;DR: This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature.
Abstract: This article gives a survey of all results on the power graphs of groups and semigroups obtained in the literature. Various conjectures due to other authors, questions and open problems are also included.
187 citations
TL;DR: This paper proposes a new method of using Cayley graphs for classification of data using the endomorphism monoids of graphs, a convenient tools expressing asymmetries of the graphs.
184 citations
TL;DR: In this article, it was shown that the undirected power graph determines the directed power graph up to isomorphism, and that two finite groups which have isomorphic undirectED power graphs have the same number of elements of each order.
Abstract: Abstract The directed power graph of a group G is the digraph with vertex set G, having an arc from y to x whenever x is a power of y; the undirected power graph has an edge joining x and y whenever one is a power of the other. We show that, for a finite group, the undirected power graph determines the directed power graph up to isomorphism. As a consequence, two finite groups which have isomorphic undirected power graphs have the same number of elements of each order.
137 citations
References
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Book•
01 Jan 1964
TL;DR: A survey of the structure and representation theory of semi groups is given in this article, along with an extended treatment of the more important recent developments of Semi Group Structure and Representation.
Abstract: This book, along with volume I, which appeared previously, presents a survey of the structure and representation theory of semi groups. Volume II goes more deeply than was possible in volume I into the theories of minimal ideals in a semi group, inverse semi groups, simple semi groups, congruences on a semi group, and the embedding of a semi group in a group. Among the more important recent developments of which an extended treatment is presented are B. M. Sain's theory of the representations of an arbitrary semi group by partial one-to-one transformations of a set, L. Redei's theory of finitely generated commutative semi groups, J. M. Howie's theory of amalgamated free products of semi groups, and E. J. Tully's theory of representations of a semi group by transformations of a set. Also a full account is given of Malcev's theory of the congruences on a full transformation semi group.
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19 Oct 2011
TL;DR: A detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative is given in this article, where the reader is provided with only a basic knowledge of modern algebra.
Abstract: This is a detailed introduction to the theory of groups: finite and infinite; commutative and non-commutative. Presupposing only a basic knowledge of modern algebra, it introduces the reader to the different branches of group theory and its principal accomplishments.
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01 Jan 1995
TL;DR: Inverse semigroups as discussed by the authors are a subclass of regular semigroup classes and can be seen as semigroup amalgamations of semigroup groups, which is a special case of regular semiigroups.
Abstract: 1. Introductory ideas 2. Green's equivalences regular semigroups 3. 0-simple semigroups 4. Completely regular semigroups 5. Inverse semigroups 6. Other classes of regular semigroups 7. Free semigroups 8. Semigroup amalgams References List of symbols
1,979 citations
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28 Dec 1984
TL;DR: Perrin and Perrin this article showed that square free words and idempotent semigroups can be expressed in terms of free monoids, and the critical factorization theorem of Van der Waerden's theorem.
Abstract: 1. Words D. Perrin 2. Square free words and idempotent semigroups J. Berstel and C. Reutenauer 3. Van der Waerden's theorem J. E. Pin 4. Repetitive mappings and morphisms G. Pirillo 5. Factorizations of free monoids D. Perrin 6. Subwords J. Sakarovitch and I. Simon 7. Unavoidable irregularities in words M. P. Schutzenberger 8. The critical factorization theorem Choffrut 9. Equations in words 10. Rearrangements of words D. Foata 11. Words and trees R. Cori.
1,941 citations
01 Apr 1997
TL;DR: This first handbook of formal languages gives a comprehensive up-to-date coverage of all important aspects and subareas of the field.
Abstract: The theory of formal languages is the oldest and most fundamental area of theoretical computer science. It has served as a basis of formal modeling from the early stages of programming languages to the recent beginnings of DNA computing. This first handbook of formal languages gives a comprehensive up-to-date coverage of all important aspects and subareas of the field. Best specialists of various subareas, altogether 50 in number, are among the authors. The maturity of the field makes it possible to include a historical perspective in many presentations. The individual chapters can be studied independently, both as a text and as a source of reference. The Handbook is an invaluable aid for advanced students and specialists in theoretical computer science and related areas in mathematics, linguistics, and biology.
1,915 citations