Structure of regular semigroups. I
01 Jan 1979-Memoirs of the American Mathematical Society (American Mathematical Society)-Vol. 22, Iss: 224, pp 0-0
TL;DR: In this paper, the authors consider a semigroup satisfying the property abc = ac and prove that it is left semi-normal and right quasi-normal, where ac is the number of variables in the semigroup.
Abstract: This paper concerned with basic concepts and some results on (idempotent) semigroup satisfying the identities of three variables. The motivation of taking three for the number of variables has come from the fact that many important identities on idempotent semigroups are written by three or fewer independent variables. We consider the semigroup satisfying the property abc = ac and prove that it is left semi- normal and right quasi-normal. Again an idempotent semigroup with an identity aba = ab and aba = ba (ab = a, ab = b) is always a semilattices and normal. An idempotent semigroup is normal if and only if it is both left quasi-normal and right quasi-normal. If a semigroup is rectangular then it is left and right semi-regular. I. PRELIMINARIES AND BASIC PROPERTIES OF REGULAR SEMIGROUPS In this section we present some basic concepts of semigroups and other definitions needed for the study of this chapter and the subsequent chapters. 1.1 Definition: A semigroup (S, .) is said to be left(right) singular if it satisfies the identity ab = a (ab = b) for all a,b in S 1.2 Definition: A semigroup (S, .) is rectangular if it satisfies the identity aba = a for all a,b in S. 1.3 Definition: A semigroup (S, .) is called left(right) regular if it satisfies the identity aba = ab (aba = ba) for all a,b in S. 1.4 Definition: A semigroup (S, .) is called regular if it satisfies the identity abca = abaca for all a,b,c in S 1.5 Definition: A semigroup (S, .) is said to be total if every element of Scan be written as the product of two elements of S. i.e, S 2
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01 Jan 2010
TL;DR: The Hodgkin-Huxley Equations are applied to the model of Neuronal Networks to describe the “spatially distributed” networks.
Abstract: The Hodgkin-Huxley Equations.- Dendrites.- Dynamics.- The Variety of Channels.- Bursting Oscillations.- Propagating Action Potentials.- Synaptic Channels.- Neural Oscillators: Weak Coupling.- Neuronal Networks: Fast/Slow Analysis.- Noise.- Firing Rate Models.- Spatially Distributed Networks.
1,170 citations
Book•
12 May 2009
TL;DR: The q-theory of finite semigroups as mentioned in this paper is a theory that provides a unifying approach to finite semigroup theory via quantization, and it is the only contemporary exposition of the complete theory of the complexity of finite semiigroups.
Abstract: Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.
325 citations
Cites background from "Structure of regular semigroups. I"
...The groupoid interpretation is ours, although it clearly relates to the viewpoint of Nambooripad [204]....
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Book•
11 Mar 2005TL;DR: Renner's extension principle and classification of linear algebraic groups is discussed in detail in this paper, where Renner's Decomposition and Related Finite Semigroups are discussed.
Abstract: 1. Abstract Semigroups 2. Algebraic Geometry 3. Linear Algebraic Semigroups 4. Linear Algebraic Groups 5. Connected Algebraic Semigroups 6. Connected Algebraic Monoids 7. Reductive Groups and Regular Semigroups 8. Diagonal Monoids 9. Cross-section Lattices 10. xi-Structure 11. Renner's Decomposition and Related Finite Semigroups 12. Biordered Sets 13. Tits Building 14. The System of Idempotents 15. J-irreducible and J co-reducible Monoids 16. Renner's Extension Principle and Classification.
228 citations
01 Jan 1986
TL;DR: Denecke and Vogel as discussed by the authors have published Tools for a Theory of Partial Algebras, which has been published in 1986 by the Akademie-Verlag Berlin, Volume 32 in the series “Mathematical Research.
Abstract: 1This book was published 1986 by the Akademie-Verlag Berlin, Volume 32 in the series “Mathematical Research”; Lector was Dr. Reinhard Hoppner. It was printed in the German Democratic Republic. It has been transferred into LATEX by Ulrich Thiemann. In connection with this translation also the notation for the direction used for the composition of homomorphisms has been changed (the first morphism is written to the right of the second one, etc.). So far most of the diagrams are still missing. Also an index is still missing. The errors and misprints in the original version have not yet been corrected. The bibliography has been extended about 1990 and not yet been updated – moreover, it may still contain many titles, which treat “partial operations” (which was the keyword for the search in the Zentralblatt), but which are not really concerned with partial algebras in the sense of this book. In order to get a better idea of the material in the book, the article: Tools for a Theory of Partial Algebras, which has been published in: General Algebra and Applications (Eds.: K.Denecke and H.-J.Vogel), Research and Exposition in Mathematics, Vol. 20, Heldermann Verlag Berlin, 1993, pp. 12–32, has been added to a revised introduction.
225 citations
TL;DR: In this article, the authors investigated the relationship between certain classes of ordered small categories, introduced by Charles Ehresmann in the course of his work on local structures, and the class of U -semiabundant semigroups, first studied by El-Qallali and by de Barros.
146 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.
3,533 citations
01 Jan 1973
TL;DR: In this article, the structure of idempotent semigroups is described in terms of semilattices Q, partial chains Q of left zero semigroup, and partial chainsQ of right zero semiigroups.
Abstract: An idempotent semigroup (band) is a semigroup in which every element is an idempotent. We describe the structure of idempotent semigroups in terms of semilattices Q, partial chains Q of left zero semigroups, and partial chains Q of right zero semigroups. We also describe bands of maximal left zero semigroups in terms of partial chains Q of left zero semigroups and semilattices Q of right zero semigroups. An idempotent semigroup (band) is a semigroup in which every element is an idempotent. We describe the structure of idempotent semigroups in terms of semilattices Q, partial chains Q of left zero semigroups, and partial chains Q2 of right zero semigroups. We also describe bands of maximal left zero semigroups in terms of partial chains ?2 of left zero semigroups and semilattices Q2 of right zero semigroups. Unless otherwise specified we employ the definitions and notation of [2]. The following theorem is a starting point in the proof of both of our structure theorems. THEOREM 1 (CLIFFORD [1], MCLEAN [3]). Let E be an idempotent semigroup. Then, E is a semilattice ?2 of rectangular bands (Es: 6 E ?2). We begin by introducing the following concepts. Let W be a partial groupoid which is a union of a collection of pairwise disjoint subsemigroups (T,:6 E A) where A is a semilattice. If x c Tv, y E T,5, and 6 6 and z E T: imply (,xy)z=x(yz), W is termed an (upper) partial chain of the semigroups (Ta: 6 E A). Received by the editors August 9, 1971. AMS (MOS) subject classifications (1970). Primary 20M10. Key wvords and phrases. Idempotent semigroup, band of left zero semigroups, band, semilattice, semilattice of right zero semigroups. (a American Mathematical Society 1973 17 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:25:52 UTC All use subject to http://about.jstor.org/terms 18 R. J. WARNE [January We are now in a position to give the first theorem. Let Q be a semilattice, let I be a (lower) partial chain Q of left zero semigroups (I,: 6 e A), and let J be an (upper) partial chain Q of right zero semigroups (J,:3 6 cA). Let a be a mapping of Jx I into I and let f be a mapping of J x I into J subject to the conditions: I. If r, s e Q, (Jr x Is) ac Irs and (Jr x Is) ,c Jrs II. If j s Js, p e It, q e Jt, and m e Ig, (, p)(]j, p)fq, m)c =j, p((q, m)x))x and (j, p((q, m)c))fl(q, m),9 = ((j, p)flq, m),f. Let (Q, I, J, oc, /B) denote U (Is x J,: s e Q) under the multiplication (i,j)(p, q) = (i((j, p)(x), (j, p)fq). THEOREM 2.1 E is an idempotent semigroup if and only ifE_?(Q, I, J, X, for some collection Q, I, J, ,#I PROOF. Let E be an idempotent semigroup. Select and fix an Yclass I, of E, and select and fix an a-class J, of E, (Y and M are Green's relations [2]). Thus every element of E may be expressed uniquely in the form x= ij where i e I and j J, for some 6 e Q. If e e I,, f e I, and v? ;r satisfying I(a). We This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:25:52 UTC All use subject to http://about.jstor.org/terms 20 R. J. WARNE [January obtain I(b) from the expression (gehed)Crureuep = ur(g h)= (u,g ,)h,, =(ge9oCr)(ure,hefl) =(g%,,xr)(h,efir,,)Ure,5e where ep-
26 citations
01 Jan 1958
10 citations