Hypergroup Theory

294 Pages·2021·3.763 MB·English

by  Bijan Davvaz, Violeta Leoreanu-fotea

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s. Other World Scientific Titles by the Author
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NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
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s. Published by
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TE ISBN 978-981-124-938-9 (hardcover)
F ISBN 978-981-124-939-6 (ebook for institutions)
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E ISBN 978-981-124-940-2 (ebook for individuals)
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omstri Hypergrouptheory isafieldof algebrathat appearedin the1930s and was
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wnln 0 The book presents an updated study of hypergroups, being structured
ory DoETH) o omna1in2:csheamptiheryspsetragrrtoiunpgsw, ihtyhptehregrporuepsesn,tcaltaisosnesofofthseubbhaysipcenrgortoiounpss,intythpeesdoo-f
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pergroY ZU tions between hypergroups and binary relations, starting from connections
yG establishedbyRosenbergandCorsini. Varioustypesofbinaryrelationsare
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O highlighted, in particular equivalence relations and the corresponding quo-
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H tient structures, which enjoy certain properties: commutativity, cyclicity,
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E solvability. A special attention is paid to the fundamental β relationship,
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OF whichleadstoagroupquotientstructure. Inthefinitecase, thenumberof
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T non-isomorphicRosenberghypergroupsofsmallordersismentioned. Also,
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T thestudyofhypergroupsassociatedwithrelationsisextendedtothecaseof
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S hypergroups associated to n-ary relations. Then follows an applied excur-
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vi Hypergroup Theory
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arti updated one in the field, followed by an index of the notions presented in
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worldscientific.cd distribution is P1.refS1a.ce1emihyHpyeprgerroouppersations and hypergroupoids . . . . . . . . . . . v11
ww.e an 1.2 Semihypergroups . . . . . . . . . . . . . . . . . . . . . . 6
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wnln 0 2.2 Conjugacy class and character hypergroups . . . . . . . . 17
y DoTH) o 2.3 Hv-groups as a generalization of hypergroups . . . . . . . 20
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group TheZURICH ( 3. S3.u1bhypCelrogsreodu,pinsvertible, ultraclosed and conjugable 23
perY subhypergroups . . . . . . . . . . . . . . . . . . . . . . . 23
HyOG 3.2 Hypergroups induced by quasiordered groups . . . . . . . 27
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viii Hypergroup Theory
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arti 5.3 Abelian groups derived from hypergroups . . . . . . . . . 60
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pt for hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . 78
e 5.7 Multisemi-direct hyperproduct of hypergroups . . . . . . 82
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mitte 6. More about the Corresponding Quotient Structures 89
ot per 6.1 The relation ζe . . . . . . . . . . . . . . . . . . . . . . . . 89
n 6.2 Transitivity condition of ζ . . . . . . . . . . . . . . . . . 92
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Contents ix
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arti 9. Hypergroups and n-ary Relations 177
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m Re- binary relations . . . . . . . . . . . . . . . . . . . . . . . 209
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groZU 12.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . 239
perY 12.2 Topological hypergroupoids . . . . . . . . . . . . . . . . . 244
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arti Chapter 1
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A Semihypergroups
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wuti hypergroupoid and a semihypergroup. We present some examples to show
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Y In [57], several examples are provided from different biological points
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SI of view, and it is shown that the theory of hyperstructures exactly fits the
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Example 1.1 ABO Blood Group Inheritance. In 1900, the Austrian
physician Karl Landsteiner realized that human blood was of different
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