Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1996 Complex algebras of semigroups Pamela Jo Reich Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Reich, Pamela Jo, "Complex algebras of semigroups " (1996). Retrospective Theses and Dissertations. 11765. https://lib.dr.iastate.edu/rtd/11765 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfihn master. UMI fibns the text du-ectly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproductioii is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and unproper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., m^s, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overiaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600 Complex fdgebras of semigroups by Pamela Jo Reich A dissertation submitted to the graduate faculty in partial ftilfiUment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Mathematics Major Professor: CliflFord Bergman Iowa State University Ames, Iowa 1996 Copyright © Pamela Jo Reich, 1996. All rights reserved. UMI Number: 9725478 Copyright 1996 by Reich, Pamela Jo All rights reserved. UMI Microform 9725478 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 ii Graduate College Iowa State University This is to certify that the Doctoral dissertation of Pamela Jo Reich has met the dissertation requirements of Iowa State University Signature was redacted for privacy. ommittee Member Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Committee Member Signature was redacted for privacy. Major Professor Signature was redacted for privacy. For the Major Program Signature was redacted for privacy. For the Graduate College iii TABLE OF CONTENTS CHAPTER 1 INTRODUCTION I CHAPTER 2 PRELEVnNARrES 4 Universal Algebraic Definitions and Theorems 4 Boolean Algebras 5 Boolean Algebras with Operators 9 Relational Structures and the Representation Theorem for BAOs 12 CHAPTER 3 COMPLEX ALGEBRAS AND DUALITY WITH RELATIONAL STRUCTURES 16 The Complex Algebra of a Semigroup 16 Bounded Morphisms and BAG Subalgebras 17 Inner Substructures and BAG Homomorphic Images 24 Direct Products and Disjoint Unions 26 CHAPTER 4 BASIC RESULTS ON COMPLEX ALGEBRAS OF SEMIGROUPS 28 A Closure Operator and Some Results 28 Subdirectly Irreducible algebras in 'HSV{S'^) 32 Discriminator Varieties 37 CHAPTER 5 REPRESENTATIONS, REPRESENTABLE ALGEBRAS AND QUA- SIVARIETIES 40 Representations 40 Integrality 46 Representable Algebras 50 Quasivarieties 52 CHAPTER 6 THE STRUCTURE OF THE LATTICE OF SUBVARIETIES OF nsv{s-^) 54 iv Atoms of the Lattice of Subvarieties of HSV{S'*') 54 Splitting algebras and conjugate varieties 71 APPENDIX A REPRESENTATIONS OF BAOS WITH TWO ATOMS 77 APPENDIX B COVERS OF ATOMS IN LATTICE OF SUBVARIETIES 86 BIBLIOGRAPHY 91 V LIST OF TABLES TableS.l The BAO E 41 Table 5.2 The BAO G 41 Table 5.3 Example of a BAO satisfying Corollary 5.5 45 Table 5.4 Example BAO B 47 Table 6.1 The semigroup B"*" from Case 1 of Theorem 6.7 61 Table 6.2 The boimded morphic image B 65 Table 6.3 The semigroup S2 67 Table 6.4 The semigroup S3 67 Table 6.5 The semigroup T2 71 Table A.l BAO number 1 77 Table A.2 The semigroup Ai 77 Table A.3 BAO number 2 78 Table A.4 The semigroup A2 78 Table A.5 BAO number 3 78 Table A.6 The semigroup ^43 78 Table A.7 BAO number 4 79 Table A.8 The semigroup 79 Table A.9 BAO number 5 79 Table A. 10 The semigroup As 79 Table A. 11 BAO number 6 80 Table A. 12 The semigroup As 80 Table A.13 BAO number 7 80 Table A. 14 The semigroup A-; 80 Table A. 15 BAO number 8 80 vi Table A.16 The semigroup 80 Table A.17 B.AO number 9 80 Table A.18 The semigroup Aq 80 Table A.19 BAO number 10 80 Table A.20 BAO number 11 81 Table A.21 BAO number 12 81 Table A.22 BAO number 13 81 Table A.23 BAO number 14 81 Table A.24 The semigroup/In 81 Table A.25 BAO number 15 81 Table A.26 The semigroup/lis 81 Table A.27 BAO number 16 82 Table A.28 The semigroup .4i6 82 Table A.29 BAO number 17 82 Table A.30 The semigroup An 82 Table A.31 BAO number 18 82 Table A.32 The semigroup Ais 82 Table A.33 BAO number 19 83 Table A.34 The semigroup .^19 83 Table A.35 BAO number 20 83 Table A.36 BAO number 21 83 Table A.37 The semigroup A21 83 Table A.38 BAO number 22 84 Table A.39 BAO number 23 84 Table A.40 BAO number 24 84 Table A.41 The semigroup A24 84 Table A.42 BAO number 25 84 Table A.43 BAO number 26 84 Table A.44 BAO number 27 84 Table A.45 BAO number 28 85 Table B.l Cover IG 86 Table B.2 Cover 2G 86 vii Table B.3 Cover 3G 87 Table B.4 Cover 4G 87 Table B.5 Cover 5G 87 Table B.6 Cover 6G 87 Table B.7 Cover 7G 87 Table B.8 Cover 8G 88 Table B.9 Cover 9G 88 Table B.IO Cover IE 89 Table B.ll Cover 2E 89 Table B.12 Cover 3E 89 Table B.13 Cover 4E 90 Table B.14 Cover 5E 90 Table B.15 Cover 6E 90 Table B.16 Cover 7E 90 Table B.17 Cover 8E 90 viii ACKNOWLEDGEMENTS I would like to thank Professor Cliff Bergman. He found a problem in which I was (and still am) very interested, and was always there to help, inspire, and encourage. I very much appreciate his patience and sense of humor. I had many other professors and friends at Iowa State who deserve thanks for all the years of help and encouragement. I would also like to thank my family and friends for putting up with me all the years I was in graduate school; in particular, my husband, Matt. He has probably suffered more during my graduate school years than I did! Thank you all. 1 CHAPTER 1 INTRODUCTION Boolean algebras have been studied for almost 150 years, beginning with G. Boole in the L850s, and ciUminating in the famous Stone Representation Theorem in the 1930s, which proves that every Boolean algebra cein be thought of as a field of sets. Stone also established the connection between Boolean algebras, Boolean spaces, and Boolean rings. This enables us to think of Boolean algebras from a topological point of view as well as from an algebraic and a logical point of view. In fact, the topological viewpoint has several advantages, and further representation theorems were obtained by looking at Boolean algebras from this direction. Boolean algebras with operators (BAOs) are Boolean Jilgebras upon which additional fimctions with special properties, called operators, have been defined. They developed out of the theory of relation algebras, which in turn evolved out of the theory of binziry relations. Each new concept is a generalization of its precursor: the set of all binary relations on a set X endowed with certain operations is a relation algebra. Furthermore, every relation algebra is a Boolean algebra with an additional binary operation, uneiry operation, and constant (so it is a BAO). Binary relations were first studied as early as the mid 1800s. See Schroder [17] for more information on the early work in binary relation theory. Tarski began looking at relation algebras from an axiomatic point of view and presented a set of axioms defining them in his 1941 paper [19]. He continued generalizing these ideas, and in 1951 Jonsson and Tarski [11] first presented the concept of Boolean algebras with operators. Boolean algebras with one additional unary operator have been studied quite a lot in the last 40 years. Included in this class of algebras (among others) are modal algebras, closure algebras, and monadic algebras. BAOs with two conjugate unary operators have also been studied.