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A Study of Green's Relations on Algebraic Semigroups

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A Study of Green's Relations on Algebraic Semigroups Western University Scholarship@Western Electronic Thesis and Dissertation Repository 8-19-2015 12:00 AM A Study Of Green’s Relations On Algebraic Semigroups Allen O'Hara The University of Western Ontario Supervisor Lex Renner The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Allen O'Hara 2015 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Algebraic Geometry Commons Recommended Citation O'Hara, Allen, "A Study Of Green’s Relations On Algebraic Semigroups" (2015). Electronic Thesis and Dissertation Repository. 3047. https://ir.lib.uwo.ca/etd/3047 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. A STUDY OF GREEN’S RELATIONS ON ALGEBRAIC SEMIGROUPS (Thesis format: Monograph) by Allen O’Hara Graduate Program in Mathematics A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Allen O’Hara 2015 Acknowledgements First and foremost, I would like to thank my supervisor, Lex Renner, who was phenomenal in his support and encouragement. Having been one of the first in the field of linear algebraic monoids, I had much to learn from him and he made it not a task but a pleasure. Our meetings have been fantastic and he was always eager to share his wisdom and enthusiasm. Many thanks go to my colleagues, not just those from Western but all those I have worked with in my academic career. Each of them helped shape the mathematician I am today and will be in the future. I would like to thank my friends and family for their love and support. Special mention goes to my parents, who taught me to dream and to never stop until I exceeded my potential. Lastly, I wish to acknowledge my darling wife, Elise. She is the light of my life, and when I am with her I feel like I can accomplish anything. I would like to dedicate this work to her. It has been a hell of a ride. I cannot wait to see where I go next! ii Abstract The purpose of this work is to enhance the understanding regular algebraic semigroups by considering the structural influence of Green’s relations. There will be three chief topics of discussion. ◦ Green’s relations and the Adherence order on reductive monoids ◦ Renner’s conjecture on regular irreducible semigroups with zero ◦ a Green’s relation inspired construction of regular algebraic semigroups Primarily, we will explore the combinatorial and geometric nature of reductive monoids with zero. Such monoids have a decomposition in terms of a Borel subgroup, called the Bruhat decomposition, which produces a finite monoid, R, the Renner monoid. We will explore the structure of R by way of Green’s relations. In particular, we will be exploring the nature of the Adherence order poset, (R; ≤) when restricted to J -, R-, L -, and H -classes. From reductive monoids we broaden the impact of Green’s relations and explore regular algebraic semigroups. Specifically, we resolve Renner’s conjecture and show that the supports, X` = J =R and Xr = J =L are projective varieties. Spurred on by the result, we use in- variant theory to generalise the Rees matrix construction for algebraic semigroups to construct irreducible regular semigroups with 0. Our construction will start with specified maximal classes, Re, Le, and He and reconstruct an entire semigroup. In a lengthy example, we will use some of our previous combinatorial results to apply the construction to a natural generalisation of determinantal varieties. Highlights include the unique “vanilla form” decomposition for elements of the Renner monoid (Definition 5.36), a proof of Renner’s conjecture on the projectiveness of supports for irreducible regular semigroups with zero (Theorem 8.40), and the construction of irreducible regular semigroups from prespecified maximal R- and L -classes (Definition 9.6). Keywords: Algebraic semigroups, reductive monoids, Green’s relations, Adherence order, Renner monoid, semigroup supports, irreducible regular algebraic semigroups with zero iii Contents Acknowlegements .................................... ii Abstract ......................................... iii List of Appendices ................................... vii 1 Introduction .................................... 1 2 Background .................................... 4 2.1 Green’s Relations ............................. 5 2.2 Normality and Regularity ......................... 8 2.3 Reductive Monoids ............................ 11 2.4 Example .................................. 15 3 A New Trichotomy ................................ 19 3.1 Previous Results ............................. 19 3.2 The Trichotomy .............................. 23 3.3 Example .................................. 28 4 Fat Green’s Relations ............................... 33 4.1 Equivalent Definitions .......................... 34 4.2 Fat J -classes ............................... 36 4.3 Fat L -Classes and Fat R-Classes .................... 38 4.4 Fat H -Classes .............................. 41 4.5 Example .................................. 43 iv 5 Vanilla Form .................................... 46 5.1 Coset Posets ................................ 46 5.2 Standard Forms .............................. 50 5.3 Vanilla Form ............................... 57 5.4 Example .................................. 63 6 Maximum and Minimum Elements ........................ 67 6.1 Maximum and Minimum Elements .................... 67 6.2 Relative Maxima ............................. 75 6.3 Relative Minima ............................. 80 6.4 Example .................................. 84 7 Parabolic Green’s Relations ............................ 87 7.1 A Series Of Equivalence Relations .................... 87 7.2 Generalized Trichotomy ......................... 94 7.3 Relative Maxima and Minima ...................... 99 7.4 Example ..................................106 8 Projective Supports ................................110 8.1 Rees Theorem And Quotients In Linear Algebraic Semigroups . 110 8.2 Geometric Invariant Theory . 112 8.3 Putcha’s Determinant . 118 8.4 The Conjecture ..............................121 8.5 Examples .................................125 9 A New Way To Construct Regular Semigroups . 127 9.1 Affinized Quotients ............................127 9.2 Constructing Semigroups With Green’s Relations . 130 9.3 Normality .................................135 9.4 Determinantal Varieties . 139 v 10 Concluding Remarks ...............................146 Bibliography .......................................148 Appendix ........................................151 A.1 Results From Other Sources . 151 A.2 Opposite Standard Form . 152 A.3 Pointed Parabolic J -classes on M3(K) . 155 Curriculum Vitae ....................................172 vi List of Appendices Results From Other Sources ................................151 Opposite Standard Form ..................................152 Pointed Parabolic J -classes on M3(K) . 155 vii 1 1 Introduction The systematic investigation of linear algebraic semigroups and reductive algebraic monoids was pioneered around 1980 by Mohan Putcha and my supervisor, Lex Renner. Since then, the discipline has blossomed into a coherent branch of algebra involving embedding theory, repre- sentation theory and algebraic combinatorics. One of the most important features of a reductive monoid is the existence of the Bruhat Decomposition. More precisely, the Bruhat decomposi- tion, which is much studied for groups, extends to a perfect analogue for reductive monoids. This monoid Bruhat decomposition allows us to express the monoid as a disjoint union of double cosets (for a given Borel subgroup) indexed by a finite structure called the Renner monoid (the monoid analogue of the Weyl group). The importance of this decomposition is that it allows many questions about the nature and structure of the monoid to become simpler questions about the Renner monoid. Unlike groups, semigroups and monoids bring an additional structure in the form of Greens relations, which characterise the elements of the semigroup in terms of the ideals they generate. So important are Greens relations that Scottish semigroup theorist, John Mackintosh Howie, once said, “on encountering a new semigroup, almost the first question one asks is ‘What are the Green relations like?’ ”. In reductive monoids, we denote these relations by J , L , R and H . A natural question is how do these relations interact with the Bruhat decomposition? What additional information can they tell us? Of particular focus is the H relation, which has many interesting and desirable proper- ties. H -classes most closely resemble groups (indeed the H -class of an idempotent element forms a group), and so their structure is the one most likely to form a bridge between the Ren- ner monoid and the information we know from the better understood Weyl groups. One way we can investigate this structure is by decomposing our monoid, not in terms of the Bruhat decomposition, which is indexed by elements of the Renner monoid, but in terms of a disjoint union of double cosets of the H -classes of the Renner monoid. These are the so-called fat H -classes. In the following section of
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