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Beyond Regular Semigroups Beyond Regular Semigroups Yan Hui Wang PhD University of York Department of Mathematics March 2012 Abstract The topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasi- adequate, concordant, abundant, restriction, Ehresmann and weakly abundant semigroups. A semigroup S with subset of idempotents U is weakly U-abundant if every RU -class and every LU -class contains an idempotent of U, where RU and LU are relations extending the well known Green’s relations R and L. We assume throughout that our semigroups satisfy a condition known as the Con- gruence Condition (C). We take several approaches to weakly U-abundant semigroups. Our first re- sults describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly U- abundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectan- gular bands of monoids. The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly B-orthodox semigroups, where B is a band. We note that if B is a semilattice then a weakly B-orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly B-orthodox semigroup S as a spined product of a fundamental weakly B- orthodox semigroup SB (depending only on B) and S/γB, where B is isomorphic to B and γB is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form δB for γB, which simplifies our result to a straightforward form. i For the above to work smoothly in the case S is weakly B-superabundant, we need to find a canonical fundamental weakly B-superabundant subsemigroup of SB. This we do, and give the corresponding answers in the case of the Hall semigroup WB and a number of intervening semigroups. We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a reg- ular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad’s work to weakly U-regular semigroups, that is, weakly U-abundant semigroups with (C) and U generating a regular subsemigroup whose set of idem- potents is U. As an intervening step we consider weakly B-orthodox semigroups in this light. We take two approaches. The first is via a new construction of an induc- tive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids. We show that the category of weakly B-orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is in- fluenced by that of Nambooripad, however, there are significant differences in strategy, the first being the introduction of generalised categories and the second being that it is more convenient to consider (generalised) categories equipped with pre-orders, rather than with partial orders. Our work may be regarded as extending a result of Lawson for Ehresmann semigroups. We also examine the trace of a weakly B-orthodox semigroup, which is a primitive weakly B-orthodox semigroup. We then take a more ‘traditional’ approach to weakly B-orthodox semigroups via band categories and weakly orthodox categories over a band, equipped with two pre-orders. We show that the category of weakly B-orthodox semigroups is equivalent to the category of weakly orthodox categories over bands. To do so we must substantially adjust Armstrong’s method for concordant semigroups. Finally, we consider the most general case of weakly U-regular semigroups. Following Nambooripad’s theorem, which establishes a correspondence between algebraic structures (inverse semigroups) and ordered structures (inductive group- oids), we build a correspondence between the category of weakly U-regular semi- groups and the category of weakly regular categories over regular biordered sets, equipped with two pre-orders. ii Contents Abstract i Contents vi List of Figures vii Preface viii Acknowledgments xiii Author’s Declaration xv 1 Basic Theory I 1 1.1 Relations................................ 1 1.2 Orderedsets.............................. 4 1.3 SemigroupsandGreen’srelations . 5 1.3.1 Basicdefinitions........................ 5 1.3.2 Green’srelations ....................... 8 1.3.3 Regularsemigroups. 11 1.3.4 Inversesemigroups . 13 1.3.5 Orthodoxsemigroups. 14 1.3.6 Completely regular semigroups . 15 1.4 Biorderedsets............................. 17 1.5 Categories............................... 26 2 Basic Theory II 32 2.1 Abundantsemigroups. 32 2.1.1 Relations L∗, R∗ ....................... 32 iii 2.1.2 Abundantsemigroups . 34 2.2 Weakly U-abundantsemigroups . 35 2.2.1 Relations LU , RU ....................... 35 2.2.2 Weakly U-abundantsemigroups . 36 2.2.3 Weakly U-regularsemigroups . 39 2.2.4 Weakly B-orthodoxsemigroups . 41 2.2.5 Ehresmannsemigroups . 41 2.3 The idempotent connected condition . 42 2.3.1 (WIC),(IC)and(PIC). 43 2.3.2 Specialcases.......................... 46 2.4 An analogue of the least inverse congruence . 50 2.5 Orders................................. 53 2.5.1 The weakly U-abundantcase . .. 53 2.5.2 The weakly U-regularcase .................. 55 2.5.3 The weakly B-orthodoxcase. 56 2.5.4 TheEhresmanncase . 56 2.6 Examples ............................... 57 3 Weakly U-superabundant semigroups with (C) 63 3.1 Weakly U-superabundant semigroups with (C) . 63 3.2 Completely JU -simplesemigroups . 67 4 Representations for generalised orthogroups 70 4.1 Fundamental inverse semigroups . 70 4.2 A fundamental orthogroup of WB .................. 72 4.3 A fundamental weakly B-superabundant subsemigroup of VB .. 84 4.4 A fundamental weakly B-superabundant subsemigroup of UB .. 89 4.5 A fundamental weakly B-superabundant subsemigroup of SB .. 97 5 Structure theorems for weakly B-orthodox semigroups 103 5.1 The least admissible Ehresmann congruence . 103 5.2 A structure theorem for weakly B-orthodox semigroups . 106 5.3 Weakly B-orthodoxsemigroupswith(N) . 110 5.4 Weakly B-superabundant semigroups with (C) . 112 5.5 Examples ............................... 113 iv 6 Correspondence between algebraic structures and ordered struc- tures 117 6.1 Inverse semigroups and inductive1 groupoids . 117 6.2 Concordant semigroups and inductive2 cancellative categories . 119 6.3 Ehresmann semigroups and Ehresmann categories . 122 7 Beyond orthodox semigroups I: weakly B-orthodox semigroups and generalised categories 126 7.1 Inductive generalised categories . 126 7.2 Construction ............................. 131 7.3 Correspondence ............................ 136 7.4 Specialcases.............................. 140 8 Trace of weakly B-orthodox semigroups 147 8.1 Preliminaries ............................. 147 8.2 BlockedReesmatrixsemigroups. 148 8.3 Primitive weakly B-orthodoxsemigroups . 152 8.4 Trace of weakly B-orthodoxsemigroups. 158 9 Beyond orthodox semigroups II: weakly B-orthodox semigroups and categories 161 9.1 Weaklyorthodoxcategories . 161 9.2 Construction ............................. 168 9.3 Correspondence ............................ 175 9.4 Specialcases.............................. 184 10 Weakly U-regular semigroups 195 10.1 Weaklyregularcategories . 195 10.2 Structuretheorems . 203 10.3 Correspondence . 222 11 Special kinds of weakly U-regular semigroups 233 11.1 Weakly U-regularsemigroupswith(WIC) . 233 11.2 Theabundantcase . .. .. 235 11.3 Theconcordantcase . 238 11.4 Theregularcase. .. .. 260 v Bibliography 262 vi List of Figures 1 Thestructureofthisthesis. xii 1.1 The τ-commutativecondition . 25 1.2 Thenaturaltransformationproperty . 31 2.1 Classesofsemigroups. 50 5.1 Thespinedproduct. .. .. 106 5.2 The structure of weakly B-orthodoxsemigroups . 108 7.1 Maps.................................. 127 9.1 A natural transformation of IWO and CS .............. 183 9.2 A natural transformation of IWOC and SC ............. 184 10.1 A natural transformation of IWRS and CS ............. 231 10.2 A natural transformation of IWRC and SC ............. 232 vii Preface The aim of this thesis is to investigate weakly U-abundant semigroups, using both techniques developed for regular and abundant semigroups, and new ones. The relevant definitions concerning the classes of semigroups in question are given in Chapter 2. Fundamental semigroups, that is, regular semigroups having no non-trivial idempotent separating congruences, have played an important role in the struc- ture theory of regular semigroups, especially, in the study of inverse semigroups. As an extension of Munn’s approach to inverse semigroups, Hall constructed the fundamental semigroup WB depending only on B, which is a subsemigroup of OP(B/L) × OP ∗(B/R), where for any partially ordered set X, OP(X) is the monoid of its order preserving selfmaps, with dual OP ∗. Also, he showed that if S is an orthodox semigroup with band of idempotents B, then there exists a morphism φ : S → WB whose kernel is µ, the maximum idempotent separating congruence on S. Consequently, an orthodox semigroup S with band of idem- potents B is fundamental if and only if it is isomorphic to a full subsemigroup of
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