On Morita equivalence of semigroups
Date
2023-05-18
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Abstract
Kaks isomorfset poolrühma on eristamatud. Siiski, kõikide poolrühmade kirjeldamine isomorfismi täpsuseni pole mõistlik ettevõtmine - neid on liiga palju. Öeldakse, et kaks poolrühma on Morita ekvivalentsed, kui kõikide püsivate parempoolsete polügoonide kategooriad üle nende poolrühmade on ekvivalentsed. Selline ekvivalentsiseos on oluliselt nõrgem isomorfismiseosest. Käesolevas töös uurime poolrühmade Morita ekvivalentsust kasutades selleks teiste autorite poolt uuritud algebralisi konstruktsioone kirjeldamaks Morita ekvivalentseid poolrühmi mitmetel poolrühmade alamklassidel. Meie eesmärgid on kirjeldada mõnede hästi tuntud poolrühmade, nagu rühmad või monoidid, tugevaid Morita ekvivalentsiklasse ning leida Morita invariante - need on parajasti kõikide ühest ja samast ekvivalentsiklassist pärit poolrühmade ühised omadused. Täpsemalt, näitame muu hulgas ära, et faktoriseeruv poolrühm on Morita ekvivalentne monoidiga parajasti siis, kui ta on selle monoidi laiend. Järeldame, et rühma laiendid on parajasti Rees'i maatrikspoolrühmad üle selle rühma ning et täiesti lihtsus on faktoriseeruvate poolrühmade Morita invariant. Kaks faktoriseeruvat poolrühma on Morita ekvivalentsed parajasti siis, kui nende vahel leidub unitaarne sürjektiivne Morita kontekst. Kahes peatükis uurime mõningaid Morita kontekstiga määratud morfisme ning võresid. Kirjeldame ära püsivate poolrühmade Morita ekvivalentsuse ning näitame, et Morita ekvivalentsetel poolrühmadel, mis mõlemad sisaldavad ühiseid nõrkasid lokaalseid ühikelemente, on isomorfsed kooskõlaliste seoste võred. Viimane peatükk on pühendatud poolrühmade perfektsuse uurimisele. Üldistame mitmeid perfektsete monoidide kirjeldused faktoriseeruvate poolrühmade juhule ning lisame ka uue kirjelduse. Saadud tulemuste põhjal järeldame, et kõik nilpotentsed ja täiesti (0-)lihtsad poolrühmad on perfektsed ning et perfektsus on Morita invariant faktoriseeruvate poolrühmade korral.
Two semigroups are regarded to be one and the same if they are isomorphic. Up to isomorphism, however, there are far too many semigroups for it to be reasonable to try and characterise them. Two semigroups are said to be Morita equivalent if the categories of all firm right acts over them are equivalent. Such a relation is a significantly weaker equivalence relation on the class of all semigroups than isomorphism. The purpose of this thesis is to study Morita equivalence of semigroups in terms of algebraic constructions used by other authors to describe Morita equivalence for various subclasses of semigroups. The objectives are to describe strong Morita equivalence classes of some well known semigroups like groups or monoids and also determine Morita invariants - these are properties that are shared by all semigroups in the same Morita equivalence class. Specifically, we shall show among other things that a factorisable semigroup is Morita equivalent to a monoid if and only if it is an enlargement of that monoid. Consequently, the enlargements of a group are precisely Rees matrix semigroups over that group and complete simplicity is a Morita invariant for factorisable semigroups. Morita equivalence of two factorisable semigroups occurs if and only if the semigroups are connected by a unitary surjective Morita context. Two chapters are reserved for the study of some morphisms and lattices induced by a Morita context. We shall obtain a description for the Morita equivalence of firm semigroups as well as show that Morita equivalent semigroups with common weak local units must have isomorphic lattices of compatible relations. The final chapter is devoted to the perfection for semigroups. We generalise many known descriptions of perfect monoids to the case of factorisable semigroups and add one more to the list. Our results allow us to conclude that the class of perfect semigroups contains all nilpotent semigroups, all completely (0-)simple semigroups and that perfection is a Morita invariant for factorisable semigroups.
Two semigroups are regarded to be one and the same if they are isomorphic. Up to isomorphism, however, there are far too many semigroups for it to be reasonable to try and characterise them. Two semigroups are said to be Morita equivalent if the categories of all firm right acts over them are equivalent. Such a relation is a significantly weaker equivalence relation on the class of all semigroups than isomorphism. The purpose of this thesis is to study Morita equivalence of semigroups in terms of algebraic constructions used by other authors to describe Morita equivalence for various subclasses of semigroups. The objectives are to describe strong Morita equivalence classes of some well known semigroups like groups or monoids and also determine Morita invariants - these are properties that are shared by all semigroups in the same Morita equivalence class. Specifically, we shall show among other things that a factorisable semigroup is Morita equivalent to a monoid if and only if it is an enlargement of that monoid. Consequently, the enlargements of a group are precisely Rees matrix semigroups over that group and complete simplicity is a Morita invariant for factorisable semigroups. Morita equivalence of two factorisable semigroups occurs if and only if the semigroups are connected by a unitary surjective Morita context. Two chapters are reserved for the study of some morphisms and lattices induced by a Morita context. We shall obtain a description for the Morita equivalence of firm semigroups as well as show that Morita equivalent semigroups with common weak local units must have isomorphic lattices of compatible relations. The final chapter is devoted to the perfection for semigroups. We generalise many known descriptions of perfect monoids to the case of factorisable semigroups and add one more to the list. Our results allow us to conclude that the class of perfect semigroups contains all nilpotent semigroups, all completely (0-)simple semigroups and that perfection is a Morita invariant for factorisable semigroups.
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Keywords
semigroups (math.), equivalence relations, category theory, monoids, algebraic structures