Morita equivalence of partially ordered semigroups
Date
2011-05-24
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Abstract
Väitekirjas uuritakse, kas ja kuivõrd on võimalik üldistada Morita ekvivalentsuse mõistet ja teooriat poolrühmadelt osaliselt järjestatud poolrühmadele. Morita teooria põhiidee on taandada ühe struktuuri uurimine teise, temaga Morita-ekvivalentse ja loodetavasti lihtsama struktuuri uurimisele.
Töös defineeritakse osaliselt järjestatud poolrühmade tugev Morita ekvivalentsus nn Morita kontekstide abil. Kahte osaliselt järjestatud poolrühma S ja T nimetatakse tugevalt Morita-ekvivalentseteks siis, kui leidub sürjektiivsete kujutustega unitaarne Morita kontekst (S, T, P, Q, <−, −>, [−,−]).
Siis esitatakse rida konstruktsioone ja näidatakse, et kui poolrühmadelt S ja T eeldada erinevate lokaalsete ühikute olemasolu, siis on tugev Morita ekvivalentsus samaväärne nii nende konstruktsioonide (Cauchy täieldid, järjestatud Reesi maatrikspoolrühmad, järjestatud Morita poolrühmad) abil saadud seostega kui ka teatud järjestatud polügoonide kategooriate vahelise ekvivalentsusega.
Peale eelmainitud tingimuste samaväärsuse uuritakse väitekirjas nn. Morita invariante ehk omadusi, mis kanduvad osaliselt järjestatud poolrühmalt temaga tugevalt Morita-ekvivalentsetele osaliselt järjestatud poolrühmadele. Sealhulgas leitakse tarvilikud ja piisavad tingimused selleks, et üks osaliselt järjestatud poolrühm oleks tugevalt Morita-ekvivalentne osaliselt järjestatud poolrühmaga mõnest tuntud klassist (nt. osaliselt järjestatud inverssed monoidid, osaliselt järjestatud ristkülikpoolrühmad, lineaarselt järjestatud poolrühmad).
The dissertation investigates whether and to what degree it is possible to generalize the notion and theory of Morita equivalence from semigroups to partially ordered semigroups. The principal idea of Morita theory is to reduce the study of a structure to the study of a potentially simpler Morita equivalent structure. In the thesis, strong Morita equivalence of partially ordered monoids is defined via so-called Morita contexts. Two partially ordered semigroups S and T are said to be strongly Morita equivalent if there exists a unitary Morita context (S, T, P, Q, <−, −>, [−,−]) with surjective mappings. Then we introduce a number of constructions and show that if the semigroups S and T have various local units, then strong Morita equivalence is equivalent both to relations defined via those structures (Cauchy completions, ordered Rees matrix semigroups, ordered Morita semigroups) and to the equivalence of certain categories of ordered acts. Besides the above equivalent conditions, the dissertation also makes a study of Morita invariants or properties that a partially ordered semigroup passes on to those partially ordered semigroups that are strongly Morita equivalent to it. Among others, we find necessary and sufficient conditions for a given partially ordered semigroup to be strongly Morita equivalent to a partially ordered semigroup from a well-known class (e.g. partially ordered inverse monoids, partially ordered rectangular bands, linearly ordered semigroups).
The dissertation investigates whether and to what degree it is possible to generalize the notion and theory of Morita equivalence from semigroups to partially ordered semigroups. The principal idea of Morita theory is to reduce the study of a structure to the study of a potentially simpler Morita equivalent structure. In the thesis, strong Morita equivalence of partially ordered monoids is defined via so-called Morita contexts. Two partially ordered semigroups S and T are said to be strongly Morita equivalent if there exists a unitary Morita context (S, T, P, Q, <−, −>, [−,−]) with surjective mappings. Then we introduce a number of constructions and show that if the semigroups S and T have various local units, then strong Morita equivalence is equivalent both to relations defined via those structures (Cauchy completions, ordered Rees matrix semigroups, ordered Morita semigroups) and to the equivalence of certain categories of ordered acts. Besides the above equivalent conditions, the dissertation also makes a study of Morita invariants or properties that a partially ordered semigroup passes on to those partially ordered semigroups that are strongly Morita equivalent to it. Among others, we find necessary and sufficient conditions for a given partially ordered semigroup to be strongly Morita equivalent to a partially ordered semigroup from a well-known class (e.g. partially ordered inverse monoids, partially ordered rectangular bands, linearly ordered semigroups).
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dissertatsioonid, osaliselt järjestatud hulgad, poolrühmad, homoloogiline algebra, partially ordered sets, homological algebra