Some classes of planar lattices and interval-valued fuzzy sets

Abstract

<p>U radu je ispitan sledeći problem: <em>Pod kojim&nbsp;uslovima se može rekonstruisati&nbsp; (sintetisati)&nbsp;intervalno-vrednosni rasplinuti skup iz&nbsp; poznate&nbsp;familije nivo skupova.</em></p><p>U tu svrhu su proučena svojstva mreža&nbsp;intervala za svaki od četiri izabrana mrežna&nbsp;<br />uređenja: poredak po komponentama, neprecizni&nbsp;poredak (skupovna inkluzija), strogi &nbsp;i leksikografski&nbsp;poredak.&nbsp;</p><p>Definisane su i-između i ili-između ravne&nbsp;mreže&nbsp;&nbsp; i ispitana njihova svojstva potrebna za&nbsp;re&scaron;avanje postavljenog problema sinteze za&nbsp;intervalno-vrednosne rasplinute skupove. Za i-između ravne mreže je dokazano da su, u svom&nbsp;konačnom slučaju, slim mreže i dualno, da su ili-između ravne mreže dualno-slim mreže.</p><p>Data je karakterizacija kompletnih konačno&nbsp;prostornih i dualno konačno prostornih mreža.&nbsp;</p><p>Određena je klasa mreža koje se mogu&nbsp;injektivno preslikati u direktan proizvod n&nbsp;<br />kompletnih lanaca tako da su očuvani supremumi i&nbsp;dualno, određena je klasa mreža koje se mogu&nbsp;injektivno preslikati u direktan proizvod n lanaca&nbsp;tako da su očuvani infimumi.&nbsp;</p><p>U re&scaron;avanju problema sinteze posmatrana su&nbsp;dva tipa nivo skupova - gornji i donji nivo skupovi.&nbsp;Potreban i dovoljan uslov za sintezu intervalno-vrednosnog rasplinutog skupa iz poznate familije&nbsp;nivo skupova određen je za mrežu intervala koja je&nbsp;uređena poretkom po komponentama, za oba tipa&nbsp;posmatranih nivo skupova.</p><p>Za mrežu intervala uređenu nepreciznim&nbsp;poretkom, problem je re&scaron;en za donje nivo skupove,&nbsp;dok su za gornje nivo skupove određeni dovoljni&nbsp;uslovi.</p><p>Za mrežu intervala koja je uređena&nbsp;leksikografskim poretkom, takođe su dati dovoljni<br />uslovi i to za oba tipa nivo skupova.&nbsp;</p><p>Za mrežu intervala uređenu strogim poretkom&nbsp;problem nije re&scaron;avan, jer izlazi izvan okvira ovog&nbsp;rada.</p><p><br />Dobijeni rezultati su primenjeni za re&scaron;avanje&nbsp;sličnog problema sinteze za intervalno-vrednosne&nbsp;intuicionističke rasplinute skupove&nbsp; za mrežu&nbsp;intervala uređenu poretkom po komponentama.&nbsp;</p><p>Rezultati ovog istraživanja su od teorijskog&nbsp;značaja u teoriji mreža i teoriji rasplinutih skupova,&nbsp;ali postoji mogućnost za primenu u matematičkoj&nbsp;morfologiji i obradi slika.</p>

<p>In this thesis&nbsp; the following problem was investigated: Under which conditions an interval-valued fuzzy set can be reconstructed from the given family of cut sets.</p><p>We consider interval-valued fuzzy sets as&nbsp; a special type of lattice-valued fuzzy sets and&nbsp; we studied properties of lattices of intervals using four different lattice&nbsp; order: componentwise ordering, imprecision ordering (inclusion of sets), strong and&nbsp;lexicographical ordering.</p><p>We proposed new definitions&nbsp; of meet-between planar and join - between planar lattices, we investigated their properties and used them for solving problem of synthesis&nbsp; in&nbsp; interval-valued fuzzy sets.</p><p>It has been proven that finite meet- between planar lattices and slim lattices are equivalent, and dually:&nbsp;&nbsp;&nbsp; finite join-&nbsp; between planar lattices and dually slim lattices are equivalent.</p><p>Complete finitely&nbsp; spatial lattices and complete dually finitely spatial lattices are fully&nbsp;characterized&nbsp; in this setting. Next, we characterized&nbsp; lattices which can be order<br />embedded into a Cartesian product of&nbsp; n&nbsp; complete chains such that all suprema are preserved under the embedding.</p><p>And dually, we characterized lattices which can be order embedded into a Cartesian product of n complete chains such that all infima are preserved under the embedding.</p><p>We considered two types of cut sets &ndash; upper cuts and lower cuts.</p><p>Solution of the&nbsp; problem of synthesis of interval-valued fuzzy sets are given for lattices of intervals under componentwise ordering for both types of cut sets. Solution of problem of synthesis of&nbsp; interval-valued fuzzy sets&nbsp; are&nbsp; given for lower cuts for lattices of intervals under imprecision ordering.&nbsp; Sufficient conditions are given for lattices of intervals under imprecision ordering and family of upper cuts.</p><p>Sufficient conditions are also given for lattices of intervals under lexicographical ordering.</p><p>The problem of synthesis of interval-valued fuzzy sets for lattices of&nbsp; intervals under strong ordering is beyond the scope of this thesis.</p><p>A similar problem of synthesis of&nbsp; interval-valued intuitionistic fuzzy sets is solved for lattices of intervals under componentwise ordering.</p><p>These results are&nbsp; mostly of theoretical importance in lattice theory and fuzzy sets theory, but also they could&nbsp; be applied in mathematical morphology and in&nbsp; image processing.</p>