Topological minors of cover graphs and dimension

We show that posets of bounded height whose cover graphs exclude a fixed graph as a topological minor have bounded dimension. This result was already proven by Walczak. However, our argument is entirely combinatorial and does not rely on structural decomposition theorems. Given a poset with large dimension but bounded height, we directly find a large clique subdivision in its cover graph. Therefore, our proof is accessible to readers not familiar with topological graph theory, and it allows us to provide explicit upper bounds on the dimension. With the introduced tools we show a second result that is supporting a conjectured generalization of the previous result. We prove that $(k+k)$-free posets whose cover graphs exclude a fixed graph as a topological minor contain only standard examples of size bounded in terms of $k$.Comment: revised versio

Dimension of posets with planar cover graphs excluding two long incomparable chains

It has been known for more than 40 years that there are posets with planar cover graphs and arbitrarily large dimension. Recently, Streib and Trotter proved that such posets must have large height. In fact, all known constructions of such posets have two large disjoint chains with all points in one chain incomparable with all points in the other. Gutowski and Krawczyk conjectured that this feature is necessary. More formally, they conjectured that for every $k\geq 1$, there is a constant $d$ such that if $P$ is a poset with a planar cover graph and $P$ excludes $\mathbf{k}+\mathbf{k}$, then $\dim(P)\leq d$. We settle their conjecture in the affirmative. We also discuss possibilities of generalizing the result by relaxing the condition that the cover graph is planar.Comment: New section on connections with graph minors, small correction

Dimension and Ramsey results in partially ordered sets.

In this dissertation, there are two major parts. One is the dimension results on different classes of partially ordered sets. We developed new tools and theorems to solve the bounds on interval orders using different number of lengths. We also discussed the dimension of interval orders that have a representation with interval lengths in a certain range. We further discussed the interval dimension and semi dimension for posets. In the second part, we discussed several related results on the Ramsey theory of grids, the results involve the application of Product Ramsey Theorem and Partition Ramsey Theore

Some classes of planar lattices and interval-valued fuzzy sets

U radu je ispitan sledeći problem: Pod kojim uslovima se može rekonstruisati  (sintetisati) intervalno-vrednosni rasplinuti skup iz  poznate familije nivo skupova. U tu svrhu su proučena svojstva mreža intervala za svaki od četiri izabrana mrežna  uređenja: poredak po komponentama, neprecizni poredak (skupovna inkluzija), strogi  i leksikografski poredak.  Definisane su i-između i ili-između ravne mreže   i ispitana njihova svojstva potrebna za rešavanje postavljenog problema sinteze za intervalno-vrednosne rasplinute skupove. Za i-između ravne mreže je dokazano da su, u svom konačnom slučaju, slim mreže i dualno, da su ili-između ravne mreže dualno-slim mreže. Data je karakterizacija kompletnih konačno prostornih i dualno konačno prostornih mreža.  Određena je klasa mreža koje se mogu injektivno preslikati u direktan proizvod n  kompletnih lanaca tako da su očuvani supremumi i dualno, određena je klasa mreža koje se mogu injektivno preslikati u direktan proizvod n lanaca tako da su očuvani infimumi.  U rešavanju problema sinteze posmatrana su dva tipa nivo skupova - gornji i donji nivo skupovi. Potreban i dovoljan uslov za sintezu intervalno-vrednosnog rasplinutog skupa iz poznate familije nivo skupova određen je za mrežu intervala koja je uređena poretkom po komponentama, za oba tipa posmatranih nivo skupova. Za mrežu intervala uređenu nepreciznim poretkom, problem je rešen za donje nivo skupove, dok su za gornje nivo skupove određeni dovoljni uslovi. Za mrežu intervala koja je uređena leksikografskim poretkom, takođe su dati dovoljni uslovi i to za oba tipa nivo skupova.  Za mrežu intervala uređenu strogim poretkom problem nije rešavan, jer izlazi izvan okvira ovog rada. Dobijeni rezultati su primenjeni za rešavanje sličnog problema sinteze za intervalno-vrednosne intuicionističke rasplinute skupove  za mrežu intervala uređenu poretkom po komponentama.  Rezultati ovog istraživanja su od teorijskog značaja u teoriji mreža i teoriji rasplinutih skupova, ali postoji mogućnost za primenu u matematičkoj morfologiji i obradi slika.In this thesis  the following problem was investigated: Under which conditions an interval-valued fuzzy set can be reconstructed from the given family of cut sets. We consider interval-valued fuzzy sets as  a special type of lattice-valued fuzzy sets and  we studied properties of lattices of intervals using four different lattice  order: componentwise ordering, imprecision ordering (inclusion of sets), strong and lexicographical ordering. We proposed new definitions  of meet-between planar and join - between planar lattices, we investigated their properties and used them for solving problem of synthesis  in  interval-valued fuzzy sets. It has been proven that finite meet- between planar lattices and slim lattices are equivalent, and dually:    finite join-  between planar lattices and dually slim lattices are equivalent. Complete finitely  spatial lattices and complete dually finitely spatial lattices are fully characterized  in this setting. Next, we characterized  lattices which can be order embedded into a Cartesian product of  n  complete chains such that all suprema are preserved under the embedding. 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. We considered two types of cut sets – upper cuts and lower cuts. Solution of the  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  interval-valued fuzzy sets  are  given for lower cuts for lattices of intervals under imprecision ordering.  Sufficient conditions are given for lattices of intervals under imprecision ordering and family of upper cuts. Sufficient conditions are also given for lattices of intervals under lexicographical ordering. The problem of synthesis of interval-valued fuzzy sets for lattices of  intervals under strong ordering is beyond the scope of this thesis. A similar problem of synthesis of  interval-valued intuitionistic fuzzy sets is solved for lattices of intervals under componentwise ordering. These results are  mostly of theoretical importance in lattice theory and fuzzy sets theory, but also they could  be applied in mathematical morphology and in  image processing

Interval orders and dimension

We show that for every interval order X, there exists an integer t so that if Y is any interval order with dimension at least t, then Y contains a subposet isomorphic to X. c ○ 2000 Publishe