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We study several classes of finite posets equipped with linear orderings. We examine these classes according to the Ramsey and the ordering property. As an application we give several new extremely amenable groups of automorphisms of countable structures and compute several new universal minimal flows for such groups. The technique that we develop is also useful for studying classes of structures related to posets, such as quasi-orderings.

We study several classes of finite posets equipped with linear orderings. We examine these classes according to the Ramsey and the ordering property. As an application we give several new extremely amenable groups of automorphisms of countable structures and compute several new universal minimal flows for such groups. The technique that we develop is also useful for studying classes of structures related to posets, such as quasi-orderings.

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

The thesis is devoted to the study of certain combinatorial objects called \emph{non-crossing partitions}. The enumeration properties of the lattice ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ of \emph{non-crossing partitions} were studied since the work of G. Kreweras in 1972. An important feature of ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$, observed by P. Biane in 1997, is that it embeds into the symmetric group $\mathfrak{S}_n$; via this embedding, ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ is canonically identified to the interval $[\varepsilon, \gamma_o] \subseteq \mathfrak{S}_n$ (considered with respect to a natural partial order on $\mathfrak{S}_n$), where $\varepsilon$ is the unit of $\mathfrak{S}_n$ and $\gamma_o$ is the forward cycle.\\ There are two extensions of the concept of non-crossing partitions that were considered in the recent research literature. On the one hand, V. Reiner introduced in 1997 the analogue of \emph{type B} for ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$. This poset is denoted \textsf{NC$^{\textsf{\,B}}$(n)} and it is isomorphic to the interval $[\varepsilon, \gamma_o]$ of the hyperoctahedral group $B_n$, where now $\gamma_o$ stands for the natural forward cycle of $B_n$. On the other hand, J. Mingo and A. Nica studied in 2004 a set of \emph{annular} non-crossing partitions (diagrams drawn inside an annulus -- unlike the partitions from ${\textsf{NC$^{\textsf{\,A}}$(n)}}\,$ or from ${\textsf{NC$^{\textsf{\,B}}$(n)}}\,$, which are drawn inside a disc).\\ In this thesis the type B and annular objects are considered in a unified framework. The forward cycle of $B_n$ is replaced by a permutation which has two cycles, $\gamma= [1,2,\ldots,p][p+1,\ldots,p+q]$, where $p+q = n$. Two equivalent characterizations of the interval $[ \varepsilon , \gamma ] \subseteq B_n$ are found -- one of them is in terms of a \emph{genus inequality}, while the other is in terms of \emph{annular crossing patterns}. A corresponding poset \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}} of \emph{annular non-crossing partitions of type B} is introduced, and it is proved that $[\varepsilon, \gamma] \simeq \mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$, where the partial order on $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ is the usual reversed refinement order for partitions.\\ The posets $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(p, q)}\,}}$ are not lattices in general, but a remarkable exception is found to occur in the case when $q=1$. Moreover, it is shown that the meet operation in the lattice $\mbox{{\textsf{NC$^{\textsf{\,B}}$\,(n-1, 1)}\,}}$ is the usual ``intersection meet'' for partitions. Some results concerning the enumeration properties of this lattice are obtained, specifically concerning its rank generating function and its M\"{o}bius function.\\ The results described above in type B are found to also hold in connection to the Weyl groups of \emph{type D}. The poset \mbox{{\textsf{NC$^{\textsf{\,D}}$\,(n-1, 1)}\,\,}} turns out to be equal to the poset {\textsf{NC$^{\textsf{\,D}}$(n)}} constructed by C. Athanasiadis and V. Reiner in a paper in 2004. The non-crossing partitions of type D of Athanasiadis and Reiner are thus identified as annular objects.\\ Non-crossing partitions of type A are central objects in the combinatorics of free probability. A parallel concept of \emph{free independence of type B}, based on non-crossing partitions of type B, was proposed by P. Biane, F. Goodman and A. Nica in a paper in 2003. This thesis introduces a concept of \emph{scarce $\mathbb{G}$-valued probability spaces}, where $\mathbb{G}$ is the algebra of Gra{\ss}man numbers, and recognizes free independence of type B as free independence in the ``scarce $\mathbb{G}$-valued'' sense.

Techniques from the theory of matrix problems have proven to be helpful for studying problems within representation theory. In particular, matrix problems are well suited to use in problems related to classifying indecomposable representations of quivers and of posets. However, throughout the literature, there are many different types of matrix problems and little clarification of the relationships between them. In this thesis, we choose six types of matrix problems, place them all within a common framework and find correspondences between them. Moreover, we show that their use in the classification of finite-dimensional representations of quivers and posets are, in general, well-founded. Additionally, we investigate a direct relationship between the problem of classifying quiver representations and the problem of classifying poset representations.

We discuss results dealing with universal cycles (ucycles) and s-overlap cycles, and contribute to the body of those results by proving existence of universal cycles of naturally labeled posets (NL posets), s-overlap cycles of words of weight k, and juggling patterns. The result on posets is, to the best of our knowledge, the first demonstration of the existence of a ucycle whose length is unknown.

This dissertation has two principal components: the dimension of posets with planar cover graphs, and the cartesian product of posets whose cover graphs have hamiltonian cycles that parse into symmetric chains. Posets of height two can have arbitrarily large dimension. In 1981, Kelly provided an infinite sequence of planar posets that shows that the dimension of planar posets can also be arbitrarily large. However, the height of the posets in this sequence increases with the dimension. In 2009, Felsner, Li, and Trotter conjectured that for each integer h at least 2, there exists a least positive integer c(h) so that if P is a poset with a planar cover graph (the class of posets with planar cover graphs includes the class of planar posets) and the height of P is h, then the dimension of P is at most c(h). In the first principal component of this dissertation we prove this conjecture. We also give the best known lower bound for c(h), noting that this lower bound is far from the upper bound. In the second principal component, we consider posets with the Hamiltonian Cycle--Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a hamiltonian cycle which parses into w symmetric chains. This definition is motivated by a proof of Sperner's theorem that uses symmetric chains, and was intended as a possible method of attack on the Middle Two Levels Conjecture. We show that the subset lattices have the HC-SCP property by showing that the class of posets with the strong HC-SCP property, a slight strengthening of the HC-SCP property, is closed under cartesian product with a two-element chain. Furthermore, we show that the cartesian product of any two posets from this strong class has the (weak) HC-SCP property.

Forbidding and enforcing systems (fe-systems) provide a new way of defining classes of structures based on boundary conditions. Forbidding and enforcing systems on formal languages were inspired by molecular reactions and DNA computing. Initially, they were used to define new classes of languages (fe-families) based on forbidden subwords and enforced words. This paper considers a metric on languages and proves that the metric space obtained is homeomorphic to the Cantor space. This work studies Chomsky classes of families as subspaces and shows they are neither closed nor open. The paper investigates the fe-families as subspaces and proves the necessary and sufficient conditions for the fe-families to be open. Consequently, this proves that fe-systems define classes of languages different than Chomsky hierarchy. This work shows a characterization of continuous functions through fe-systems and includes results about homomorphic images of fe-families. This paper introduces a new notion of connecting graphs and a new way to study classes of graphs. Forbidding-enforcing systems on graphs define classes of graphs based on forbidden subgraphs and enforced subgraphs. Using fe-systems, the paper characterizes known classes of graphs, such as paths, cycles, trees, complete graphs and k-regular graphs. Several normal forms for forbidding and enforced sets are stated and proved. This work introduces the notion of forbidding and enforcing to posets where fe-systems are used to define families of subsets of a given poset, which in some sense generalizes language fe-systems. Poset fe-systems are, also, used to define a single subset of elements satisfying the forbidding and enforcing constraints. The latter generalizes graph fe-systems to an extent, but defines new classes of structures based on weak enforcing. Some properties of poset fe-systems are investigated. A series of normal forms for forbidding and enforcing sets is presented. This work ends with examples illustrating the computational potential of fe-systems. The process of cutting DNA by an enzyme and ligating is modeled in the setting of language fe-systems. The potential for use of fe-systems in information processing is illustrated by defining the solutions to the k-colorability problem.

Boij-Söderberg theory shows that the Betti table of a graded module can be written as a linear combination of pure diagrams with integer coefficients. In chapter 2 using Ferrers hypergraphs and simplicial polytopes, we provide interpretations of these coefficients for ideals with a d-linear resolution, their quotient rings, and for Gorenstein rings whose resolution has essentially at most two linear strands. We also establish a structural result on the decomposition in the case of quasi-Gorenstein modules. These results are published in the Journal of Algebra, see [25]. In chapter 3 we provide some further results about Boij-Söderberg decompositions. We show how truncation of a pure diagram impacts the decomposition. We also prove constructively that every integer multiple of a pure diagram of codimension 2 can be realized as the Betti table of a module. In chapter 4 we introduce the idea of a c-polar self-dual polytope. We prove that in dimension 2 only the odd n-gons have an embedding which is polar self-dual. We also define the family of Ferrers polytopes. We prove that the Ferrers polytope in dimension d is d-polar self-dual hence establishing a nontrivial example of a polar self-dual polytope in all dimension. Finally we prove that the Ferrers polytope in dimension d supports a cellular resolution of the Stanley-Reisner ring of the (d+3)-gon.