The breadth of a Lie algebra $L$ is defined to be the maximal dimension image $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into three families algebras by posets and provide combinatorial formulas members each family.

We proceed with the study of net bundles over partially ordered sets (posets), defined as the analogues of ordinary fibre bundles. To this end, we analyze the connection between homotopy, net homology and net cohomology of a poset, providing versions of classical Hurewicz theorems. Focusing our attention to Hilbert net bundles, we define the K-theory of a poset and introduce functions over the ...

For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of ...

Contents Poset Topology: Tools and Applications 1 Introduction 3 Lecture 1. Basic definitions, results, and examples 5 1.1. Order complexes and face posets 5 1.2. The Möbius function 9 1.3. Hyperplane and subspace arrangements 11 1.4. Some connections with graphs, groups and lattices 16 1.5. Poset homology and cohomology 17 1.6. Top cohomology of the partition lattice 19 Lecture 2. Group action...

The dimension of a poset (X, P) is the minimum number of linear extensions of P whose intersection is P. A poset is irreducible if the removal of any point lowers the dimension. If A is an antichain in X and X A # B’, then dim X < 2 width (X A) + 1. We construct examples to show that this inequality is best possible; these examples prove the existence of irreducible posets of arbitrarily large ...

In this paper we study quotients of posets by group actions. In order to define the quotient correctly we enlarge the considered class of categories from posets to loopfree categories: categories without nontrivial automorphisms and inverses. We view group actions as certain functors and define the quotients as colimits of these functors. The advantage of this definition over studying the quoti...

We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the Möbius function and topology of such pattern posets. We prove our results using a poset fibration based on the embeddings of the poset, where embeddings are representations of occurrences. We show that the Möbius ...