We consider the class Pn of labeled posets on n elements which avoid certain three-element induced subposets. We show that the number of posets in Pn is (n+1) by exploiting a bijection between Pn and the set of regions of the arrangement of hyperplanes in R of the form xi&xj=0 or 1 for 1 i< j n. It also follows that the number of posets in Pn with i pairs (a, b) such that a<b is equal to the nu...

We study the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point ...

One of central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of mo...

We introduce a special type of graph embedding called book embedding and apply it to posets. A book embedding scheme for bipartite graphs is given, and is used to extend the embeddings to general k-partite graphs. Finally, we view the Hasse diagram of a poset as a directed k-partite graph and use this scheme to derive a book embedding for arbitrary posets. Using this book embedding scheme, we a...

In this paper we attempt to find and investigate the most general class of posets which satisfy a properly generalized version of the Hofmann-Mislove theorem. For that purpose, we generalize and study some notions (like compactness, the Scott topology, Scott open filters, prime elements, the spectrum etc.), and adjust them for use in general posets instead of frames. Then we characterize the po...

The closure of the convex cone generated by all flag f -vectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chain-enumeration functionals on this class of posets. These are in one-to-one correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove ...

We introduce a new type of dcpo-completion of posets, called D-completion. For any poset P , the D-completion exists, and P and its Dcompletion have the isomorphic Scott closed set lattices. This completion is idempotent. A poset P is continuous (algebraic) if and only if its D-completion is continuous(algebraic). Using the D-completion, we construct the local dcpocompletion of posets, that rev...

Recently a new kind of approximation to continuum topological spaces has been introduced, the approximating spaces being partially ordered sets (posets) with a finite or at most a countable number of points. The partial order endows a poset with a nontrivial non-Hausdorff topology. Their ability to reproduce important topological information of the continuum has been the main motivation for the...

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 ...

Stanely, R.P., f-vectors and h-vectors of simplicial posets, Journal of Pure and Applied Algebra 71 (1991) 319-331. A simplicial poset is a (finite) poset P with d such that every interval [6, x] is a boolean algebra. Simplicial posets are generalizations of simplicial complexes. The f-vector f(P) = (f,, f,, , ,f_,) of a simplicial poset P of rank d is defined by f; = #{x E P: [6, x] g B,, I}, ...