For instance, for real and complex hyperplane face monoids, the subposets B[X, Y) are the face posets
of spheres (see [section][section] 5.3-5.4).
 Das, Tarun, On posets
of certain classes of maps, Math.
Theorem 3.4 s[[LAMBDA].sub.y]s that if a permutation has a principal order ideal that can decompose nontrivially into a direct product of posets
, then there are ways for that principal order ideal to appear as an interval in an "interesting" way, as described in Remark 1.6.
Quasisymmetric functions were first defined by Gessel [Ges84] in the 1980s as weight enumerators for labelled posets
. Since then, the Hopf algebra of quasisymmetric functions, QSym, has arisen in a variety of contexts including the study of Lie representations, riffle shuffles, random walks, and the representation theory of Hecke algebras.
Matveev presents a set of problems in combinatorics, combinatorial optimization, posets
, graphs, elementary number theory, and other areas that represent a far-reaching extension of the arsenal of committee methods in pattern recognition.
The following notion of regularity is borrowed from Alfuraidan and Khamsi in  that considered it for posets
Subsequently, a new proof was found by Bjorner and Wachs [2, Theorem 6.1] using their theory of CL-shellable posets
. A simpler proof of Farmer's result was given by Kerz [6, Theorem 1] in 2004.
In section 2, one considers examples and properties of the Stokes posets
. In particular, one proves that the Tamari lattice is recovered as a special case, for a very regular quadrangulation.
In a standard way, every poset
can be considered as a category, and monotone mappings between posets
can be considered as functors.
The former class is associated to finite posets
, while the latter corresponds to dedekind complete finite posets
further generalized the concept of countably approximating lattices to the concept of countably approximating posets
and characterized countably approximating posets
via the cr-Scott topology.