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Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size
$n^{1/(r+1)}$
.
This bound is known to be tight for
$r=1$
. The question whether it is optimal for
$r\ge 2$
was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for
$r=2$
, as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size
$n^{1/3+o(1)}$
.
Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size
$cn/{(log n)^r}$
. With this, we improve a result of Fox and Pach.
Let
$X$
be a topological space. We consider certain generalized configuration spaces of points on
$X$
, obtained from the cartesian product
$X^{n}$
by removing some intersections of diagonals. We give a systematic framework for studying the cohomology of such spaces using what we call ‘twisted commutative dg algebra models’ for the cochains on
$X$
. Suppose that
$X$
is a ‘nice’ topological space,
$R$
is any commutative ring,
$H_{c}^{\bullet }(X,R)\rightarrow H^{\bullet }(X,R)$
is the zero map, and that
$H_{c}^{\bullet }(X,R)$
is a projective
$R$
-module. We prove that the compact support cohomology of any generalized configuration space of points on
$X$
depends only on the graded
$R$
-module
$H_{c}^{\bullet }(X,R)$
. This generalizes a theorem of Arabia.
We characterize the topological spaces of minimum cardinality which are weakly contractible but not contractible. This is equivalent to finding the non-dismantlable posets of minimum cardinality such that the geometric realization of their order complexes are contractible. Specifically, we prove that all weakly contractible topological spaces with fewer than nine points are contractible. We also prove that there exist (up to homeomorphism) exactly two topological spaces of nine points which are weakly contractible but not contractible.
We introduce a notion of ‘hereditarily antisymmetric’ operator algebras and prove a structure theorem for them in finite dimensions. We also characterize those operator algebras in finite dimensions which can be made upper triangular and prove matrix analogs of the theorems of Dilworth and Mirsky for finite posets. Some partial results are obtained in the infinite dimensional case.
In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.
A new class of functions with a unique identification minor is introduced: functions determined by content and singletons. Relationships between this class and other known classes of functions with a unique identification minor are investigated. Some properties of functions determined by content and singletons are established, especially concerning invariance groups and similarity.
In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc.89 (2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.
Given a poset
$P$
and a standard closure operator
$\unicode[STIX]{x1D6E4}:{\wp}(P)\rightarrow {\wp}(P)$
, we give a necessary and sufficient condition for the lattice of
$\unicode[STIX]{x1D6E4}$
-closed sets of
${\wp}(P)$
to be a frame in terms of the recursive construction of the
$\unicode[STIX]{x1D6E4}$
-closure of sets. We use this condition to show that, given a set
${\mathcal{U}}$
of distinguished joins from
$P$
, the lattice of
${\mathcal{U}}$
-ideals of
$P$
fails to be a frame if and only if it fails to be
$\unicode[STIX]{x1D70E}$
-distributive, with
$\unicode[STIX]{x1D70E}$
depending on the cardinalities of sets in
${\mathcal{U}}$
. From this we deduce that if a poset has the property that whenever
$a\wedge (b\vee c)$
is defined for
$a,b,c\in P$
it is necessarily equal to
$(a\wedge b)\vee (a\wedge c)$
, then it has an
$(\unicode[STIX]{x1D714},3)$
-representation.
To a pair
$P$
and
$Q$
of finite posets we attach the toric ring
$K[P,Q]$
whose generators are in bijection to the isotone maps from
$P$
to
$Q$
. This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets
$P$
and
$Q$
we show that
$K[P,Q]$
is normal and that their defining ideal admits a quadratic Gröbner basis.
The jaggedness of an order ideal
$I$
in a poset
$P$
is the number of maximal elements in
$I$
plus the number of minimal elements of
$P$
not in
$I$
. A probability distribution on the set of order ideals of
$P$
is toggle-symmetric if for every
$p\in P$
, the probability that
$p$
is maximal in
$I$
equals the probability that
$p$
is minimal not in
$I$
. In this paper, we prove a formula for the expected jaggedness of an order ideal of
$P$
under any toggle-symmetric probability distribution when
$P$
is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.
We study a natural generalization of the noncrossing relation between pairs of elements in
$[n]$
to
$k$
-tuples in
$[n]$
that was first considered by Petersen et al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on
$\binom{[n]}{k}$
induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product
$[k]\times [n-k]$
of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for
$k=2$
. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex the noncrossing complex, and the polytope derived from it the dual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism
$G_{k,n}\cong G_{n-k,n}$
. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.
We characterize those partially ordered sets I for which the canonical maps Mi → colim Mj into colimits of abstract sets are always injective, provided that the transition maps are injective. We also obtain some consequences for colimits of vector spaces.
Higgins [‘The Mitsch order on a semigroup’, Semigroup Forum49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.
In this paper we shall give characterizations of the closed subsemigroups of a Clifford semigroup. Also, we shall show that the class of all Clifford semigroups satisfies the strong isomorphism property and so is globally determined. Thus the results obtained by Kobayashi [‘Semilattices are globally determined’, Semigroup Forum29 (1984), 217–222] and by Gould and Iskra [‘Globally determined classes of semigroups’ Semigroup Forum28 (1984), 1–11] are generalized.
For a semigroup
$S$
, let
${S}^{1} $
be the semigroup obtained from
$S$
by adding a new symbol 1 as its identity if
$S$
has no identity; otherwise let
${S}^{1} = S$
. Mitsch defined the natural partial order
$\leqslant $
on a semigroup
$S$
as follows: for
$a, b\in S$
,
$a\leqslant b$
if and only if
$a= xb= by$
and
$a= ay$
for some
$x, y\in {S}^{1} $
. In this paper, we characterise the natural partial order on some transformation semigroups. In these partially ordered sets, we determine the compatibility of their elements, and find all minimal and maximal elements.
In this paper it is shown how nonpointed exactness provides a framework which allows a simple categorical treatment of the basics of Kurosh–Amitsur radical theory in the nonpointed case. This is made possible by a new approach to semi-exactness, in the sense of the first author, using adjoint functors. This framework also reveals how categorical closure operators arise as radical theories.
An interval in a combinatorial structure R is a set I of points that are related to every point in R∖I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes—this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: an arbitrary structure S of size n belonging to a class 𝒞 can be embedded into a simple structure from 𝒞 by adding at most f(n) elements. We prove such results when 𝒞 is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than two. The functions f(n) in these cases are 2, ⌈log 2(n+1)⌉, ⌈(n+1)/2⌉, ⌈(n+1)/2⌉, ⌈log 4(n+1)⌉, ⌈log 3(n+1)⌉ and 1, respectively. In each case these bounds are the best possible.
The partition monoid is a salient natural example of a *-regular semigroup. We find a Galois connection between elements of the partition monoid and binary relations, and use it to show that the partition monoid contains copies of the semigroup of transformations and the symmetric and dual-symmetric inverse semigroups on the underlying set. We characterize the divisibility preorders and the natural order on the (straight) partition monoid, using certain graphical structures associated with each element. This gives a simpler characterization of Green’s relations. We also derive a new interpretation of the natural order on the transformation semigroup. The results are also used to describe the ideal lattices of the straight and twisted partition monoids and the Brauer monoid.
Marques-Smith and Sullivan [‘Partial orders on transformation semigroups’, Monatsh. Math.140 (2003), 103–118] studied various properties of two partial orders on P(X), the semigroup (under composition) consisting of all partial transformations of an arbitrary set X. One partial order was the ‘containment order’: namely, if α,β∈P(X) then α⊆β means xα=xβ for all x∈dom α, the domain of α. The other order was the so-called ‘natural order’ defined by Mitsch [‘A natural partial order for semigroups’, Proc. Amer. Math. Soc.97(3) (1986), 384–388] for any semigroup. In this paper, we consider these and other orders defined on the symmetric inverse semigroup I(X) and the partial Baer–Levi semigroup PS(q). We show that there are surprising differences between the orders on these semigroups, concerned with their compatibility with respect to composition and the existence of maximal and minimal elements.