Related papers: J\'onsson posets
This paper studies three natural pre-orders of increasing generality on the set of all completely non-unitary partial isometries with equal defect indices. We show that the problem of determining when one partial isometry is less than…
We consider the coset poset associated with the families of proper subgroups, proper subgroups of finite index, and proper normal subgroups of finite index. We investigate under which conditions those coset posets have contractible…
We show that a linearly ordered topological space is initially \lambda-compact if and only if it is \lambda-bounded, that is, every set of cardinality $\leq \lambda$ has compact closure. As a consequence, every product of initially…
We introduce a new invariant for triangulated categories: the poset of spherical subcategories ordered by inclusion. This yields several numerical invariants, like the cardinality and the height of the poset. We explicitly describe…
Hereditarily finite sets (sets which are finite and have only hereditarily finite sets as members) are basic mathematical and computational objects, and also stand at the basis of some programming languages. This raises the need for…
Let $P$ be an arbitrary finite partially ordered set. It will be proved that the number of edges of the order polytope ${\mathcal O}(P)$ is equal to that of the chain polytope ${\mathcal C}(P)$. Furthermore, it will be shown that the degree…
Let $A$ and $B$ be finite ordered sets. We show that if the ordered sets of isotone self-maps $A^A$ and $B^B$ (ordered pointwise) are isomorphic, then $A$ and $B$ are isomorphic. This resolves a question originating with D. Duffus in 1978,…
Order dimension theory measures the complexity of partially ordered sets by quantifying how far they are from being linearly ordered. In this paper we study classical bounding results for order dimension within the framework of reverse…
The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner, arXiv:2007.06142]. We study this poset from the perspective of the…
In 1982, J\'onsson and McKenzie posed the following problem: "Find counter examples (or prove that none exist) to the refinement of $A^C\cong B^D$ [$A$, $B$, $C$, and $D$ non-empty posets] under" the condition "$C$, $D$, and $A^C$ are…
A poset is $(\omega,C)$-representable if it can be embedded into a field of sets in such a way that all existing joins, and all existing \emph{finite} meets are preserved. We show that the class of $(\omega,C)$-representable posets cannot…
In this paper, we extend the notion of orthogonality to the general elements of an absolute matrix order unit space and relate it to the orthogonality among positive elements. We introduce the notion of a partial isometry in an absolute…
Assume $\mathsf{ZF + AD^+ + V = L(\mathscr{P}(\mathbb{R}))}$. Let $\approx$ denote the relation of being in bijection. Let $\kappa \in \mathrm{ON}$ and $\langle E_\alpha : \alpha < \kappa\rangle$ be a sequence of equivalence relations on…
We determine the finite groups $G$ in which every subset $A \subseteq G$ of cardinality dividing the order of $G$ is a \emph{factor}, i.e. has a complement $B \subseteq G$ of cardinality $|G|/|A|$ such that $G = A \cdot B$ or $G = B \cdot…
A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived…
It is known when we call a poset P, a $\mathcal{P}$-chain permutational poset, given a subset of permutations $\mathcal{P}$ of the symmetric group $S_{n}$. In this work, we use the same idea to study subsets of words of length $n$, that are…
Earlier an arbitrary poset $P$ was proved to be isomorphic to the collection of subsets of a space $M$ with two closures which are closed in the first closure and open in the other. As a space $M$ for this representation an algebraic dual…
We study two new parameters for finite posets motivated by the problem of efficiently determining the set of successors of a given element. A plane map of a poset $P=(X,\leq)$ is an injective mapping of $X$ into the Cartesian plane…
We establish formulas for the number of all downsets (or equivalently, of all antichains) of a finite poset P. Then, using these numbers, we determine recursively and explicitly the number of all posets having a fixed set of minimal points…
Given a poset $P$ we say a family $\mathcal{F}\subseteq P$ is centered if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the centeredness property if for any $M$, among all families of…