The following equivalent results in the Boolean lattice 2 are proven. (a) Every fibre of 2 contains a maximal chain. (b) Every cutset of 2 contains a maximal antichain. (c) Every red-blue colouring of the vertices of 2 produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in 2 there is a disjoint maximal antichain. Statement (a) is then improved to: (a') Ev...

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be in PSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerate configuration of n points can be associated with a maximal chain in the weak Bruhat order of the symmetric group...

References for these notes are Sections 2.7 and 2.8 as well as Appendicies A2.3, A2.4, and A2.5. of Bjorner and Brenti. There are several supplementary sources listed at the end. Section 2.7 explores the combinatorial structure of the interval between two elements of a coxeter group W , or more generally of a quotient W J . First of all, Corollary 2.5.6 assures us that every maximal chain in [w...

We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses seat next to each other and no k consecutive persons are of the same gender. While the case of k = 2 corresponds to the classical problème des ménages with a well-studied solution, no closed-form expression for number of arrangements is known when k ≥ 3. We propose a nov...

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every st...

given a non-abelian finite group $g$, let $pi(g)$ denote the set of prime divisors of the order of $g$ and denote by $z(g)$ the center of $g$. thetextit{ prime graph} of $g$ is the graph with vertex set $pi(g)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $g$ contains an element of order $pq$ and the textit{non-commuting graph} of $g$ is the graph with the vertex s...

In this paper, we show how maximal entanglement between boundary qubits in the open spin chain of an XX model is realized. This creation of maximal entanglement could be used for phase covariant quantum cloning in a spin chain. The maximal entanglement is achieved with specially engineered couplings. We compare our realization with alternative methods and find that the method of pre-engineered ...