The Fixed Point Property for Posets of Small Width

The fixed point property for finite posets of width 3 and 4 is studied in terms of forbidden retracts. The ranked forbidden retracts for width 3 and 4 are determined explicitly. The ranked forbidden retracts for the width 3 case that are linearly indecomposable are examined to see which are minimal automorphic. Part of a problem of Niederle from 1989 is thus solved

Chain Decomposition Theorems for Ordered Sets (and Other Musings)

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets

Does the endomorphism poset PPP^P determine whether a finite poset PP is connected? An issue Duffus raised in 1978

summary:Duffus wrote in his 1978 Ph.D. thesis, ``It is not obvious that $P$ is connected and $P^P\cong Q^Q$ imply that $Q$ is connected'', where $P$ and $Q$ are finite nonempty posets. We show that, indeed, under these hypotheses $Q$ is connected and $P\cong Q$

The Transcription Factor Pebbled/RREB1 Regulates Injury-Induced Axon Degeneration

Neurons establish complex networks within the nervous system allowing for rapid cell-cell communication via their long, thin axonal processes. These wire-thin projections are susceptible to a number of insults or injuries, and axonal damage can lead to disruption in signal propagation and an overall dysfunction of the neural network. Recent research focused on investigating the underlying mechanisms of injury-induced axon degeneration led to the discovery of a number of endogenous, pro-degenerative molecules such as dSarm/Sarm1, Highwire/Phr1, and Axundead. These signaling molecules are thought to execute axon degeneration in response to injury locally within the distal severed axon, but the exact mechanism of action is unclear. To further identify novel participants of the axon death signaling cascade, we performed an unbiased forward genetic mutagenesis screen using the sensory neurons within the adult wing of Drosophila melanogaster. We identified a novel role for the C2H2 zinc finger transcription factor, Pebbled (Peb)/Ras-responsive element binding protein 1 (RREB1) in partially suppressing injury-induced axon degeneration. Loss of function peb mutant glutamatergic neurons present two distinct axon degeneration defects: either complete protection from axotomy, or they exhibit a novel phenotype in which axons fragment into long, continuous pieces instead of undergoing complete degeneration. Additionally, we show an enhancement of the peb protective phenotype when dSarm levels are decreased, but not with reduced levels of axundead. These data provide the first evidence of a transcription factor involved in regulating injury-induced axon degeneration signaling in vivo