Small Representations of Permutation Groups by Embedding the Domain in an Ordered Set

We present a representation for permutation groups as the automorphism group of an ordered set $U$ such that the automorphism group's action on a subset $T\subseteq U$ is the permutation group itself. For many imprimitive permutation groups, the representation uses fewer points than the smallest representations to date

The Fixed Vertex Property for Graphs

Analogous to the fixed point property for ordered sets, a graph has the fixed vertex property if each of its endomorphisms has a fixed vertex. The fixed point theory for ordered sets can be embedded into the fixed vertex theory for graphs. Therefore, the potential for cross-fertilization should be explored

Towards the Automorphism Conjecture I: Combinatorial Control and Compensation for Factorials

This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the structures which currently prevent the proof of such an exponential bound, or which indeed inflate the number of automorphisms beyond such a bound. This is a first step towards a possible resolution of the Automorphism Conjecture for ordered sets

The Automorphism Conjecture for Ordered Sets of Width ≤11\leq 11 (Version 2)

We introduce a recursive method to deconstruct the automorphism group of an ordered set. By connecting this method with deep results for permutation groups, we prove the Automorphism Conjecture for ordered sets of width less than or equal to $11$. Subsequent investigations show that the method presented here could lead to a resolution of the Automorphism Conjecture.Comment: arXiv admin note: substantial text overlap with arXiv:2209.0931

Performance guarantees and applications for Xia's algorithm

AbstractXia's algorithm consists of a reduction algorithm and a translation procedure both originally used to tackle the fixed point property for ordered sets. We present results that show that the translation procedure allows access to a much wider range of problems and results showing that the algorithm is very efficient when applied to the fixed point property in ordered sets or for order/graph isomorphism/rigidity

Structure and function of SPP/SPPL proteases: insights from biochemical evidence and predictive modeling

More than 20 years ago, signal peptide peptidase (SPP) and its homologues, the signal peptide peptidase-like (SPPL) proteases have been identified based on their sequence similarity to presenilins, a related family of intramembrane aspartyl proteases. Other than those for the presenilins, no high-resolution structures for the SPP/SPPL proteases are available. Despite this limitation, over the years bioinformatical and biochemical data have accumulated, which altogether have provided a picture of the overall structure and topology of these proteases, their localization in the cell, the process of substrate recognition, their cleavage mechanism, and their function. Recently, the artificial intelligence-based structure prediction tool AlphaFold has added high-confidence models of the expected fold of SPP/SPPL proteases. In this review, we summarize known structural aspects of the SPP/SPPL family as well as their substrates. Of particular interest are the emerging substrate recognition and catalytic mechanisms that might lead to the prediction and identification of more potential substrates and deeper insight into physiological and pathophysiological roles of proteolysis

Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general