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Full-Text Articles in Physical Sciences and Mathematics
Morphisms And Order Ideals Of Toric Posets, Matthew Macauley
Morphisms And Order Ideals Of Toric Posets, Matthew Macauley
Publications
Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, …
Coxeter Groups And Asynchronous Cellular Automata, Matthew Macauley, Henning S. Mortveit
Coxeter Groups And Asynchronous Cellular Automata, Matthew Macauley, Henning S. Mortveit
Publications
The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.