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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, …
Toric Partial Orders, Mike Develin, Matthew Macauley, Victor Reiner
Toric Partial Orders, Mike Develin, Matthew Macauley, Victor Reiner
Publications
We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.