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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.
Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have bee
An interval $k$-graph is the intersection graph of a family $mathcal{I}$ of intervals of the real line partitioned into at most $k$ classes with vertices adjacent if and only if their corresponding intervals intersect and belong to different classes.
We consider the Graver basis, the universal Groebner basis, a Markov basis and the set of the circuits of a toric ideal. Let $A, B$ be any two of these bases such that $A ot subset B$, we prove that there is no polynomial on the size or on the maxima
Toric posets are cyclic analogues of finite posets. They can be viewed combinatorially as equivalence classes of acyclic orientations generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. I
We introduce the notion of tau-like partial order, where tau is one of the linear order types omega, omega*, omega+omega*, and zeta. For example, being omega-like means that every element has finitely many predecessors, while being zeta-like means th