poset

Definitions

A partially ordered set.

Equivalents

Examples

“1973, Barbara L. Osofsky, Homological Dimensions of Modules, American Mathematical Society, →ISBN, page 76, 42. Definition. A poset (partially ordered set⟳) (X, ≤) (usually written just X) is a set⟳ X together with a transitive, antisymmetric relation ≤ on X. 43. Definition. A linearly ordered set⟳ or chain is a poset (X, ≤), such that ∀a, b ∈ X, either a ≤ b or b ≤ a or a = b.”

“1998, Yuri A. Drozd, Representations of bisected posets and reflection functors, Idun Reiten, Sverre O. Smalø, Øyvind Solberg (editors), Algebras and Modules II, American Mathematical Society (for Canadian Mathematical Society), page 153, We construct a complete⟳ set⟳ of reflection functors for the representations of posets and prove⟳ that they really have⟳ the usual properties. In particular, when the poset is of finite representation type⟳, all of its indecomposable representations can be obtained from some "trivial" ones via relations. To define⟳ such reflection functors, a wider class of matrix problem is introduced, called "representations of bisected posets".”

“The combinatorial interest⟳ in posets is largely due to two unoriented graphs associated with a given poset: the comparability graph (which we shall not consider⟳ here) and the covering graph.”

CEFR level