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This is a glossary of some terms used in various branches of mathematics that are related to the fields of order, lattice, and domain theory. Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles: * completeness properties of partial orders * distributivity laws of order theory * preservation properties of functions between posets. In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, ≤ will suffice to denote the corresponding relational symbol, even without prior introduction. Furthermore, < will denote the strict order induced by ≤. __NOTOC__ == A == * Acyclic. A binary relation is acyclic if it contains no "cycles": equivalently, its transitive closure is antisymmetric.〔 * Adjoint. See ''Galois connection''. * Alexandrov topology. For a preordered set ''P'', any upper set ''O'' is Alexandrov-open. Inversely, a topology is Alexandrov if any intersection of open sets is open. * Algebraic poset. A poset is algebraic if it has a base of compact elements. * Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements ''x'' and ''y'' such that ''x'' ≤ ''y''. In other words, the order relation of an antichain is just the identity relation. * Approximates relation. See ''way-below relation''. * A relation ''R'' on a set ''X'' is antisymmetric, if ''x R y'' and ''y R x'' implies ''x = y'', for all elements ''x'', ''y'' in ''X''. * An antitone function ''f'' between posets ''P'' and ''Q'' is a function for which, for all elements ''x'', ''y'' of ''P'', ''x'' ≤ ''y'' (in ''P'') implies ''f''(''y'') ≤ ''f''(''x'') (in ''Q''). Another name for this property is ''order-reversing''. In analysis, in the presence of total orders, such functions are often called monotonically decreasing, but this is not a very convenient description when dealing with non-total orders. The dual notion is called ''monotone'' or ''order-preserving''. * Asymmetric. A relation ''R'' on a set ''X'' is asymmetric, if ''x R y'' implies ''not y R x'', for all elements ''x'', ''y'' in ''X''. * An atom in a poset ''P'' with least element 0, is an element that is minimal among all elements that are unequal to 0. * A atomic poset ''P'' with least element 0 is one in which, for every non-zero element ''x'' of ''P'', there is an atom ''a'' of ''P'' with ''a'' ≤ ''x''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Glossary of order theory」の詳細全文を読む スポンサード リンク
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