Editing Glossary of order theory (section)

== A ==
* '''Acyclic'''.  A [[binary relation]] is acyclic if it contains no "cycles": equivalently, its [[transitive closure]] is [[Antisymmetric relation|antisymmetric]].<ref name=BosSuz/>
* '''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''.
* '''Antisymmetric relation'''. A [[homogeneous relation]] ''R'' on a set ''X'' is '''[[Antisymmetric relation|antisymmetric]]''', if ''x R y'' and ''y R x'' implies ''x = y'', for all elements ''x'', ''y'' in ''X''.
* '''Antitone'''. 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 [[Mathematical analysis|analysis]], in the presence of [[total order]]s, 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 relation]]'''. A [[homogeneous relation]] ''R'' on a set ''X'' is asymmetric, if ''x R y'' implies ''not y R x'', for all elements ''x'', ''y'' in ''X''.
* '''[[Atom (order theory)|Atom]]'''. An atom in a poset ''P'' with least element 0, is an element that is minimal among all elements that are unequal to 0.
* '''Atomic'''. An 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''.