Glossary of Order Theory - F

F

• Filter. A subset X of a poset P is called a filter if it is a filtered upper set. The dual notion is called ideal.

• Filtered. A non-empty subset X of a poset P is called filtered, if, for all elements x and y of X, there is an element z of X such that z ≤ x and z ≤ y. The dual notion is called directed.

• Finite element. See compact.

• Frame. A frame F is a complete lattice, in which, for every x in F and every subset Y of F, the infinite distributive law x ∧ Y = {x ∧ y | y in Y} holds. Frames are also known as locales and as complete Heyting algebras.