Ideal (association theory)

In abstract algebra is an ideal of a federation a subset of the respect of both lattice operations and respect even with elements from the entire association completed is. The name is based on the concept of the ideal in ring theory . ${\displaystyle (V,\vee ,\wedge )}$ ${\displaystyle I,}$ ${\displaystyle \wedge }$

Definition of associations

Be an association . An ideal of is a non-empty subset of which holds: ${\displaystyle V}$ ${\displaystyle I}$ ${\displaystyle V}$ ${\displaystyle V}$

• ${\displaystyle I}$is a subgroup of and ${\displaystyle V}$
• for everyone and is ${\displaystyle a\in I}$ ${\displaystyle b\in V}$ ${\displaystyle a\wedge b\in I.}$

general definition

A partial order is related upper half bandage ( English conditional upper semi-lattice , in short CUSL ) if each couple has a supremum limited, so if all the following applies: If there with , then there exists a least upper bound . ${\displaystyle (P,\leq )}$ ${\displaystyle a,b\in P}$ ${\displaystyle u\in P}$ ${\displaystyle a,b\leq u}$ ${\displaystyle a\vee b=\sup\{a,b\}\in P}$

Be a Cusl. A non-empty subset is called an ideal of if: ${\displaystyle (P,\leq )}$ ${\displaystyle I\subseteq P}$ ${\displaystyle P}$

• ${\displaystyle I}$is closed at the bottom, d. H. for , and holds . ${\displaystyle a\in P}$ ${\displaystyle b\in I}$ ${\displaystyle a\leq b}$ ${\displaystyle a\in I}$
• Are in limited, d. H. there is one with so is . ${\displaystyle a,b\in I}$ ${\displaystyle P}$ ${\displaystyle u\in P}$ ${\displaystyle a,b\leq u}$ ${\displaystyle a\vee b\in I}$

properties

Obviously every association is a cusl; in an association is defined as . ${\displaystyle a\leq b}$ ${\displaystyle a\wedge b=a}$

The general definition includes that of associations.

See also

• Filter (math)

credentials

1. ^ Viggo Stoltenberg-Hansen, Ingrid Lindstrom and Edward R. Griffor: Mathematical theory of domains. Cambridge Tracts in Theoretical Computer Science, 1994.