An element in a lattice is said to be join irreducible iff is not a bottom element, and, whenever , then or . Dually, is meet irreducible iff is not a top element, and, whenever , then or . If is both join and meet irreducible, then is said to be irreducible. Any atom in a lattice is join irreducible.
is meet irreducible but not join irreducible, is join irreducible but not meet irreducible, while are irreducible.
From this, we make the observations that in any chain, all the elements except the bottom one are join irreducible. Dually, all the elements except the top one are meet irreducible. An element is join irreducible iff it covers (http://planetmath.org/CoveringRelation) at most one other element. An element is meet irreducible iff it is covered by at most one other element.
Remark. If a lattice satisfies the descending chain condition, then every element can be expressed as a join of join irreducible elements. This statement can be dualized: if a lattice satisfies the ascending chain condition, then every element is the meet of meet irreducible elements.
- 1 B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, 2nd Edition, Cambridge (2003)
|Date of creation||2013-03-22 15:47:29|
|Last modified on||2013-03-22 15:47:29|
|Last modified by||CWoo (3771)|