## Quick Reference

An algebraic structure, such as a Boolean algebra, in which there are two dyadic operations that are both commutative and associative and satisfy the absorption and idempotent laws. The two dyadic operators, denoted by ∧ and ∨, are called the **meet** and the **join** respectively.

An alternative but equivalent view of a lattice is as a set *L* on which there is a partial ordering defined. Further, every pair of elements has both a greatest lower bound and a least upper bound. The least upper bound of {*x*,*y*} can be denoted by *x* ∨ *y* and is referred to as the **join** of *x* and *y*. The greatest lower bound can be denoted by *x* ∧ *y* and is called the **meet** of *x* and *y*. It can then be shown that these operations satisfy the properties mentioned in the earlier definition, since a partial ordering ≤ can be introduced by defining *a* ≤ *b* iff *a* ∨ *b* = *a*

*a* ≤ *b* iff *a* ∨ *b* = *a*

Lattices in the form of Boolean algebras play a very important role in much of the theory and mathematical ideas underlying computer science. Lattices are also basic to much of the approximation theory underlying the ideas of denotational semantics.

**From:**
lattice
in
A Dictionary of Computing »

*Subjects:*
Computing.