Lattice Theory Introduction

The theory of lattices is one of the most important theories in algebra. Lattices are made up of partially ordered sets in which any of their elements represent a least upper bound and a greater lower bound. Lattice structures are used in satisfying axiomatic identities, and they are also helpful in defining order theory and universal algebraic descriptions.

Characteristics

Algebra is composed of various forms of sets. Lattices are partially ordered sets (posets), but not all partially ordered sets can be lattices. Such sets are lattices when they satisfy two defining axioms. The first is when the posets have binary joins (i.e., for any two elements [A, B] of L, the set {a, b} has a join or least upper bound). The second axiom that has to be satisfied is the existence of binary meets (i.e., for any two elements in the set of L [A, B], the set {a,b} has a meet or greatest lower bound).

Structure

As algebraic structures, lattices satisfy common mathematical laws such as commutative, associative, idempotent and absorption laws. The elements in which the joins and meets are present are called semilattices, and when bound, they form a complete lattice structure with a distinct, appropriate interaction between the two lattices. A lattice whose join operation satisfies the identity property of algebra is called "bound," and from the said structure all other forms of lattices can take place.

Types

There are different types of lattices. Complete lattices are present when the subsets have a join and a meet present, while distributive lattices are those subsets which are composed of two binary operations, with either operation following dual laws and in turn satisfy the first and second axiom by means of distribution. Modular lattices are those which follow a weaker identity as compared with the distributive lattice.

Applications

The lattice theory is used in defining and understanding common mathematical laws and theorems, including that of natural numbers where the "min" is zero, and the "max" is absent. The Cartesian square of natural numbers also follows the same definition. Lattices are also found in the study of sets, where the subsets included in the power set of "A" which represents the "union" and "intersection" are the same as the "joins" and "meets."

Exclusions

While lattices are posets, not all posets are lattices. Examples of non-lattice posets include two elements which lack a join and a meet. Two elements in a subset that meet at more than one element (e.g., x­<y) are also not considered a lattice, regardless of being partially ordered by means of divisibility.

References

• Introduction to Lattice Theory