Limited Lattice

up:: Lattice

Given a lattice , we say it is limited if there are elements and which act as its global maximum (denoted ) and minimum (denoted ), respectively.

A lattice is said to have a unity if

A lattice is said to have a neutral element if

Examples

Given a set , its powerset is a limited lattice, with unity and neutral element .

In a Boolean Algebra, (denoted ) is absolute truth and (denoted ) is absolute falsehood.

Properties

• Given a limited lattice, one can say two elements are complements if¹

This is very similar to the definition of a Disconnected Topological Space.

• When every element of a lattice has at least one complement, it is said to be a Complemented Lattice. Note that it need not be unique.

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References

• Notas para um Curso de Física-Matemática, João Carlos Alves Barata. Cap. 02

Footnotes

1. Yes, I’m using as the logical “and”, as well as the join operator. ↩