Limited Lattice

up:: Lattice

Given a lattice $(X,\wedge ,\vee )$, we say it is limited if there are elements $1$ and $0$ which act as its global maximum (denoted $1$) and minimum (denoted $0$), respectively.

A lattice is said to have a unity $1\in X$ if

$$\forall x\in X:x\wedge 1=x⟺x\vee 1=x$$

A lattice is said to have a neutral element $0\in X$ if

$$\forall x\in X:x\wedge 0=0⟺x\vee 0=x$$

Examples

Given a set $X$, its powerset $\mathbb{P}(X)$ is a limited lattice, with unity $X$ and neutral element $\varnothing$.

In a Boolean Algebra, $1$ (denoted $\top$) is absolute truth and $0$ (denoted $\perp$) is absolute falsehood.

Properties

• Given a limited lattice, one can say two elements $x,y\in X$ are complements if¹

$$(x\wedge y=0)\wedge (x\vee y=1)$$

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 $\wedge$ as the logical “and”, as well as the join operator. ↩