Hilbert Book Model Project/Relational Structures

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Relational structures

Relational structures are sets that restrict the types of their internal relations. Mathematics applies "lattice" as the type name of a category of relational structures.

Lattices

A lattice is a set of elements $a,b,c,....$ that is closed for the connections $\cap$ and $\cup$.

These connections obey

• The set is partially ordered.
  • This means that with each pair of elements $a,b$ belongs an element $c$, such that $a\subset c$ and $b\subset c$.
• The set is a $\cap$ half lattice.
• This means that with each pair of elements $a,b$ an element $c$ exists, such that $c=a\cap b$.
• The set is a $\cup$ half lattice.
  • This means that with each pair of elements $a,b$ an element $c$ exists, such that $c=a\cup b$.
• The set is a lattice.
  • This means that the set is both a $\cap$ half lattice and a $\cup$ half lattice.

The following relations hold in a lattice

$a\cap b=b\cap a$

(1)

$(a\cap b)\cap c=a\cap (b\cap c)$

(2)

$a\cap (a\cup b)=a$

(3)

$a\cup b=b\cup a$

(4)

$(a\cup b)\cup c=a\cup (b\cup c)$

(5)

$a\cup (a\cap b)=a$

(6)

The lattice has a partial order inclusion $\subset$:

$a\subset b\Leftrightarrow a\cap b=a$

(7)

A complementary lattice contains two elements $n$ and $e$ with each element $a$ a complementary element $a^{'}$ such that:

$a\cap a^{'}=n$

(8)

$a\cap n=n$

(9)

$a\cap e=a$

(10)

$a\cup a^{'}=e$

(11)

$a\cup e=e$

(12)

$a\cup n=a$

(13)

An orthocomplemented lattice contains two elements $n$ and $e$ and with each element $a$ an element $a^{''}$ such that:

$a\cup a^{''}=e$

(14)

$a\cap a^{''}=n$

(15)

$(a^{''})^{''}=a$

(16)

$a\subset b\Leftrightarrow b^{''}\subset a^{''}$

(17)

$e$ is the unity element; $n$ is the null element of the lattice

A distributive lattice supports the distributive laws:

$a\cap (b\cup c)=(a\cap b)\cup (a\cap c)$

(18)

$a\cup (b\cap c)=(a\cup b)\cap (a\cup c)$

(19)

A modular lattice supports:

$(a\cap b)\cup (a\cap c)=a\cap (b\cup (a\cap c))$

(20)

A weak modular lattice supports instead:

There exists an element $d$ such that

$a\subset c\Leftrightarrow (a\cup b)\cap c=a\cup (b\cap c)\cup (d\cap c)$

(21)

where $d$ obeys:

$(a\cup b)\cap d=d$

(22)

$a\cap d=n$

(23)

$b\cap d=n$

(24)

$(a\subset g)$ and $(b\subset g)\Leftrightarrow d\subset g$

(25)

In an atomic lattice holds

$\exists _{(p\epsilon S)}\forall _{(x\in L)}\{x\subset p\Rightarrow x=n\}$

(26)

$\forall _{(a\in L)}\forall _{(x\in L)}\{(a<x<a\cap p)\Rightarrow (x=a$ or $x=a\cap p)\}$

(27)

p is an atom

Well known lattices

Classical logic has the structure of an orthocomplemented distributive modular and atomic lattice.

Quantum logic has the structure of an orthocomplemented weakly modular and atomic lattice. It is also called an orthomodular lattice.

Both lattices are atomic lattices.A complete set of atoms spans the complete lattice.

The orthomodular lattice finds a realization in the set of closed subspaces of a separable Hilbert space. The lattice structure of this set is isomorphic to the orthomodular lattice.

The set of rays that are spanned by the members of an orthonormal base of the Hilbert space form a full set of atoms of the orthomodular lattice.

The modular configuration lattice

The set of closed subspaces of the Hilbert space is an orthomodular lattice. The underlying vector space contains a set of rays, which each represent an elementary module, These rays span a subspace whose closed subspaces represent a modular configuration lattice. The atoms of this lattice represent elementary modules.

All modules own a private mechanism that provides the locations of the elementary modules that the module contains. The mechanism applies a stochastic process that owns a characteristic function. This characteristic function equals the superposition of the characteristic functions of the processes that supply the locations for the individual elementarymodules that constitute the module.

In the Hilbert space, some of the rays in the scanning subspace represent the elementary modules. These rays are mutually orthogonal. At each instant, a private mechanism provides elementary modules with a new location.

In the creator's view tubes that zigzag with the progression value, contain the locations of a corresponding elementary module.