**From "Cybernetics & Human Knowing", Vol. 24 (2017), No. 3-4, pp. 161-188**

*4. Fixedpoint Lattices*
Any diamond-logic function

__F__(__x__) has the extreme fixedpoints:__F__^{n}(__i__),__F__^{n}(__j__), the*leftmost*and*rightmost*fixedpoints. But often this is not all.
In general,

__F__has an entire*lattice*of fixedpoints.**Lemma:**For any diamond-logic f;

f(x) min f(y)

**>**f(x min y)
f(x) max f(y)

**<**f(x max y)
Proof is an exercise for the student.

**Theorem:**If

__F__is diamond-logic, and has fixedpoints

__a__and

__b__, then these fixedpoints exist:

__a__min

_{F}

__b__= the

*rightmost*fixedpoint

*left*of both

__a__and

__b__=

__F__

^{2n}(

__a__min

__b__)

__a__max

_{F}

__b__= the

*leftmost*fixedpoint

*right*of both

__a__and

__b__=

__F__

^{2n}(

__a__max

__b__)

Proof. Let

__a__and__b__be fixedpoints, and let__c__be a fixedpoint left of both__a__and__b__. Then (__a__min__b__)**>**__c__; so (__a__min__b__) =__F__(__a__) min__F__(__b__)**>**__F__(__a__min__b__)**>**__F__(__c__) =__c__
Ergo (

__a__min__b__) is a left seed greater than__c__:
(

__a__min__b__)**>**__F__(__a__min__b__)**>**__F__^{2}(__a__min__b__)**>**...
>

__F__^{2n}(__a__min__b__) =__F__(__F__^{2n}(__a__min__b__))**>**__c__
Therefore

__F__^{2n}(__a__min__b__) is a fixedpoint left of__a__and of__b__, and is also the rightmost such fixedpoint. Thus,__F__^{2n}(__a__min__b__) =__a__min_{F}__b__. QED.
Similarly,

__F__^{2n}(__a__max__b__) =__a__max_{F}__b__. QED.
For instance, consider the following Brownian form:

**______________________________d**

**_________________c |**

**| _____a _____b | |**

**| | | | |**

**| | |____| | | |**

**| |________| | |**

**|_______________________|**

This is equivalent to this bracket-form system:

a = [b] ; b = [a]
; c = [ab] ; d =
[cd] .

In the standard interpretation, (a,b,c,d) is a
fixedpoint for:

F(a,b,c,d) = ( ~
b, ~ a,
~(a Ú b), ~(c Ú
d) )

In the nand-gate interpretation:

d =
~(d Ù c) =
~d Ú ~c =
(d Þ da )

Sentence d says “If I’m not mistaken, then A is both
true and false”.

In the nor-gate interpretation:

d = Da
- d ;

Sentence d says “A is true or false, and I am a liar.”

F has this fixedpoint lattice:

**tffi ----- tffj**

**/ \**

**/ \**

**iiii jjjj**

**\ /**

**\ /**

**ftfi ----- ftfj**

To find min

_{F}of tffj and ftfj, first take their minimum, then apply F three times: (tffj min ftfj) = iifj è iiij è iiit è iiii
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