Many fractals, in particular L-systems (Lindenmayer systems), can be described by recursive forms. For example, the structure of Koch fractal can be expressed by a recursive form K=.

A lattice is an algebraic structure (L,Ù,Ú) consisting from a set L and two algebraic operations Ù and Ú ("meet" and "join", or "and" and "or"), such that for any a,b,c Î L hold:

1. aÙa=a, aÚa=a (idempotent laws),

2. aÙb=bÙa, aÚb=bÚa (commutativity laws),

3. (aÙb)Ùc=aÙ(bÙc), (aÚb)Úc=aÚ(bÚc) (associativity laws),

4. aÚ(aÙb)=a, aÙ(aÚb)=a (absorption laws).

A Boolean algebra is a lattice (B,Ù,Ú) that satisfies four additional properties:

1. there exists an element 0 Î B that aÚ0=a for all a Î B (lower bound),

2. there exists an element 1 Î B that aÙ1=a for all a Î B (upper bound),

3. for all a,b,c Î B, (aÙb)Úc=(aÚc)Ù(bÚc) (distributivity law)

4. for every a Î B there exists an element Øa in B such that aÙØa=0 and aÚØa=1 (complement law).

Instead of using classical Boolean algebra with only two discrete values 0 and 1, we can consider more general case where values belong to the continuous set [0,1]. Then we can construct a new model of polyvalent Boolean algebra that will be called the square-free polynomial model. We can define pÙq=pq, Øp=1-p. A square-free polynomial is any polynomial in which a degree of each variable is 1. By introducing the idempotent law p²=p for all variables, we can model a Boolean algebra working in the continuous interval [0,1], with square-free polynomials, standard polynomial multiplication, and idempotency. This model is very similar to a Boolean ring, but Øp is defined as Øp=1-p, and not as Øp=1+p. In this way the structure of standard truth tables is preserved and extended to the set [0,1].

A concept of square-free polynomials is implicitly given in the original works of G. Boole (2003).

By using De Morgan laws, we obtain pÚq=p+q-pq. Then, we can check if our model is consistent with regard to the complete set of axioms. We simply conclude that holds

aÙa=a²=a, aÚa=a+a-a²=a+a-a=a (idempotency),

aÙb=ab=ba=bÙa, aÚb=a+b-ab=b+a-ba=bÚa (commutativity),

(aÙb)Ùc=(ab)c=a(bc)=aÙ(bÙc), (aÚb)Úc=(a+b-ab)+c-(a+b-ab)c=a+b-ab+c-ac-bc+abc=a+b+c-bc-ab-ac+abc = a+(b+c-bc)-a(b+c-bc) = aÚ(bÚc) (associativity),

aÚ(aÙb)=a+ab-a²b=a+ab-ab=a, aÙ(aÚb)=a(a+b-ab)=a²+ab-a²b=a+ab-ab=a (absorption),

aÚ0=a+0-a0=a, aÙ1=a1=a (lower and upper bound),

(aÙb)Ùc=(ab)c=a(bc)=(aÚc)Ù(bÚc) (distributivity),

aÙØa=a(1-a)=a-a²=a-a=0, aÚØa=a+(1-a)-a(1-a)=a+1-a-a+a²=a+1-a-a+a=1 (complement).

Hence, our model is compatible with Boolean logic.

In order to simplify, we can denote Øp by , pÙq by pq, and continue to work in the line segment notation. In this case we are working in a model of Spencer-Brown calculus of indications, based on one symbol , where this symbol is replaced with a single line segment (Spencer-Brown, 1969). Hence, two forms of equality will be expressed as:

1. =  (form of condensation), and

2. =      (form of cancellation).

In the form of cancellation appears an unmarked state     . These two rules can be replaced with = and =a.