- Multiplication of reflexive polynomials is
non-commutative.
It's easy to see since "a from the point of view of b" is apparently not the same as "b from the point of view of a", so ab
ba.
- Distributive property: we can "open
brackets", e.g.: (T+b)a = Ta+ba. Due
to non-commutative multiplication we need
explicitely declare that we can do that on both
sides - left and right. Thus
T(a+b)=Ta+Tb.
We can intperpret brackets in (T+b)a as if "an agent a has a whole image of the scene T with b in it" in it. The right-hand side Ta+ba then corresponds to a having separate images of T and b. While sometimes this distinction is important, for most purposes we will ignore it, so the distributive rule (in both - left and right - versions) will be one of the axioms.
- Reduction rule: duplicate members can be
eliminated, e.g. T+Ta+Ta=T+2Ta=T+Ta.
An agent does not receive any new information by having second copy of his image of the scene or other agents.
Because of the rule 3 reflexive polinomials can be always reduced to the form with only unit multipliers at all non-zero terms. That leads to the definition of Lefebvre polynomials as general expressions of the form
i
iTai,
where ai is a word in an the alphabet {a,b,...}
with letters designating agents. The mutlipliers i
take values in the binary set {0,1}. They obey
the rules of Boolean algebra:
0+0=0, 0+1=1, 1+0=1, 1+1=1, 0*0=0, 0*1=0, 1*0=0, 1*1=1.
The following polynomials are examples of elements of Lefebvre algebra.
0, 1, T, a, 1+a, 1+ab, Ta, T+Ta+Tb+Tc, etc.
Q = T+Ta+Tb+Taa.
In Q a scene T can be factored out:
Q = T(1+a+b+aa)=Tq.
We will call the polynomial q, which contains no scene symbol T, the awareness operator. Factoring out and eliminating the scene T allows to study mutual awareness of agents in a context independent of the particular scene. Thus awareness operator describes inherent, invariant properties of mutual awareness of agents.
Example 1. It is easy to see that a single act of awareness by agent a corresponds to the operator 1+a. Applying this operator to T we will obtain T+Ta, interpreted as "agent a is aware of the scene T".
Example 2. If two agents a and b are present and their awareness acts are performed consequently, one after another, then the awareness operator for such two-agent system is (1+a)(1+b). Applying this operator to scene gives T+Ta+Tb+Tab, interpreted as "both a and b are aware of the scene T and b is aware of a being aware of the scene."
An important remark. Instead of saying "b is aware of a being aware of the scene" we can say a bit more verbously that "b knows that a is aware of itself in the scene and b assumes that a has a model of the scene T". It is important to note that we don't make any assumptions on the "quality" or adequacy of such awareness or situational models created by the agents. Lefebvre polynomial only states that such awareness or model exists but does not say anything about its validity.
Example 3. When two agents perform act of awareness simulteniously the operator takes form 1+a+b.