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I imagine an algebra on the set of three values with an addition operation like this:

0 + 0 = 0 0 + 1 = random value 0 or 1 with 50/50 probability 1 + 0 = random value 0 or 1 with 50/50 probability 1 + 1 = 1 

The idea is that if this were implemented as an electronic gate where the two inputs were binary 0 or 1, the output is off, half-on, or full-on, representing a kind of summation. The half-on value would vary randomly very fast to represent a PWM-like half-amplitude.

Without having thought far, I suspect that (since the actual output values are just 0 and 1) this output could be fed into other similar gates, resulting in a variety of intermediate random-PWM-amplitude levels.

Is there a name for this mathematical concept? I would like to read about it.

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    It sounds like you want some kind of [fuzzy logic](http://en.wikipedia.org/wiki/Fuzzy_logic) or [probabilistic logic](http://en.wikipedia.org/wiki/Probabilistic_logic).2011-09-05
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    The addition is supposed to be commutative, and $a+0=0+a=a$ for every element of the algebra.2011-09-05
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    The set up you describe sounds very much like Markov chain...2011-09-05
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    Thank you for clarifying this is not a "boolean algebra". And, probabilistic logic and Markov chain are generally the correct idea for me to dig into. I figured out that I could make a real Markov chain by replacing the probability with oscillating states 0 and 1.2011-09-06

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