# Name for this type of space over a boolean algebra?

The structure has two carrier sets $E$ and $A$, operators $({}^*, \wedge)$ over $E$, and a ternary "decision" operator $D:E \times A \times A \to A$, written infix $(p?a:b)$, whose intended meaning is "if p then a, else b."

Axioms:

• $(p?a:a) =a$
• $(p?(p?a:b):(p?c:d)) = (p?a:d)$
• $(p?(q?a:b):(q?c:d)) = (q?(p?a:c):(p?b:d))$
• $(p^*?a:b) = (p?b:a)$
• $(p \wedge q ? a:b) = (p?(q?a:b):b)$

If we quotient $E$ by equivalence $p \sim q :\iff \forall a,b (p?a:b)=(q?a:b)$ then it follows this quotient is a boolean algebra.

I'm sure lots of people have thought of this, or something equivalent. Does it have a name? I was going to call it a "decision space over a boolean algebra".

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Asked on MO: mathoverflow.net/questions/82246/… –  The Decider Nov 30 '11 at 4:03