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Can someone point me to an article concerning the "inversion" of formulas?

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closed as not a real question by Andres Caicedo, Andreas Blass, Pete L. Clark, Franz Lemmermeyer, Andy Putman Apr 18 '11 at 0:01

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Let me clarify since this is getting voted down. and and or form Boolean operations on formulas. Specifically, does anyone know of any research which formally inverts the or operation. As an example, in rings, localization, is given by inverting formulas of the form x where x is an element of A. In other words, add the sentence $\exists y, \ xy = 1$. –  Andrew Stout Apr 16 '11 at 19:40
Your question does not make any sense. You should first explain the terms that you use (what do you mean by "invert 'or'"?!?) and then ask a well defined question about those. –  André Henriques Apr 16 '11 at 21:48
@REX: please click on the "How to ask" option at the top of the page and take a look at some of the suggestions there. At the moment it looks like you are relying on people to read your mind in order to answer the question. –  Pete L. Clark Apr 17 '11 at 0:14
So, you think I should re-ask the question? –  Andrew Stout Apr 17 '11 at 6:33

1 Answer 1

up vote 3 down vote accepted

Although the question mentions inversion of formulas and the comment mentions inversion of an operation ("or"), the example leads me to guess that the proposer actually wants neither of these but rather to invert elements of some algebraic structure, probably Boolean algebras. In case my guess is correct, let me point out that, for any element $a$ in a Boolean algebra, all of the following are equivalent: (1) invert $a$ with respect to "and", i.e., adjoin $y$ such that $a\wedge y=1$, (2) invert $\neg a$ with respect to "or", i.e., adjoin $z$ such that $(\neg a)\vee z=0$, (3) identify $a$ with 1, (4) identify $\neg a$ with 0. As a result, localization (in the sense of inverting an element) amounts to forming a quotient Boolean algebra (dividing by the ideal generated by $\neg a$ or equivalently dividing by the filter generated by $a$).

If, on the other hand, my guess about the intended question is wrong, then I'd like to see the intended question formulated clearly. To encourage that, I'll vote to close the present form of the question.

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Actually, I want both of those. I want to form a Boolean algebra of formulas and then adjoin inverses with respect to the operation $\vee$. At least, in my mind, that is what the inverse of a formula would be; however, I am not sure. –  Andrew Stout Apr 17 '11 at 6:24

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