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• CommentRowNumber1.
• CommentAuthorMike Shulman
• CommentTimeJun 6th 2010
• CommentRowNumber2.
• CommentAuthorTobyBartels
• CommentTimeJun 6th 2010

Thanks for Kleene equality; I have added some examples to it. (Teaching junior-college algebra makes one think about this sort of thing, at least if one has a mind like mine.)

• CommentRowNumber3.
• CommentAuthorMike Shulman
• CommentTimeJun 6th 2010

Is there a word for an equality which means “if the LHS is defined, then so is the RHS and they are equal”?

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJun 6th 2010
Freyd calls it "directed equality" on page 3 of Categories, Allegories.
• CommentRowNumber5.
• CommentAuthorTobyBartels
• CommentTimeJun 6th 2010

Directed equality is particularly important for rewriting rules. So if you ever see $x^2/x$ and are confident that it arises legitimately (that is, that it must be defined in the context at hand, or it would never have been written down), then you are allowed to simplify it to $x$. But if you write down $x^2/x$ yourself, without first checking that it is defined, then you do not know that it is equivalent to $x$ (but only to $x, x \ne 0$).

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJun 6th 2010

“Directed equality” isn’t quite ideally evocative for me, although I see the intent, but calling it a “rewrite rule” is pretty good.

• CommentRowNumber7.
• CommentAuthorTobyBartels
• CommentTimeJun 7th 2010
• (edited Jun 7th 2010)

But the proposition that $A = B$ if $A$ is defined is different from the rewrite rule that $A$ can be replaced by $B$. Rewrite rules are usually rules for simplifying expressions, or at least for getting them into a more useful form, and one is selective about them. Given a collection of terms and a collection of rewrite rules between them, there is a derived notion of equality (two terms are equal iff they can be rewritten to the same result), which is not the same thing; and even this (which is symmetric after all) doesn’t match the directed equality.

For example, nobody would propose $7 \rightsquigarrow 3 + 4$ as a rewrite rule for elementary arithmetic, even though adding it to the usual rewrite rules would preserve the derived notion of equality and $7 \gt\!\!\!= 3 + 4$ (there should be no space at all between ‘$\gt$’ and ‘$=$’, to make a single symbol for directed equality) is certainly true; that’s because it’s not a useful step to take.

• CommentRowNumber8.
• CommentAuthorTobyBartels
• CommentTimeJun 7th 2010

I added a note about models which also alludes to directed equality.

• CommentRowNumber9.
• CommentAuthorMike Shulman
• CommentTimeJun 7th 2010

Ok, fair enough.