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Any search for "theory of equations" leads to material on polynomial equations and on how to find solutions to them. This is not what I'm looking for.

I want something (I'm not sure it's broad enough to merit being regarded a "theory") which I believe would be a subtopic in algebra that systematizes how equations and systems of equations work, the algebraic actions one can carry on them, how each step in the solution of an equation or a system of equations relates to the previous one in terms of logical implication, etc. Anyone know if there are any texts on this and what this is usually referred to as?

jvf
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I suspect that what you're looking for is universal algebra, and I think you're looking for the topic of equational logic in particular (although this old post of mine may also be of interest). The freely-and-legally-avialable book by Burris and Sankappanavar has a good presentation; see in particular the area around the Birkhoff completeness theorem which provides an explicit proof system for reasoning about equations. Note that you'll want to distinguish between equations which are asserting universal behavior (e.g. "$x+y=y+x$" holds for all $x,y$ usually) and equations describing specific-but-unknown objects (e.g. "$x+2=7$" as an expression which we want to solve); syntactically in equational logic this separation is captured by the distinction between variables (which are implicitly universally quantified) and constant symbols.

This should be compared with model theory and first-order logic specifically. Basically, FOL is what we get by augmenting equational logic by allowing Booleans (like "not" and "or") and quantifiers. There's a ton of difference between the two logics, and accordingly between the two subjects built around them.

Noah Schweber
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