Actual Question
I need to describe the relation of summing the parts of geometric figures such as line segments and angles: for example, the two angles on either side of a line that intersects another line sum to the same as the sum of two right angles. However, not all sums exist because summing is an operation that joins two parts of a larger whole into the whole. Two angles can only be summed if they are disjoint and adjacent. Summing two angles in this sense basically means ignoring the line between them to get the larger angle. Summing two line segments is only allowed when they are part of the same line, disjoint, and meet in a point.
So I'm hoping there is an abstract algebra with a predicate $Sxy$ (saying that $x$ and $y$ can be summed) and an operation $+$ such that
- $Sxy\rightarrow x+y=y+x$
- $(x+y)+z=x+(y+z)$ assuming all the sums exist
There is no identity element.
Explanation in case anyone cares
I've been trying to map Euclid's theory of magnitudes into modern math. It's interesting how many modern ideas he prefigured--in fact, I think the theory of magnitudes might be the first abstract algebra. The ToM is a set of rules that applies to magnitudes of several types: length of a straight line segment, areas of a closed plane figure, volume of a solid, and angle. I also think it probably applies to Archimedes proof of the law of the lever.
Two concepts Euclid uses are parthood and sum:
The whole is greater than the part
Equals added to equals are equal
Euclid did not distinguish between object and magnitude. For example, he does not distinguish between the line segment and its length. He says two line segments are equal, meaning that their length is equal. What I am trying to do is rewrite this into a modern theory beginning with a mereology (a theory of parthood), adding an equivalence relation "equi-magnitude", and then using an abstraction principle to define magnitudes from the equivalence relation, and the order of the magnitudes based on parthood.
In a sense I am trying to do for the positive real numbers what Frege did for the natural numbers, using Euclid's theory of magnitudes as the starting point.