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I have a question probably related to pure mathematics, but I'm not sure. I have some line segments (in 1-dimensional space), where I want to define a function over them to make a union of them. However, my definition of union means a new line segment starting from the minimum position between line segments, and finishing at the maximum position of end positions between line segments. Is there any algebra where some operator (for example + or $\cup$) in it satisfies my requirement.

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There is certainly an algebraic structure you can use. Let $A_{min}$ be the min-plus algebra and $A_{max}$ be the max-plus algebra (don't be fooled by the names, these are technically semirings). Consider the semiring $A_{min}\times A_{max}$, which consists of pairs $(a,b)\in [-\infty,\infty)\times (-\infty,\infty]$. Then you are looking for the subsemiring of $A_{min}\times A_{max}$ with $a<b$ (or $\leq$ if you allow degenerate intervals).

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