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text: 'An **associative operation** $\\bullet : X \\times X \\to X$ is a [3kb binary] [3h7 operation] such that for all $x, y, z$ in $X$, $x \\bullet (y \\bullet z) = (x \\bullet y) \\bullet z$. For example, $+$ is an associative function, because $(x + y) + z = x + (y + z)$ for all values of $x, y,$ and $z$. When an associative function is used to combine many elements in a row, parenthesis can be dropped, because the order of application is irrelevant.\n\nImagine that you're trying to use $f$ to combine 3 elements $x, y,$ and $z$ into one element, via two applications of $f$. $f$ is associative if $f(f(x, y), z) = f(x, f(y, z)),$ i.e., if the result is the same regardless of whether you apply $f$ to $x$ and $y$ first (and then apply that result to $z$), or whether you apply $f$ to $y$ and $z$ first (and then apply $x$ to that result).\n\nVisualizing $f$ as a [3mb physical mechanism], there are two different ways to hook up two copies of $f$ together to create a function $f_3 : X \\times X \\times X \\to X,$ which takes three inputs and produces one output:\n\n\n\nAn associative function $f$ is one where the result is the same no matter which way the functions are hooked up, which means that the result of using $f$ twice to turn three inputs into one output yields the same output regardless of the order in which we combine adjacent inputs.\n\n\n\nBy similar argument, an associative operator $f$ also gives rise (unambiguously) to functions $f_4, f_5, \\ldots,$ meaning that [3ms associative functions can be seen as a family of functions on lists].\n\nThis justifies the omission of parenthesis when writing expressions where an associative operator $\\bullet$ is applied to many inputs in turn, because the order of application does not matter. For example, multiplication is associative, so we can write expressions such as $2 \\cdot 3 \\cdot 4 \\cdot 5$ without ambiguity. It makes no difference whether we compute the result by first multiplying 2 by 3, or 3 by 4, or 4 by 5.\n\nBy contrast, the function `prependx` that sticks its inputs together and puts an `x` on the front is not associative: `prependx(prependx("a","b"),"c") = "xxabc"`, but `prependx("a",prependx("b","c"))=xaxbc`.',
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