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Are “$S$-monoids” known to be good for anything?

I came up with the following... ...Definition. Let $(S,\wedge,1_S)$ denote a fixed but arbitrary monoid. (In the examples I have in mind, $S$ is always commutative and idempotent. But ...
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How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
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Seeking more information regarding the “hybriation function.”

Definition 0. Given a pair of finite sets $Y$ and $X$, write $Y_X$ for the set of all collections $\mathcal{K}$ of functions $f : Y \leftarrow X$ that are closed under "hybridization", by which I ...
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Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
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How to get intuition in topology concerning the definitions?

Most topology texts go on directly to give definition of topology, then they give some examples and that's it, like they directly tell you right Let $X$ be a set and let $τ$ be a family of subsets ...