Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \rightarrow X$. A quotient of $X$ consists of a $T$-algebra $Y$ together with a surjective homomorphism $X \rightarrow Y$. Also: A congruence on $X$ is […]

Background. Let $x$ denote an arbitrary real number. Then $x^n$ can be defined for each $n \in \mathbb{N}$ as follows: $$x^n = \underbrace{x \times \cdots \times x}_n$$ If $x$ is furthermore non-zero, then we can extend the above definition by declaring that $x^k$ makes sense for each $k \in \mathbb{Z},$ by defining: $$x^{-n} = \underbrace{(1/x) […]

(Reminder: A $T_0$ topological space, also known as a Kolmogorov space, is a space where the topological structure “recognizes” that different points are different: No two points have exactly the same open sets around them.) For a space that is not $T_0$, we can uniquely form a $T_0$ space from it by taking the Kolmogorov […]

I have recently been studying different separation and countability axioms in topology. I am looking for a motivation for why such a refined division of different axioms was made and is studied. I am namely talking about the $T_{k}$ and $N_{k}$ hierarchy. I understand the Hausdorff property, i.e. $T_{2}$-property, is important in analysis and limits; and […]

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered already on StackExchange: note that $(x,y) = (x+y+xy)$ and then use induction. However, is there any clear motivation for using $x+y+xy$ as the […]

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What is the applications of the fractional boundary value problem?

I think of adjoint functors as some sort of inverses. So, the first part of the definition looks reasonable that there exists natural transformations $$\epsilon : FG \rightarrow 1_C$$ $$\eta : 1_D \rightarrow GF$$ But what is the motivation behind the second part of definition? $$1_F = \epsilon F \circ F \eta $$ $$1_G = […]

Disclaimer: My apologies for making such a long question. The question is possibly also rather specific, but I hope that (some parts of) it might be useful in general. Background: I have recently learnt some non-standard analysis and generalisations of limits introduced via (ultra-)filters. The way I see things is strongly inspired by Tao’s short […]

I was discussing some ideas with my professor and he always says that before you work on something in mathematics, you need to know the motivation for studying/working on it. A better way to put this would be, Why were sequences studied and sought after?

Can someone explain what’s the motivation behind the definition of a prime ideal? Or why is it exactly called a prime ideal? Has it anything to do it prime numbers?

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