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The axiom of choice was something people had used without always noting that there is an assumption to be made in order to justify making infinitely many choices at once. On the other hand, Cantor felt that there shouldn't be intermediate cardinals between the naturals and the reals, so he hypothesized that this is the case and spent a considerable amount ...


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Suppose you have a function $f:A \to C$, but you regard $A$ as a subset of some other set $i:A \hookrightarrow B$, and you do all your work from $B$. It is natural to ask if the function $f$ extends to a function $\tilde f:B \to C$ such that $f = \tilde f \circ i$. Now, when dealing with subsets, we like to abuse notation and talk about $x \in i(A)$ and ...


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I believe that such a matrix is said to be persymmetric.


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Given an algebra $\mathbf{A}=(A,F)$, the flat (one-point) extension of $\mathbf{A}$ is the algebra $\mathbf{A}^\flat=(A\cup\{0\},F\cup\{\wedge\})$ where each $f$ is extended to $A\cup\{0\}$ by setting all undefined values to 0 and $x\wedge y=0$ for distinct $x,y$ and $x\wedge x=x$. Since sets can be considered as algebras with no operations, your ...


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There are two possible meanings: For any pullback square as below, $$\require{AMScd} \begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \end{CD}$$ if $X \to Y$ is a monomorphism, then $X' \to Y'$ is also a monomorphism. For any commutative diagram of the form below, $$\begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \\ @VVV ...


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I know this question is more than a year old, but for the sake of posterity, the correct term for "the opposite of idempotent" (at least in computer science) is non-idempotent. For example, see section 9.1.2 of Hypertext Transfer Protocol -- HTTP/1.1 (RFC 2616): 9.1.2 Idempotent Methods Methods can also have the property of "idempotence" in that ...



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