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Sophomore's dream is especially the statement that the sum, let me call it $s$, of the (convergent) real series $\sum_{n \ge 1} n^{-n}$ is equal to the (improper) integral $\int_0^1 x^{-x} dx$. A few …

Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the cou …

This is basically another reference request. Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that t …

This is basically a reference request, but the post is going to be relatively long (and a little bit verbose): I apologize in advance for that. Premise. There are several examples of "ordered structu …

Let $\mathbf Z_K$ be the ring of integers of an algebraic number field $K$. It is well known that $\mathbf Z_K$ has infinitely many non-associated atoms (and hence is not a Cohen-Kaplansky domain). …

Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that an element $u \in S$ is regular if (quote) "[...] it can be cancelled, i. …

Q1. Is there any standard name for a (multiplicatively written) monoid $H$ with the property that, for all $x, y \in H \setminus H^\times$, there exist $m, n \in \mathbf N^+$ and $u, v \in H^\times …

Introduction. Allow me to use the NBG axiomatic system as a foundation (*). Charles Ehresmann is acknowledged as the first one to have introduced the idea of multiplicative graphs as a further level o …

I like Shakespeare and Greek tragedy, so let me word it as I'm doing: I desperately need J.H.B. Kemperman's 1956 paper On complexes in a semigroup, but the online archive of Indagationes Mathematicae, …

This is again a request for references. I'd appreciate a pointer to any published proof of the following: Proposition. Given $n \in \mathbb{N}^+$, let $\Phi$ be a function $\mathbb{C}^n > \to \m …

Hi there. Suppose ${\bf C}_1$, ${\bf C}_2$ and $\bf D$ are categories and $F_i$ is a functor ${\bf C}_i \to \bf D$. Consider the subcategory of the comma category $( F_1 \downarrow F_2)$ whose objects …

The question is fairly dry: Is there any semigroup analogue of Lagrange's theorem for groups (counting as a generalization of the latter)? Let me guess the answer: Obviously yes. So the real question …

As pointed out by Mark Sapir in his answer to a related question, every residually finite divisible semigroup is idempotent (hence uniquely divisible). On another hand, it is not difficult to prove th …

Let $X = \{x, y\}$ be a two-element set, and let $H$ be the monoid defined by the presentation $$ \langle x, y \mid x y^k x = y x y^{k+1} x y, \text{ for } k = 0, 1, 2, \ldots\rangle. $$ That is, $H$ …

This is primarily a request for references and advices. Question (edited on 10/29/2011). What's known about comprehensive generalisations of Gelfand's spectral theory for unital [associative] …