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I suspect that I'm missing some category theoretic/universal algebraic terminology which would make this question more efficient. Feel free to edit to include that!
Given a variety (in the sense of universal algebra) $\mathbb{V}$ and a set $X$, let $\mathbb{V}(X)$ be the set of $\mathbb{V}$-algebras with underlying set $X$. Say that a a $(\mathbb{V},X)$-umbrella is a pair $(M,(f_A)_{A\in\mathbb{V}(X)})$ where $M...

Let $p_1, p_2, p_3, p_4, p_5 ... = 2, 3, 5, 7, 11, ...$ be theprime numbers in increasing order. Let $k > 5$ and$A_k = {1, 4, 5, 6, 7, 9, 11, p_6, ..., p_k}$.In $A_k$ all Least Common Multiples are distinctand $A_k$ contains $k+2$ elements, so one...

Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach algebra with involution (or, we can say, the structure...

For each positive integer $N$, define the set$$\mathcal S_N=\{(m,n,a,b):m,n\ge1,\ a,b\ge0,\ 2mn+m+an+b(2m-1)=2N\}$$and the associated sum$$a_N:=\sum_{(m,n,a,b)\in\mathcal S_N}(-1)^{m+a}.$$
I am interested with the below problem.
QUETSION. Is there a combinatorial proof that $a_N=0$?...