Hugh Denoncourt
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 Mar31 comment Lemma regarding subgroup of group of automorphisms The automorphism $\tau$ is applied to the coefficients of $P(x)$: If $P(x) = a_n x^n + \cdots + a_0$, then $\tau P(x) = \tau(a_n) x^n + \cdots + \tau(a_0)$. Since $\tau$ extends to a ring homomorphism from $A[x]$ to $A[x]$, we can apply $\tau$ to each factor in the product representation. The point is that $\tau P(x) = P(x)$ implies $\tau(a_i) = a_i$ for any coefficient $a_i$ of $P(x)$. Since this holds for all $\tau \in G$, the coefficients of $P(x)$ are in $A^G$. Mar26 answered 4D Geometry Book Mar22 comment What happens if singleton set is not closed The trivial topology (only $\emptyset$ and $X$ are open) is an example of a topology on $X$ such that singleton sets are not closed, assuming $|X| > 1$. Mar6 answered Reference request: calculus of variations Mar3 revised How would I put these recurrence relation terms into a summation? Inserted \left and \right on parentheses that needed them. Mar3 suggested approved edit on How would I put these recurrence relation terms into a summation? Feb23 comment Recognizing subgroups With the last two examples, don't use the same ordered pair to check closure. Use $(x,y)$ and $(x', y')$ or some variant of that. Feb22 comment Mutual difference of vectors squared, does it have a name? I suspect (but don't know as it wasn't me) the issue is with $\overrightarrow{v}^2$, which is not standard for any known vector operation. If you mean dot product, then $\overrightarrow{v} \cdot \overrightarrow{v}$ or $\Vert \overrightarrow{v} \Vert^2$ fixes this issue. Feb19 answered Name of a particular “sum of products” function Feb15 comment Cycle structures of $S_6$ Yes, that is probably the simplest approach. Feb12 comment General Classification of finite simple ternary groups? Universal algebra provides natural definitions that generalize those you find for groups, rings, etc. "Simple algebra" is probably the definition you seek. Unlike groups and rings, the congruence relations (eq-relations preserving the operations) may not correspond to a sub-(ternary group), so there may not be a natural definition of "normal". But there will be one for "simple". Standard results in universal algebra also imply isomorphism theorems for your structure, substructure theorems, etc. (If you include the inversion operation into your signature, Birkhoff's HSP theorem applies.) Jan18 comment linear independence on polynomials No. None of the polynomials have to be the zero polynomial. The hypothesis $p_j(2) = 0$ affects the coefficients of the polynomials, though it's not obvious how in the standard basis. (This is why I suggest using an alternative basis.) Jan18 comment linear independence on polynomials One possibility is to use a change of basis: Instead of using $x^0,x^1,\ldots,x^m$ as a basis, you might use $(x-2)^0,(x-2)^1,\ldots,(x-2)^m$ as a basis. Jan18 comment linear independence on polynomials Given the way it's worded, the field $\mathbb{F}$ is arbitrary, so it is not sufficient to prove it for a specific field. Jan13 comment Axioms as recreational mathematics A given closed subset of $\text{End}(X)$ is necessarily a monoid, but of course, it could satisfy additional axioms. (e.g. $\text{Aut}(X) \subset \text{End}(X)$ and is closed under $\circ$ and $e$ and satisfies the group axioms.) But if you add an axiom that isn't a consequence of the monoid axioms, there will exist a closed subset of some $\text{End}(X)$ that doesn't satisfy that axiom. I understood your question as asking whether the class of subsets of $\text{End}(X)$'s of a particular special form have additional axioms and its own "Cayley representation theorem". This is possible. Jan13 comment Axioms as recreational mathematics Related, but I don't think this contains your $\mathcal{P}(X)$ question: math.stackexchange.com/questions/218353/… Jan13 comment Axioms as recreational mathematics Given any monoid $M$, there is an embedding of $M$ into $\text{End}(M)$. The result and proof is similar to Cayley's theorem for groups. In other words, abstract monoids are concrete monoids. The monoid axioms suffice to study algebraically closed subsets of $\text{End}(X)$. The paragraph regarding the image function Im led me to believe you are looking to characterize monoids arising from submonoids (ie algebraically closed subsets with e) of $\text{End}(M)$ via Im. I guess I'm unclear what concrete structure you seek axioms for in that paragraph. Jan13 comment Axioms as recreational mathematics Just to clarify: End and Aut are relative to unstructured sets? i.e. $\text{End}(X)$ is the set of all functions from $X$ to $X$ and $\text{Aut}(X)$ is the set of all invertible functions from $X$ to $X$, right? Also, when you speak of special monoids induced by a submonoid of $\text{End}(X)$, is the result a submonoid of $\text{End}(\mathcal{P}(X))$? I think it's clear from context, but I am uncertain! Dec22 comment Prove $X\times Y$ is an equivalence relation No, that's okay. It's not about the votes or anything for me anyway. It'd be weird to edit the question to what I think is being asked and then to answer it. I really think you should re-post with the exact text of the problem instead of referring users to my comment. (More ambiguities! Which comment?) Dec22 comment Prove $X\times Y$ is an equivalence relation If the relation is on $X \times Y$, then you have to specify how two pairs are related. When is $(x,y)$ related to $(x',y')$? This is nowhere in your question.