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4

You can define the period of an element $a$ : the smallest integer $n>0$ such that there is $k\in \mathbb{N}$ with $a^{k+n} = a^k$. Of course like the (finite) order of an element it may not exist (the element is aperiodic and generates an infinite sub-semigroup) ; and for a group it coincides with the order of the element.


3

The following definitions were given in the reference book [1], p. 19 and seem to have been accepted since then: The order of an element $a$ is the cardinal of the semigroup $\langle a \rangle$ generated by $a$. If $\langle a \rangle$ is finite, there exist integers $i, p > 0$ such that $a^{i+p} = a^i$. The minimal $i$ and $p$ with this property are ...


2

I have seen these rings discussed, but never a never with a name attached. The reason is probably because this class of rings is a disjoint union of the following two classes of rings which do not need complicated names and which have rather divergent properties compared to each other: Rings of Krull dimension 0 Domains of Krull dimension 1


2

Yes! It is called a 2-cell; in general, a k-cell is a closed set of points (meaning that the boundaries are included) in k dimensions with "straight" boundaries. So a 3-cell is a filled in rectangular prism, a 1-cell is a closed interval, [a,b], and a 0-cell is a point.


2

$A=\frac{1}{n}+\frac{1}{n^2}+...+\frac{1}{n^s}$ $nA=1+\frac{1}{n}+\frac{1}{n^2}+...$ $nA-A=1$ $A=\frac{1}{n-1}$ So the sum equals $\frac{1}{n-1}$. "summation from n equals one to infinity of one over n to the s."


1

It's often called the positive (if open) or non-negative (if closed) orthant.


1

If you are looking for a topological version, a filled square without borders is an $\ell_\infty$ open ball, $\max (|x|,|y|)<1$. Same with the rectangle, with some anisotropy.


1

The authors are partitioning the set $\mathscr{F}$ of the $27$ functions from $S=\{0,1,2\}$ to $S$. For brevity we denote by $abc$ the function $f\in\mathscr{F}$ such that $f(0)=a,f(1)=b$, and $f(2)=c$. Two functions $f,g\in\mathscr{F}$ are equivalent if there is a permutation $\varphi$ of $S$ such that $g=\varphi\circ f$. This really is an equivalence ...


1

I can suggest some names, which I have learned from others who research these things. A 4-simplex, or 4D analogue of a triangle, is called a Pentachoron , describing a regular, 5-sided 4D polytope. Also called a 5-cell. These are the n-simplex. There are 4 types of ring torus objects in 4D, which can be seen visualized here. The general name of ...



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