Articles of magma

Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by $(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots$). Now consider the set S of all such sequences, $$S=\left\{(e^{2i1\pi/n}, e^{2i2\pi/n},\dots, e^{2ik\pi/n},\dots,e^{2in\pi/n}, 0, 0, 0,\dots) : […]

Efficient algorithm for calculating the tetration of two numbers mod n?

I’m trying to study the algebraic properties of the magma created by defining the binary operation $x*y$ to be: $ x*y = (x \uparrow y) \bmod n $ where $ \uparrow $ is the symbol for tetration. Doing so, the hardest commutation necessary for this kind of magma of order n is: $ (n-1) \uparrow […]

Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$

When we can define a binary operation $\cdot:M\times M\rightarrow M$ on an algebraic structure $(M,*)$ such that $$a*(b*c)=(a\cdot b)*c$$ If $*$ is associative then $\cdot=*$ even if I’m not sure about the uniqueness (But In right-invertible associative structures this is provable) If $*$ is right-invetible then $a\cdot b=(a*(b*c))\setminus c$ only if $a\cdot b$ doesn’t depends […]

Properties of finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$?

I am considering finite magmas $(S,\cdot)$ with $\forall(x,y,z)\in S^3, x\cdot(y\cdot z)=y\cdot(x\cdot z)$. Any finite commutative group is an example of such thing. But in the context (this question on crypto.stackexchange.com), I am not interested in groups; or at least, not in groups with an efficiently computable inverse. I am wondering if/how the classical magma-to-group classification […]

Can you give me some concrete examples of magmas?

I’ve seen the following (e.g. here): I’ve learned a bit about groups and I could give examples of groups, but when reading the given table, I couldn’t imagine of what a magma would be. It has no associativity, no identity, no divisibility and no commutativity. I can’t imagine what such a thing would be. Can […]

Elegant approach to coproducts of monoids and magmas – does everything work without units?

From this answer by Martin Brandenburg I learned an elegant way of constructing and describing elements of coproducts of monoids. There he writes this construction shows the existence of colimits in every finitary algebraic category. I’m trying to understand the significance of having units, and it seems to me units are actually used in the […]

A finite, cancellative semigroup is a group

Let $G$ be a finite, nonempty set with an operation $*$ such that $G$ is closed under $*$ and $*$ is associative Given $a,b,c \in G$ with $a*b=a*c$, then $b=c$. Given $a,b,c \in G$ with $b*a=c*a$, then $b=c$. I want to prove that $G$ is a group, but I don’t know how to show that […]

Is there an easy way to see associativity or non-associativity from an operation's table?

Most properties of a single binary operation can be easily read of from the operation’s table. For example, given $$\begin{array}{c|ccccc} \cdot & a & b & c & d & e\\\hline a & e & d & b & a & c\\ b & d & c & e & b & a\\ c & […]

Associativity for Magma

Say I have the operation table for a magma. I want to know whether or not the operation is associative. However, associativity is defined for an operation on 3 elements, and the operation table deals only with two. So it is not clear to me how to determine whether operation is associative by looking only […]