Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [magma]

A magma is a set together with a binary operation on this set. (For questions about the computer algebra system named Magma, use the [magma-cas] tag instead.)

A magma (also called groupoid) is a set $M$ together with a binary operation $M\times M\to M$.

For questions about the Magma computer algebra system, use the tag .

202 questions
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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 & b & e & a &…
celtschk
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55
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9 answers

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…
Red Banana
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5 answers

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,…
user39794
20
votes
5 answers

Why did I never learn about magmas?

While I’ve never taken an actual abstract algebra course, there are some things I know about the typical curriculum structure: First, define an algebraic structure. Explain groups. Everything else. But we seem to skip the most fundamental…
user1286095
17
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2 answers

What is this algebraic object called?

I was playing around with the following object: Let $Q$ be a set with a binary operator $\cdot$ obeying the axioms: $a \cdot a = a$ (idempotence) $a \cdot (b \cdot c) = (a \cdot b) \cdot (a \cdot c)$ (left self-distributivity) Examples of this…
Malper
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16
votes
6 answers

Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like "brackets actually exist" and "associativity is really…
Shaun
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13
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6 answers

Associativity test for a 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…
Matthew Matic
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11
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2 answers

Is there a term for a magma with identity (only)?

If we start from magmas and consider: associativity, identity, invertibility (divisibility). We will theoretically get $2^3=8$ structures by regarding whether such structure possess these properties. As is shown in the picture:…
Upc
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9
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What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the following forms a subset of the functions $X…
goblin GONE
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8
votes
3 answers

Which properties are inherited by the Cartesian product of two sets equipped with a binary operation?

Let $G$ and $H$ denote sets equipped with a binary operation (aka magmas). We can form the Cartesian product magma $G \times H$ in the obvious way. I'm interested in which properties of $G$ and $H$ transfer to $G \times H$. For instance, if both $G$…
goblin GONE
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8
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0 answers

Subtraction Magmas

I was looking at a collection of related closed binary operations on sets (magmas): Subtraction on the integers, reals, etc. Set difference Set symmetric difference Saturating subtraction on the nonnegative integers Ceilinged division on the…
izzyg
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8
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2 answers

Structures with $x*(y*z) = y*(x*z)$

In reading http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=182143561104878BDABB72258DA254D0?doi=10.1.1.18.2521&rep=rep1&type=pdf , they mentioned an interesting relation -- they had a magma $(X,*)$ with the property $x*(y*z) =…
Alex Meiburg
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8
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2 answers

Associanize a magma

This is a thing I have been thinking on and gotten a bit frustrated so I share my thoughts here in hope for clarification. Let $M$ be a magma, that is a set with an underlying binary operation which we denote $\cdot$. The binary operation is not…
Zelos Malum
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7
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1 answer

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…
Nathan BeDell
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7
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1 answer

Is there a term for an algebraic structure with two binary operators that are closed under a set?

For example, let's say we're using the operators +, and *, and the set {0,1,2} The Cayley tables look like this: * 0 1 2 + 0 1 2 0 0 0 1 0 1 2 0 1 1 2 1 1 0 1 0 2 0 0 2 2 1 2 2 These Cayley tables are totally random, but the point is…
Nathan BeDell
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