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.)
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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 &...
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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$, ...
user39794
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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 ...
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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 ...
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Is there a name for an element $z$ such that $\,zx = z = xz\,$ for all $x$?
Is there a general name for the following properties, (similar to the properties of existence of an additive identity, existence of multiplicative identity etc):
For any given set, the intersection ...
user124485
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5
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Inverse elements in the absence of identities/associativity.
Lets view groups as consisting of a binary operation, a distinguished element $e$, and unary operation $x \mapsto x^{-1}$. Then the group axioms can be stated as follows.
$(xy)z=x(yz).$
$xe=ex=x.$
$...
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Finite magmas representing all unary functions by terms
Say that a magma $\mathcal{M}=(M;*)$ is unary-rich iff for every function $f:M\rightarrow M$ there is a (one-variable, parameter-free) term $t_f$ such that $t_f^\mathcal{M}=f$. For example:
The one-...
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Are epimorphisms in the category of magmas surjective?
The question says it all, but let me recall the definitions.
A magma $(X, \cdot)$ is a set $X$ with a binary operation $\cdot \colon X \times X \to X$ (without any further assumptions like ...
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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 ...
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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. ...
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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.
...
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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:
...
- 3,285
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The ratio of finitely based magmas to all magmas
Let $n$ be a positive integer. By $S_n$, I denote the set of positive integers from $1$ to $n$. By $F_n$, I denote the cardinality of the set of magmas on $S_n$ which are finitely based, that is, ...
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Rings with primal term reducts
This question is a follow-up to this one.
Say that a term reduct of a ring $\mathcal{R}=(R; +,\times,0,1)$ is a magma $\mathcal{M}$ whose domain is $R$ and whose magma operation is $(x,y)\mapsto t(x,y)...
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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 ...
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