Questions tagged [quasigroups]
A quasigroup is a grouplike structure $(Q, \ast)$, that satisfies the Latin square property but need not have an identity element, nor need it be associative. It coincides with the notion of a divisible magma.
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Practical example of differences between associativity and alternativity (and the in-between Bol loop)? [closed]
Associativity is:
$$(a * b) * c = a * (b * c)$$
Alternativity is:
$$a * (a * b) = (a * a) * b$$
$$(a * b) * b = a * (b * b)$$
Bol loop is:
$${\displaystyle a(b(ac))=(a(ba))c}$$
$${\displaystyle ((ca)b)...
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Term for a Set Equipped With a Binary Operation Which Contains Inverses
Let $A$ be a set and let $\circ:A\times A\rightarrow B,$ $A\subseteq B$ be a binary operation ($A$ is not necessarily closed under $\circ$). If there exists some unique $e\in A$ such that $e\circ a=a\...
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Salzmann's topological loop example
I've been looking for loop examples and stumbled upon this post.
The answer describes a specific operation on $\mathbb{R} $, namely this:
$$x\ast t =
\begin{cases}
\ x+\frac{1}{2}t, &\text{ if }\ ...
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Can we characterize the “associate classes” of a unipotent quasi-commutative quasigroup as some combinatorial design?
$I_n$ is the $n \times n$ or order $n$ identity matrix, $J_n$ is the order $n$ matrix of all ones, and $n \in \mathbb{Z}^+$.
We define a Latin square $\mathcal{L_n}$ to be a set of $n$ permutation ...
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How can a quasigroup have a division operation if a group has only one operation?
I'm teaching myself abstract algebra.
We can define a magma $\left(M,\cdot\right)$ as $a,b\in M\implies a\cdot b\in M$.
I get confused when we talk about quasigroups.
A group must satisfy all ...
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References to for Quasigroup Theory
Can anyone suggest some references (books or articles) to understand Quasigroup Theory. I need very easy to understand reference if there is one. Thank you very much.
Edit:
Note: I have checked ...
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Uniqueness of quotients in semigroup with divisibility?
Let $G$ be a semigroup, e.g., a set with an associative binary operation. Suppose further that $G$ has the divisibility property, e.g., for all $x,y\in G$ there exist $\ell,r\in G$ such that $\ell x=y$...
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Matrix representation of Quasigroups
This paper says that, each quasigroup of order 4 can be represented in matrix form using the following equation,
\begin{equation} x \ast y \equiv m^T +Ax^T +By^T +CA\cdot x^T \circ CB\cdot y^T \end{...
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Can $S^2$ be given a (semi-)topological quasigroup structure?
It is known from this question that $S^2$ cannot be made into a topological group. Indeed, $S^2$ cannot even be made into an $H$-Space, a much looser requirement than a topological group.
However, an $...
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Do the Moufang identities *themselves* imply diassociativity / Moufang's theorem / Artin's theorem?
A Moufang loop is a loop satisfying the Moufang identities. Famously, these are diassociative -- the subloop generated by any two elements is associative (is a group) -- and more generally, they ...
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Why is the formal definition of Latin square equivalent with the informal?
Informally, a latin square is a table where each element appears exactly once in each row and each column.
I know that this is probrably not an official definition of, however, it should somehow match ...
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Is there a proof that the total number of idempotent elements over all quasigroups of order n equals the number of quasigroups of that order?
To clarify, as requested by Community
I am looking for a proof that the total number of idempotent elements over all quasigroups of order n equals the number of quasigroups of that order.
Could be ...
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Inverse element of a magma
It is accepted that two elements are inverse to each other if their product is equal to the identity element:
Inverse element in a magma
https://en.wikipedia.org/wiki/Inverse_element
The definition ...
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Example of a pair of non-isomorphic quasi-groups with parastrophic Latin squares?
A Latin square $\Lambda$ over an alphabet $A$ is a set of triples of elements of $A$ such that for every $\alpha,\beta\in A$, there is exactly one $\gamma\in A$ for which $(\alpha,\beta,\gamma)\in \...
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An example of an algebraic loop which has different L and R inverses?
Can anyone point me toward a simple example of a non-associative algebraic loop (i.e. a quasigroup with an identity) for which at least one element has a left inverse which is not equal to its right ...
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