# Associativity in multiary quasigroups: the way of biased expansions

```@article{Zaslavsky2004AssociativityIM,
title={Associativity in multiary quasigroups: the way of biased expansions},
author={Thomas Zaslavsky},
journal={Aequationes mathematicae},
year={2004},
volume={83},
pages={1-66}
}```
• T. Zaslavsky
• Published 11 November 2004
• Mathematics
• Aequationes mathematicae
A multiary (polyadic, n-ary) quasigroup is an n-ary operation which is invertible with respect to each of its variables. A biased expansion of a graph is a kind of branched covering graph with an additional structure similar to the combinatorial homotopy of circles. A biased expansion of a circle with chords encodes a multiary quasigroup, the chords corresponding to factorizations, i.e., associative structure. Some but not all biased expansions are constructed from groups (group expansions…
9 Citations
Quasigroup associativity and biased expansion graphs
We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We
Projective planarity of matroids of 3-nets and biased graphs
• Mathematics
Australas. J Comb.
• 2020
Criteria for embeddability of biased-graphic matroids in Desarguesian projective spaces is established, that is, embeddable in an arbitrary projective plane that is not necessarily Desargue'sian.
Biased Expansions of Biased Graphs and Their Chromatic Polynomials
A biased graph is a graph with a distinguished set of circles, such that if two circles in the set are contained in a theta graph, then so is the third circle of the theta graph. We introduce a new
n-Ary Quasigroups of Order 4
• Mathematics
SIAM J. Discret. Math.
• 2009
Every n-ary quasigroups of order 4 is permutably reducible or semilinear, which means that an \$n\$-aryQuasigroup can be represented as a composition of \$k-ary and \$(n-k+1)\$-aries for some \$k\$ from 2 to \$n-1\$, where the order of arguments in the representation can differ from the original order.
On one test for the switching separability of graphs modulo q
• Mathematics
• 2016
We consider graphs whose edges are marked by numbers (weights) from 1 to q - 1 (with zero corresponding to the absence of an edge). A graph is additive if its vertices can be marked so that, for
A ug 2 01 9 Constructions of transitive latin hypercubes
• Mathematics
• 2019
A function f : {0, ..., q−1}n → {0, ..., q−1} invertible in each argument is called a latin hypercube. A collection (π0, π1, ..., πn) of permutations of {0, ..., q − 1} is called an autotopism of a
Biased graphs. VI. synthetic geometry
• Mathematics
Eur. J. Comb.
• 2019

## References

SHOWING 1-10 OF 63 REFERENCES
Quasigroup associativity and biased expansion graphs
We present new criteria for a multary (or polyadic) quasigroup to be isotopic to an iterated group operation. The criteria are consequences of a structural analysis of biased expansion graphs. We
Noncrossing Partitions in Surprising Locations
The noncrossing partition lattice is introduced in three of its many guises: as a way to encode parking functions, as a key part of the foundations of noncommutative probability, and as a buildingblock for a contractible space acted on by a braid group.
Biased graphs. I. Bias, balance, and gains
Some properties of partly-associative operations
Three properties of a group-operation are (i) it is associative: (xy)z=x(yz); (ii) it is regular: a=b if ax=bx or if ya=yb; and (iii) it is reversible: ax= ya =b is solvable for x and y. These
Varieties of combinatorial geometries
• Mathematics
• 1982
A hereditary class of (finite combinatorial) geometries is a collection of geometries which is closed under taking minors and direct sums. A sequence of universal models for a hereditary class 'S of
Balanced identities in algebras of quasigroups
This section consists of self-contained 25-100 line short communications which are prepublications of results, the details of which are to be published in either aequationes mathematicae or other
Contributions to the theory of loops
Introduction. It is not altogether easy to single out the dominant idea of the present paper; perhaps this may be said to be the notion of a ir-series, which is introduced in Chapter I and recurs in
Solution of Belousov's problem
• Mathematics
• 2000
The authors prove that a local \$n\$-quasigroup defined by the equation x_{n+1} = F (x_1, ..., x_n) = [f_1 (x_1) + ... + f_n (x_n)]/[x_1 + ... + x_n], where f_i (x_i), i, j = 1, ..., n, are arbitrary
Connectivity in graphs
a b x u y w v c d Typical question: Is it possible to get from some node u to another node v ? Example: Train network – if there is path from u to v , possible to take train from u to v and vice