# 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