# 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} }

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

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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…

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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

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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

- MathematicsSIAM J. Discret. Math.
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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

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- 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…

## References

SHOWING 1-10 OF 63 REFERENCES

Quasigroup associativity and biased expansion graphs

- Mathematics
- 2006

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

- MathematicsAm. Math. Mon.
- 2006

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.

Some properties of partly-associative operations

- Mathematics
- 1954

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

- Mathematics
- 1971

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

- Mathematics
- 1946

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

- Mathematics
- 1966

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…