Ary Quasigroups Research Articles

Embedding in MDS codes and Latin cubes

AbstractAn embedding of a code is a mapping that preserves distances between codewords. We prove that any code with code distance and length can be embedded into an maximum distance separable (MDS) code with the same code distance and length but under a larger alphabet. As a corollary we obtain embeddings of systems of partial mutually orthogonal Latin cubes and ‐ary quasigroups.

Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube.
 We prove that if a group $G$ of order $n$ satisfies the Hall–Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.

Transversals, Plexes, and Multiplexes in Iterated Quasigroups

A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$. It is common to call 1-plexes transversals. In this paper we prove that there exists a constant $c(G,k)$ such that if a $d$-iterated quasigroup $G$ of order $n$ has a $k$-multiplex then for large $d$ the number of its $k$-multiplexes is asymptotically equal to $c(G,k) \left(\frac{(kn)!}{k!^n}\right)^{d-1}$. As a corollary we obtain that if the number of transversals in the Cayley table of a $d$-iterated quasigroup $G$ of order $n$ is nonzero then asymptotically it is $c(G,1) n!^{d-1}$. In addition, we provide limit constants and recurrence formulas for the numbers of transversals in two iterated quasigroups of order $5$, characterize a typical $k$-multiplex and estimate numbers of partial $k$-multiplexes and transversals in $d$-iterated quasigroups.

On extensions of partial n-quasigroups of order 4

We prove that every collection of pairwise compatible (nowhere coinciding) n-ary quasigroups of order 4 can be extended to an (n + 1)-ary quasigroup. In other words, every Latin 4×…×4 × l-parallelepiped, where l = 1, 2, 3, can be extended to a Latin hypercube.

On connection between the switching separability of a graph and its subgraphs

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: if removing any one or two vertices of a graph always results in a switching separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order greater than 4, there is a graph that is not switching separable, but removing any vertex always results in a switching separable subgraph. We show a connection with similar facts on the separability of Boolean functions and reducibility of $n$-ary quasigroups. Keywords: two-graph, reducibility, separability, graph switching, Seidel switching, graph connectivity, $n$-ary quasigroup

On connection between reducibility of an [formula omitted]-ary quasigroup and that of its retracts

An n -ary operation Q : Σ n → Σ is called an n -ary quasigroup of order | Σ | if in the equation x 0 = Q ( x 1 , … , x n ) knowledge of any n elements of x 0 , … , x n uniquely specifies the remaining one. An n -ary quasigroup Q is (permutably) reducible if Q ( x 1 , … , x n ) = P ( R ( x σ ( 1 ) , … , x σ ( k ) ) , x σ ( k + 1 ) , … , x σ ( n ) ) where P and R are ( n − k + 1 ) -ary and k -ary quasigroups, σ is a permutation, and 1 < k < n . An m -ary quasigroup R is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n − m > 0 arguments. We show that every irreducible n -ary quasigroup has an irreducible ( n − 1 ) -ary or ( n − 2 ) -ary retract; moreover, if the order is finite and prime, then it has an irreducible ( n − 1 ) -ary retract. We apply this result to show that all n -ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z 5 or Z 7 are reducible for n ≥ 4 .

N-Ary Quasigroups of Order 4

We characterize the set of all N-ary quasigroups of order 4: every N-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an N-ary quasigroup can be represented as a composition of K-ary and (N-K+1)-ary quasigroups for some K from 2 to N-1, where the order of arguments in the representation can differ from the original order. The set of semilinear N-ary quasigroups has a characterization in terms of Boolean functions. Keywords: Latin hypercube, n-ary quasigroup, reducibility

Frame cellular automata: Configurations, generating sets and related matroids

We introduce a frame cellular automaton as a broad generalization of an earlier study on quasigroup-defined cellular automata. It consists of a triple ( F , R , E F ) where, for a given finite set X of cells, the frame F is a family of subsets of X (called elementary frames, denoted S i , i = 1 , … , n ), which is a cover of X . A matching configuration is a map c ¯ : X → G which attributes to each cell a state in a finite set G under restriction of a set of local rules R = { R i ∣ i = 1 , … n } , where R i holds in the elementary frame S i and is determined by an ( | S i | -1)-ary quasigroup over G . The frame associated map E F models how a matching configuration can be grown iteratively from a certain initial cell-set. General properties of frames and related matroids are investigated. A generating set S ⊂ X is a set of cells such that there is a bijection between the collection of matching configurations and G S . It is shown that for certain frames, the algebraically defined generating sets are bases of a related geometric-combinatorially defined matroid.

On irreducible [formula omitted]-ary quasigroups with reducible retracts

An n -ary operation Q : Σ n → Σ is called an n -ary quasigroup of order | Σ | if in x 0 = Q ( x 1 , … , x n ) knowledge of any n elements of x 0 , … , x n uniquely specifies the remaining one. An n -ary quasigroup Q is permutably reducible if Q ( x 1 , … , x n ) = P ( R ( x σ ( 1 ) , … , x σ ( k ) ) , x σ ( k + 1 ) , … , x σ ( n ) ) where P and R are ( n − k + 1 ) -ary and k -ary quasigroups, σ is a permutation, and 1 < k < n . For even n we construct a permutably irreducible n -ary quasigroup of order 4 r such that all its retracts obtained by fixing one variable are permutably reducible. We use a partial Boolean function that satisfies similar properties. For odd n the existence of permutably irreducible n -ary quasigroups with permutably reducible ( n − 1 ) -ary retracts is an open question; however, there are nonexistence results for 5-ary and 7-ary quasigroups of order 4.

Generalized Hilton construction for embedding d-ary quasigroups

It is shown that if a partial idempotent d-ary quasigroup P of order n is embedded in a d-ary quasigroup of order 2 n, then P can also be embedded in an idempotent d-ary quasigroup of order 4 n.