An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible if Q(x1, . . . , xn) = P ( R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ) where P and R are (n− k+1)-ary and kary quasigroups, σ is a permutation, and 1 < k < n. Anm-ary ...

A non-decreasing sequence of positive integer weights P ={p1, p2,..., pn} (n = N*(m-1) + 1, N is number of non-leaves of m-ary tree) is called absolutely ordered if the intermediate sequences of weights produced by m-ary Huffman algorithm for initial sequence P on i-th step satisfy the following conditions 2 , , ) ( 1 ) ( − = < + N 0 i p p i m i m . Let T be an m-ary tree of size n and M=M(T) b...

In this paper we study nearly perfect sequences (NPS) via their connection to direct product difference sets (DPDS). We prove the connection between a p-ary NPS of period n and type γ and a cyclic (n, p, n, n−γ p + γ, 0, n−γ p )-DPDS for an arbitrary integer γ. Next, we present the necessary conditions for the existence of a p-ary NPS of type γ. We apply this result for excluding the existence ...

In this paper‎, ‎we give the notions of crossed polymodule and cat$^1$-polygroup‎ ‎as a generalization of Loday's definition‎. ‎Then‎, ‎we define the pullback cat$^1$-polygroup and we obtain some results in this respect‎. ‎Specially‎, ‎we prove that by a pullback cat$^1$-polygroup we can obtain a cat$^1$-group‎.

The family tree of a Galton-Watson branching process may contain N -ary subtrees, i.e. subtrees whose vertices have at least N ≥ 1 children. For family trees without infinite N -ary subtrees, we study how fast N -ary subtrees of height t disappear as t → ∞.

By adopting natural generalizations of relational frame and relational model we showed in [2] that the deontic law D, Dp -» Op, is not universally first-order definable. In effect, we showed there that for n > 2, there is no n-adic first-order sentence p such that for every n-ary frame F,F ^D iff F ^(3 in the first-order sense. The notion of n-ary frame and model employed there may be summarize...

The interconnection network considered in this paper is the k-ary n-cube that is an attractive variance of the well-known hypercube. Many interconnection networks, including the ring, the torus and the hypercube, that are attractive in both theoretical interests and practical systems can be regarded as the subclasses of the k-ary n-cubes. In this paper, we investigate the fault tolerant cycle e...

When n is odd, a cohomology of type Hochschild for n-ary partially associative algebras has been defined in Gnedbaye’s thesis. Unfortunately, the cohomology definition is not valid when n is even. This fact is found again in the computations of the n-ary partially associative free algebra. In this work, we define in a first time two approachs of an Hochschild cohomology for n-ary partially asso...

We count all latin cubes of order n ≤ 6 and latin hypercubes of order n ≤ 5 and dimension d ≤ 5. We classify these (hyper)cubes into isotopy classes and paratopy classes (main classes). For the same values of n and d we classify all d-ary quasigroups of order n into isomorphism classes and also count them according to the number of identity elements they possess (meaning we have counted the d-a...