This is a survey of the results obtained by K. G lazek and his co-workers. We restrict our attention to the problems of axiomatizations of n-ary groups, classes of n-ary groups, properties of skew elements and homomorphisms induced by skew elements, constructions of covering groups, classifications and representations of n-ary groups. Some new results are added too.

For $$m\ge 4$$ even, the duals of p-ary codes, for any prime p, from adjacency matrices m-ary 2-cube $$Q^m_2$$ are shown to have subcodes with parameters $$[m^2,2m-2,m]$$ which minimal PD-sets size $$\frac{m}{2}$$ constructed, hence attaining full error-correction capabilities code, and, as such, most efficient sets permutation decoding.

We suggest a general method of the construction of secret-sharing schemes based on orthogonal systems of partial (in particular, everywhere determined) k-ary operations which generalizes some known methods of the construction of such schemes by a nite elds and point the orthogonal systems of k-ary operations respecting to these known schemes. The di erent transformations of orthogonal systems o...

We design 4-ary codebooks using Reed Muller codes for multiple-input multiple-output (MIMO) beamforming systems. An upperbound on the maximum correlation magnitude metric of a 4-ary codebook is derived from the minimum Hamming distance of the corresponding binary linear block code. The 4-ary MIMO beamforming codebooks for the number of transmit antennas equal to 4, 8, 16, and 32 are tabulated. ...

We present an algorithm for finding a system of recurrence relations for the number of k-ary words of length n that satisfy a certain set of conditions. A rewriting of these relations automatically gives a system of functional equations satisfied by the multivariate generating function that counts k-ary words by their length and the indices of the corresponding recurrence relations. We propose ...

In this paper we present a construction of every k-ary tree using a forest of (k− 1)ary trees satisfying a particular condition. We use this method recursively for the construction of the set of k-ary trees from the set of (k−1)-Dyck paths, thus obtaining a new bijection φ between these two sets. Furthermore, we introduce a new order on [k]∗ which is used for the full description of this biject...

It is well known that the k-ary n-cube has been one of the most efficient interconnection networks for distributed-memory parallel systems. A k-ary n-cube is bipartite if and only if k is even. Let (X,Y ) be a bipartition of a k-ary 2-cube (even integer k ≥ 4). In this paper, we prove that for any two healthy vertices u ∈ X, v ∈ Y , there exists a hamiltonian path from u to v in the faulty k-ar...

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...