Dixon’s famous theorem states that the group generated by two random permutations of a finite set is generically either the whole symmetric group or the alternating group. In the context of random generation of finite groups this means that it is hopeless to wish for a uniform distribution – or even a non-trivial one – by drawing random permutations and looking at the generated group. Mealy aut...

The main aim of this work is to introduce and study the notions ideal direct product d-algebras, d-ideal sub-direct edge positive implicative d-algebras investigate their characterizations.

Let α(G) denote the independence number of a graph G and let G ×H be the direct product of graphs G and H. Set α(G ×H) = max{α(G) · |H|, α(H) · |G|}. If G is a path or a cycle and H is a path or a cycle then α(G×H) = α(G×H). Moreover, this equality holds also in the case when G is a bipartite graph with a perfect matching and H is a traceable graph. However, for any graph G with at least one ed...

We classify the orbits of coquasi-triangular structures for the Hopf algebra E(n) under the action of lazy cocycles and the Hopf automorphism group AutHopf (E(n)). Using this result we show that for any triangular structure R on E(n) the Brauer group BM(k,E(n), R) is the direct product of the Brauer-Wall group of the base field k and the group Symn(k) of symmetric matrices of order n with entri...