A description of a class of finite semigroups that are near to being Malcev nilpotent

In this paper we continue the investigations on the algebraic structure of a finite semigroup $S$ that is determined by its associated upper non-nilpotent graph $\mathcal{N}_{S}$. The vertices of this graph are the elements of $S$ and two vertices are adjacent if they generate a semigroup that is not nilpotent (in the sense of Malcev). We introduce a class of semigroups in which the Mal'cev nilpotent property lifts through ideal chains. We call this the class of \B\ semigroups. The definition is such that the global information that a semigroup is not nilpotent induces local information, i.e. some two-generated subsemigroups are not nilpotent. It turns out that a finite monoid (in particular, a finite group) is \B\ if and only if it is nilpotent. Our main result is a description of \B\ finite semigroups $S$ in terms of their associated graph ${\mathcal N}_{S}$. In particular, $S$ has a largest nilpotent ideal, say $K$, and $S/K$ is a 0-disjoint union of its connected components (adjoined with a zero) with at least two elements

Finite nilpotent semigroups of small coclass

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.Comment: 11 page

Smooth Loops and Fiber Bundles: Theory of Principal Q-bundles

A nonassociative generalization of the principal fiber bundles with a smooth loop mapping on the fiber is presented. Our approach allows us to construct a new kind of gauge theories that involve higher ''nonassociative'' symmetries.Comment: 20 page

Hyperk\"ahler torsion structures invariant by nilpotent Lie groups

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex structures are of a special kind, called abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from abelian hypercomplex structures. Furthermore, we use a correspondence between abelian hypercomplex structures and subspaces of ${\frak sp}(n)$ to produce continuous families of compact and noncompact of manifolds carrying non isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.Comment: LateX, 12 page

Global behavior of solutions to the static spherically symmetric EYM equations

The set of all possible spherically symmetric magnetic static Einstein-Yang-Mills field equations for an arbitrary compact semi-simple gauge group $G$ was classified in two previous papers. Local analytic solutions near the center and a black hole horizon as well as those that are analytic and bounded near infinity were shown to exist. Some globally bounded solutions are also known to exist because they can be obtained by embedding solutions for the $G=SU(2)$ case which is well understood. Here we derive some asymptotic properties of an arbitrary global solution, namely one that exists locally near a radial value $r_{0}$, has positive mass $m(r)$ at $r_{0}$ and develops no horizon for all $r>r_{0}$. The set of asymptotic values of the Yang-Mills potential (in a suitable well defined gauge) is shown to be finite in the so-called regular case, but may form a more complicated real variety for models obtained from irregular rotation group actions.Comment: 43 page

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v