Planar Lyapunov Algebras

We study complex involutive algebras generated by a single nonselfadjoint idempotent and use them to construct a family of algebras, which we call planar Lyapunov algebras. As our main result, we prove that every 2-dimensional commutative real algebra whose homogeneous Riccati differential equation is stable at the origin must be isomorphic either to an algebra with zero multiplication or to some planar Lyapunov algebra.

A construction of a finitely presented semigroup containing an infinite square-free ideal with zero multiplication

This work provides an example of a finitely presented semigroup [Formula: see text] with zero containing an infinite ideal of the form [Formula: see text], where [Formula: see text] is a generator of [Formula: see text], such that every word in generators representing an element of [Formula: see text] is square free (i.e. any word of the type [Formula: see text], for non-empty [Formula: see text], equals zero in [Formula: see text]).

ON THE CENTER OF THE MODULAR GROUP ALGEBRA OF A FINITE p-GROUP

Let G be a finite nonabelian p-group and F a field of characteristic p and let [Formula: see text] be the subalgebra spanned by class sums [Formula: see text], where C runs over all conjugacy classes of noncentral elements of G. We show that all finite p-groups are subgroups and homomorphic images of p-groups for which [Formula: see text]. We also give the description of abelian-by-cyclic groups for which [Formula: see text] is an algebra with zero multiplication or is nil of index 2.

Semisimple classes containing no trivial near- rings

Beside near-rings with zero-multiplication or constant multiplication also other near- rings can be considered as near-rings with trivial multiplication. From a structure theoret- ical point of view it is reasonable to consider Kurosh{Amitsur radical classes which contain all near-rings with trivial multiplication. In the present note, continuing the investigations of [2], [4] and [5], we shall characterize the semisimple classes of such radical classes. Our characterizations look clumsier than those in [2], [4] and [5], but we exhibit that exact analogues of the mentioned characterizations cannot be achieved.

Quasi-o-minimal structures

A structure (M, <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1.