Related papers: One-sided inverses in noncommutative infinitary se…

We introduce the notion of a combinatorial inverse system in non-commutative variables. We present two important examples, some conjectures and results. These conjectures and results were suggested and supported by computer investigations.

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…

We define noncommutative binary forms. Using the typical representation of Hermite we prove the fundamental theorem of algebra and we derive a noncommutative Cardano formula for cubic forms. We define quantized elliptic and hyperelliptic…

Let $\mathbb{N}$ be a set of the natural numbers. Symmetric inverse semigroup $R_\infty$ is the semigroup of all infinite 0-1 matrices $[g_{ij}]$ with at most one 1 in each row and each column such that $g_{ii}=1$ on the complement of a…

A set $B$ is a basis for a vector space $V$ if every element of $V$ can be uniquely written as a linear combination of the elements of $B$. There is a similar definition of a basis for a finite group. We show that certain semidirect…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We show that a linear group without unipotent elements of infinite order possesses properties akin to those held by groups of non positive curvature. Moreover in positive characteristic any finitely generated linear group acts properly and…

We describe the $C^*$-algebra of an $E$-unitary or strongly 0-$E$-unitary inverse semigroup as the partial crossed product of a commutative $C^*$-algebra by the maximal group image of the inverse semigroup. We give a similar result for the…

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense,…

We give a complete list of indecomposable characters of the infinite symmetric semigroup. In comparison with the analogous list for the infinite symmetric group, one should introduce only one new parameter, which has a clear combinatorial…

We classify the twisted tensor products of a finite set algebra with a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. This provides also a classification of entwining structures…

We give an algebraic characterisation of ordered groupoids, namely, we show that there is a categorical isomophism between the category of ordered groupoids and the category of $D$-inverse constellations. Here constellations are partial…

In this work we characterize shift spaces over infinite countable alphabets that can be endowed with an inverse semigroup operation. We give sufficient conditions under which zero-dimensional inverse semigroups can be recoded as shift…

For semigroup $S$, a commutative congruence $\sigma_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we…

In 1981/82, Hickin \& Plotkin and McKenzie both proved that a finite group has only countably many non-isomorphic subdirect powers if and only if it is abelian. In this paper, we prove that a finite commutative semigroup has only countably…

We consider a conservative Markov semigroup on a semi-finite $W^*$-algebra. It is known that under some reasonable assumptions it is enough to determine a kind of differential structure on such a 'noncommutative space'. We construct an…

In these notes we briefly consider various situations related to infinite commutative semigroups, connected to convolutions and Fourier transforms.

We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product…

Let $\mathbb{A} = (A, \cdot)$ be a semigroup. We generalize some recent results by G. A. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable…

Semifields are semirings in which every nonzero element has a multiplicative inverse. A rough classification uses the characteristic of the semifield, that is the isomorphism type of the semifield generated by the two neutral elements. For…