We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating finite case by various structural observations counterexamples, also highlighting role played positive orderability. Moreover, we construct a non-left-orderable, positively right-orderable submonoid of free product cyclic group order 7 with on 4 generators. Any extension that mon...

Cancellative residuated lattices are a natural generalization of lattice-ordered groups (`-groups). Although cancellative monoids are defined by quasi-equations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative intege...

We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang–Baxter equations in terms properties canonically associated braided monoid $S(X,r)$, the quadratic algebra $A= A(\mathbf{k}, X, r)$ over a field $\mathbf{k}$, and its Koszul dual $A^{!}$. More generally, we continue our systematic nondegenerate sets their algebraic objects. Next investigate class (noninvolutive) square-free...

A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is bound λ(a) on the lengths such factorizations. The factorization property one most basic finiteness properties in study non-unique Every commutative noetherian domain factorizations, but it open whether result holds noncommutative setting. We provide sufficient conditions for pr...

This article is the second of two presenting a new approach to left adequate monoids. In the first, we introduced the notion of being T -proper, where T is a submonoid of a left adequate monoid M . We showed that the free left adequate monoid on a set X is X∗-proper. Further, any left adequate monoid M has an X∗-proper cover for some set X , that is, there is an X∗proper left adequate monoid M̂ ...

We prove the so-called Addition Theorem for algebraic entropy of actions cancellative right amenable monoids S on discrete abelian groups A by endomorphisms, under hypothesis that is locally monotileable (that is, admits a Følner sequence (Fn)n?N such Fn monotile Fn+1 every n?N). study in details class groups, also relation with already existing notions monotileability introduced B. Weiss and d...

We study both Morita cancellative and skew properties of noncommutative algebras as initiated recently in several papers explore which classes are (respectively, cancellative). Several new results concerning these two types cancellations, well the classical cancellation, proved.

A semigroup S is called left ample if it can be embedded in the symmetric inverse IX of partial bijections a non-empty set X such that image closed under unary operation α → αα⁻¹, where α⁻¹ IX. Right semigroups are defined dually. both and right ample. monoid (left, right) as semigroup. We observe dominion an subsemigroup coincides with generated by it. then determine dominions certain submonoi...

In this paper we study some combinatorial aspects of the singular Artin monoids. Firstly, we show that a singular Artin monoid SA can be presented as a semidirect product of a graph monoid with its associated Artin group A. Such a decomposition implies that a singular Artin monoid embeds in a group. Secondly, we give a solution to the word problem for the FC type singular Artin monoids. Afterwa...