scholarly journals On properties of square-free elements in commutative cancellative monoids

2019 ◽  
Vol 100 (3) ◽  
pp. 850-870
Author(s):  
Piotr Jędrzejewicz ◽  
Mikołaj Marciniak ◽  
Łukasz Matysiak ◽  
Janusz Zieliński
Keyword(s):  
Cancellative Monoids

scholarly journals A correspondence between a class of monoids and self-similar group actions II

2015 ◽  
Vol 25 (04) ◽  
pp. 633-668
Author(s):  
Mark V. Lawson ◽  
Alistair R. Wallis
Keyword(s):  
Group Actions ◽  
Similar Group ◽  
Universal Group ◽  
Length Functions ◽  
Group Of Units ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Self Similar ◽  
Cancellative Monoids

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa–Szép products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych. In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be significant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.

scholarly journals GRAPH PRODUCTS OF RIGHT CANCELLATIVE MONOIDS

2009 ◽  
Vol 87 (2) ◽  
pp. 227-252 ◽  
Author(s):  
JOHN FOUNTAIN ◽  
MARK KAMBITES
Keyword(s):  
Graph Products ◽  
Graph Product ◽  
General Characterization ◽  
Inverse Hull ◽  
Cancellative Monoids

AbstractOur first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialize to the case of the inverse hulls of graph monoids, obtaining what we call ‘polygraph monoids’. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterization of those right cancellative monoids with inverse hulls that are F*-inverse.

scholarly journals Abundant Rees matrix semigroups

1987 ◽  
Vol 42 (1) ◽  
pp. 132-142 ◽  
Author(s):  
M. V. Lawson
Keyword(s):  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Abundant Semigroups ◽  
Multiplicative Semigroups ◽  
Matrix Semigroups ◽  
Cancellative Monoids

AbstractThe class of abundant semigroups originally arose from ‘homological’ considerations in the theory of S-systems: they are the semigroup theoretic counterparts of PP-rings. Cancellative monoids, full subsemigroups of regular semigroups as well as the multiplicative semigroups of PP-rings are abundant. In this paper we investigate the properties of Rees matrix semigroups over abundant semigroups. Some of our results generalise McAlister's work on regular Rees matrix semigroups.

Some New Characterizations of Right Cancellative Monoids by Condition (PWP )

2005 ◽  
Vol 71 (1) ◽  
pp. 134-139 ◽  
Author(s):  
Qiao HuSheng
Keyword(s):  
Cancellative Monoids

ON WEAKLY AMPLE SEMIGROUPS

2014 ◽  
Vol 97 (3) ◽  
pp. 404-417 ◽  
Author(s):  
MARIO PETRICH
Keyword(s):  
Left Congruence ◽  
Left And Right ◽  
The One ◽  
Cancellative Monoids

AbstractLet$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$be a semigroup. Elements$a,b$of$S$are$\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then$S$is weakly left ample if (1) idempotents of$S$commute, (2) $\widetilde{\mathscr{R}}$is a left congruence, (3) for any$a \in S$,$a$is$\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say$a^+$, and (4) for any element$a$and idempotent$e$of$S$,$ae=(ae)^+a$. Elements$a,b$of$S$are$\mathscr{R}^*$-related if, for any$x,y \in S^1$,$xa=ya$if and only if$xb=yb$. Then$S$is left ample if it satisfies (1), (3) and (4) relative to$\mathscr{R}^*$instead of$\widetilde{\mathscr{R}}$. Further,$S$is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open.

scholarly journals Infinite Examples of Cancellative Monoids That Do Not Always Have Least Common Multiple

2014 ◽  
Vol 42 (3) ◽  
pp. 305-326
Author(s):  
Tadashi Ishibe
Keyword(s):  
Common Multiple ◽  
Least Common Multiple ◽  
Cancellative Monoids

Gradations of cyclic rings and homogeneity of their nil and Jacobson radicals

2019 ◽  
Vol 19 (02) ◽  
pp. 2050035 ◽  
Author(s):  
Mohammad Obiedat
Keyword(s):  
Cancellative Monoids

A formula for the number of gradations, up to equivalence, of cyclic rings by cancellative monoids is given. As an application, the nil and Jacobson radicals of cyclic rings are shown to be homogeneous.

A characterization of left cancellative monoids by flatness properties

1990 ◽  
Vol 40 (1) ◽  
pp. 109-112 ◽  
Author(s):  
Sydney Bulman-Fleming ◽  
Kenneth McDowell
Keyword(s):  
Cancellative Monoids

scholarly journals Primary ideals of finitely generated commutative cancellative monoids

2001 ◽  
Vol 336 (1-3) ◽  
pp. 219-230
Author(s):  
J.C. Rosales ◽  
J.I. García-García
Keyword(s):  
Finitely Generated ◽  
Primary Ideals ◽  
Cancellative Monoids

scholarly journals On the atomic structure of Puiseux monoids

2016 ◽  
Vol 16 (07) ◽  
pp. 1750126 ◽  
Author(s):  
Felix Gotti
Keyword(s):  
19Th Century ◽  
Atomic Structure ◽  
Natural Generalization ◽  
The Other ◽  
Numerical Semigroups ◽  
Finitely Generated ◽  
Other Hand ◽  
The Family ◽  
Finite Set ◽  
Cancellative Monoids

In this paper, we study the atomic structure of the family of Puiseux monoids, i.e. the additive submonoids of [Formula: see text]. Puiseux monoids are a natural generalization of numerical semigroups, which have been actively studied since mid-19th century. Unlike numerical semigroups, the family of Puiseux monoids contains non-finitely generated representatives. Even more interesting is that there are many Puiseux monoids which are not even atomic. We delve into these situations, describing, in particular, a vast collection of commutative cancellative monoids containing no atoms. On the other hand, we find several characterization criteria which force Puiseux monoids to be atomic. Finally, we classify the atomic subfamily of strongly bounded Puiseux monoids over a finite set of primes.

Sign in / Sign up

Export Citation Format

Share Document