scholarly journals Limit Elements in the Configuration Algebra for a Cancellative Monoid

2010 ◽  
pp. 37-113 ◽  
Author(s):  
Kyoji Saito
Keyword(s):  
Cancellative Monoid

Graded integral domains in which each nonzero homogeneous t-ideal is divisorial

2019 ◽  
Vol 18 (01) ◽  
pp. 1950018 ◽  
Author(s):  
Gyu Whan Chang ◽  
Haleh Hamdi ◽  
Parviz Sahandi
Keyword(s):  
Integral Domain ◽  
Nonzero Ideal ◽  
Integral Domains ◽  
Cancellative Monoid ◽  
Integrally Closed ◽  
Graded Integral Domain

Let [Formula: see text] be a nonzero commutative cancellative monoid (written additively), [Formula: see text] be a [Formula: see text]-graded integral domain with [Formula: see text] for all [Formula: see text], and [Formula: see text]. In this paper, we study graded integral domains in which each nonzero homogeneous [Formula: see text]-ideal (respectively, homogeneous [Formula: see text]-ideal) is divisorial. Among other things, we show that if [Formula: see text] is integrally closed, then [Formula: see text] is a P[Formula: see text]MD in which each nonzero homogeneous [Formula: see text]-ideal is divisorial if and only if each nonzero ideal of [Formula: see text] is divisorial, if and only if each nonzero homogeneous [Formula: see text]-ideal of [Formula: see text] is divisorial.

scholarly journals Recent results on weakly factorial domains

2018 ◽  
Vol 20 ◽  
pp. 01001
Author(s):  
Chang Gyu Whan
Keyword(s):  
Polynomial Ring ◽  
Quotient Group ◽  
Integral Domain ◽  
Laurent Polynomial ◽  
Semigroup Ring ◽  
Prime Element ◽  
Cancellative Monoid ◽  
Factorial Domain ◽  
Laurent Polynomial Ring

In this paper, we will survey recent results on weakly factorial domains base on the results of [11, 13, 14]. LetD be an integral domain, X be an indeterminate over D, d ∈ D, R = D[X,d/X] be a subring of the Laurent polynomial ring D[X,1/X], Γ be a nonzero torsionless commutative cancellative monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. Among other things, we show that R is a weakly factorial domain if and only if D is a weakly factorial GCD‐domain and d = 0, d is a unit of D or d is a prime element of D. We also show that if char(D) = 0 (resp., char(D) = p > 0), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD domain, Γ is a weakly factorial GCD semigroup, and G is of type (0,0,0,…) (resp., (0,0,0,…) except p).

THE CATENARY AND TAME DEGREES ON A NUMERICAL MONOID ARE EVENTUALLY PERIODIC

2014 ◽  
Vol 97 (3) ◽  
pp. 289-300 ◽  
Author(s):  
SCOTT T. CHAPMAN ◽  
MARLY CORRALES ◽  
ANDREW MILLER ◽  
CHRIS MILLER ◽  
DHIR PATEL
Keyword(s):  
Periodicity Condition ◽  
Numerical Monoid ◽  
Cancellative Monoid ◽  
Krull Monoids ◽  
Catenary Degree

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M$ be a commutative cancellative monoid. For $m$ a nonunit in $M$, the catenary degree of $m$, denoted $c(m)$, and the tame degree of $m$, denoted $t(m)$, are combinatorial constants that describe the relationships between differing irreducible factorizations of $m$. These constants have been studied carefully in the literature for various kinds of monoids, including Krull monoids and numerical monoids. In this paper, we show for a given numerical monoid $S$ that the sequences $\{c(s)\}_{s\in S}$ and $\{t(s)\}_{s\in S}$ are both eventually periodic. We show similar behavior for several functions related to the catenary degree which have recently appeared in the literature. These results nicely complement the known result that the sequence $\{\Delta (s)\}_{s\in S}$ of delta sets of $S$ also satisfies a similar periodicity condition.

GENERALIZATION OF F-INVERSE SEMIGROUPS BY USING MCALISTER'S APPROACH

2008 ◽  
Vol 01 (04) ◽  
pp. 535-553
Author(s):  
Xiaojiang Guo ◽  
Xiangfei Ni ◽  
K. P. Shum
Keyword(s):  
Direct Product ◽  
Inverse Semigroups ◽  
New Method ◽  
Regular Band ◽  
Cancellative Monoid ◽  
Left Regular ◽  
Left Regular Band

We generalize the F-inverse semigroups within the class of lpp-semigroups by using McAlister's approach and FGC-systems. Consider a left GC - lpp monoid M. If M is lpp, then M is called a left F-pseudo group and for brevity, we call the semi-direct product of a left regular band and a cancellative monoid a twisted left cryptic group. In this paper, the structures of left F-pseudo groups are investigated. It is shown that a left F-pseudo group whose minimum right cancellative monoid congruence is cancellative can be embedded into a twisted left cryptic group. This result generalizes a number of known results in F-inverse semigroups previously given by C. C. Edwards, R. B. McFadden, L. O'Carrol, X. J. Guo and others. In particular, a new method constructing F -right inverse semigroups is provided.

One-Dimensional Monoid Rings with n-Generated Ideals

1993 ◽  
Vol 36 (3) ◽  
pp. 344-350 ◽  
Author(s):  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Commutative Ring ◽  
Artinian Ring ◽  
Krull Dimension ◽  
One Dimensional ◽  
Monoid Ring ◽  
Cancellative Monoid

AbstractA commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

scholarly journals A ŠVARC–MILNOR LEMMA FOR MONOIDS ACTING BY ISOMETRIC EMBEDDINGS

2011 ◽  
Vol 21 (07) ◽  
pp. 1135-1147 ◽  
Author(s):  
ROBERT GRAY ◽  
MARK KAMBITES
Keyword(s):  
Group Theory ◽  
Cayley Graph ◽  
Semigroup Theory ◽  
Geometric Group Theory ◽  
Distance Functions ◽  
Cancellative Monoid ◽  
Isometric Embeddings ◽  
Key Techniques

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

IF-AUTOMORPHISMS OF A FREE SOLVABLE Sp-PERMUTABLE ALGEBRA OF A FINITE RANK OVER A LEFT CANCELLATIVE MONOID ARE PSEUDO-TAME

2012 ◽  
Vol 05 (03) ◽  
pp. 1250047
Author(s):  
P. B. Zhdanovich
Keyword(s):  
Present Article ◽  
Finite Rank ◽  
Abelian Extension ◽  
Cancellative Monoid ◽  
Left Cancellative Monoid

Consider a variety V of acts over a left cancellative monoid S that have a ternary Maltsev operation p(〈p, S〉- algebras ). Using the Magnus–Artamonov representation, the construction of the free Abelian extension of an arbitrary V-algebra was obtained and explored by the author. In the present article we prove that each IF-automorphism of a free k-step solvable V-algebra of a finite rank (k > 1) is pseudo-tame, i.e. it is presented as a product of elementary automorphisms of a special module over a ring with several objects.

FAITHFUL FUNCTORS FROM CANCELLATIVE CATEGORIES TO CANCELLATIVE MONOIDS WITH AN APPLICATION TO ABUNDANT SEMIGROUPS

2005 ◽  
Vol 15 (04) ◽  
pp. 683-698 ◽  
Author(s):  
VICTORIA GOULD ◽  
MARK KAMBITES
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
The Other ◽  
Identity Matrix ◽  
Rees Matrix Semigroup ◽  
Cancellative Monoid ◽  
Faithful Functor ◽  
Ample Semigroup ◽  
Abundant Semigroups ◽  
Matrix Semigroup

We prove that any small cancellative category admits a faithful functor to a cancellative monoid. We use our result to show that any primitive ample semigroup is a full subsemigroup of a Rees matrix semigroup [Formula: see text] where M is a cancellative monoid and P is the identity matrix. On the other hand a consequence of a recent result of Steinberg is that it is undecidable whether a finite ample semigroup embeds as a full subsemigroup of an inverse semigroup.

scholarly journals Congruences on *-Simple Type A I-Semigroups

2019 ◽  
Vol 16 ◽  
pp. 8199-8207
Author(s):  
Ugochukwu Ndubuisi ◽  
Asibong-Ibe U.I ◽  
Udoaka O.G
Keyword(s):  
Type A ◽  
Simple Type ◽  
Cancellative Monoid

This paper obtains a characterisation of the congruences on *-simple type A I-semigroups. The *-locally idempotent-separating congruences, strictly *-locally idempotent-separating congruences and minimum cancellative monoid congruences, are characterised.

GRAPH EXPANSIONS OF RIGHT CANCELLATIVE MONOIDS

1996 ◽  
Vol 06 (06) ◽  
pp. 713-733 ◽  
Author(s):  
VICTORIA GOULD
Keyword(s):  
Inverse Semigroup ◽  
Cayley Graph ◽  
Semigroup Theory ◽  
Type A ◽  
Inverse Monoids ◽  
Cancellative Monoid ◽  
Group Presentations ◽  
Free Left ◽  
Left And Right ◽  
Graph Expansions

The relations ℛ* and [Formula: see text] on a monoid M are natural generalizations of Green’s relations ℛ and [Formula: see text], which coincide with ℛ and [Formula: see text] if M is regular. A monoid M in which every ℛ*-class [Formula: see text] contains an idempotent is called left (right) abundant; if in addition the idempotents of M commute, that is, E(M) is a semilattice, then M is left (right) adequate. Regular monoids are obviously left (and right) abundant and inverse monoids are left (and right) adequate. Many of the well known results of regular and inverse semigroup theory have analogues for left abundant and left adequate monoids, or at least to special classes thereof. The aim of this paper is to develop a construction of left adequate monoids from the Cayley graph of a presentation of a right cancellative monoid, inspired by the construction of inverse monoids from group presentations, given by Margolis and Meakin in [10]. This technique yields in particular the free left ample (formerly left type A) monoid on a given set X.

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