Cancellative Monoid Research Articles

Non-singular covers over monoid rings

We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever we replace the strongly cancellative monoid $G$ by the totally ordered cancellative monoid or by the totally ordered group.

Integral Closure of Graded Integral Domains

Let Γ be a torsion-free cancellative commutative monoid and let R = ⨁α∈ΓRα be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.

The loop problem for monoids and semigroups

AbstractWe propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We also study the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid.

An Application of Model Theory to Semimodules

In this note, we prove that the theory T of cancellative semimodules over a semiring R has the amalgamation property. If R is an entire cancellative zerosumfree semiring, then T has no model-companion. In particular, the theory of commutative additively cancellative monoids forms an example of a non-companionable theory.

Parabolic subgroups of Garside groups

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains all Artin–Tits groups of spherical type. We show that the well-known notion of a parabolic subgroup of an Artin–Tits group can be extended to the framework of Garside groups so that most of the standard properties known in the Artin–Tits groups case are preserved. The extension is not trivial and it requires a new approach. We also define the more general notion of a Garside subgroup of a Garside group that nicely extends the notion of an LCM-homomorphism between Artin–Tits groups.

PRODUCTS OF IDEMPOTENT ENDOMORPHISMS OF RELATIVELY FREE ALGEBRAS WITH WEAK EXCHANGE PROPERTIES

AbstractIf $A$ is a stable basis algebra of rank $n$, then the set $S_{n-1}$ of endomorphisms of rank at most $n-1$ is a subsemigroup of the endomorphism monoid of $A$. This paper gives a number of necessary and sufficient conditions for $S_{n-1}$ to be generated by idempotents. These conditions are satisfied by finitely generated free modules over Euclidean domains and by free left $T$-sets of finite rank, where $T$ is cancellative monoid in which every finitely generated left ideal is principal.

Factorizable Rpp monoids

It is well known that the rpp semigroups are generalized regular semigroups. In order to further investigate the structure of rpp semigroups, we introduce the type-(I, L ∗ ) factorizable monoids. In this paper, we study such factorizable rpp monoids admitting a left univocal type-(I, L ∗ ) factorization. In particular, we prove that a semigroup S is a factorizable rpp monoid admitting a left univocal type-(I, L ∗ ) factorization if and only if S is isomorphic to a semidirect product of a band and a left cancellative monoid. Some results of Catino and Tolo on factorizable semigroups are extended and amplified.

PROPER COVERS OF AMPLE MONOIDS

AbstractProper ample monoids are described by means of a certain category acted upon on both sides by a cancellative monoid. Making use of this characterization, we show that every ample monoid $S$ has a proper ample cover, which can be taken to be finite whenever $S$ is finite.

The Catenary and Tame Degree in Finitely Generated Commutative Cancellative Monoids

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.

Presentations of finitely generated cancellative commutative monoids and nonnegative solutions of systems of linear equations

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.

The Extraction Degree of Cale Monoids

The extraction degree measures commonality of factorization between any two elements in a commutative, cancellative monoid. Additional properties of the extraction degree are developed for monoids possessing a Cale basis. For block monoids, the complete set of extraction degrees is calculated between any two elements, between any two irreducible elements, and between any irreducible element and any general element.

Chains of Factorizations and Factorizations with Successive Lengths

ABSTRACT Let H be a commutative cancellative monoid. H is called atomic if every nonunit a ∈ H decomposes (in general in a highly nonunique way) into a product of irreducible elements (atoms) u i of H. The integer n is called the length of (†), and L(a) = {n ∈ ℕ | a decomposes into n irreducible elements of H} is called the set of lengths of a. Two integers k < l are called successive lengths of a if L(a) ∩ {m ∈ ℕ | k ≤ m ≤ l} = {k, l}. For a ∈ H we denote by Z n (a) the set of factorizations of a with length n. Suppose now that H is one of the following monoids: i. A congruence monoid in a Dedekind domain with finite residue fields. ii. H = D\\ {0}, where D is a Noetherian domain having the following properties: is a Krull domain with finite divisor class group, and is a finite ring. iii. H = D\\ {0}, where D is a one-dimensional Noetherian domain with finite normalization and finite Picard group (but possibly infinite residue fields). Let a ∈ H. In the present article, we investigate the structure of concatenating chains in Z n (a) as well as the relation between Z k (a) and Z l (a) if k and l are successive lengths of a. The work continues earlier investigations in Foroutan (2003), Foroutan and Geroldinger (2004), and Hassler (to appear).

Non-singular covers over ordered monoid rings

Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g < h$ there is $l\in G$ such that $lg = h$. For a totally ordered cancellative monoid the equalities $Z(RG) = Z(R)G$ and $\sigma (RG) = \sigma (R)G$ hold, $\sigma $ being Goldie’s torsion theory.

Regular F-Abundant Semigroups

ABSTRACT The investigation of regular F-abundant semigroups is initiated. In fact, F-abundant semigroups are generalizations of regular cryptogroups in the class of abundant semigroups. After obtaining some properties of such semigroups, the construction theorem of the class of regular F-abundant semigroups is obtained. In addition, we also prove that a regular F-abundant semigroup is embeddable into a semidirect product of a regular band by a cancellative monoid. Our result is an analogue of that of Gomes and Gould on weakly ample semigroups, and also extends an earlier result of O'Carroll on F-inverse semigroups.

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

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

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

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.

A Geometric Characterization of Automatic Monoids

It is well known that automatic groups can be characterized using geometric properties of their Cayley graphs. Along the same line of thought, we provide a geometric characterization of automatic monoids. This involves working with a slightly strengthened definition of an automatic monoid which is still a proper generalization of the concept of an automatic group. The two definitions coincide in the case of right cancellative monoids for which a particularly simple characterization is obtained.

A Note on Almost GCD Monoids

A commutative cancellative monoid H (with 0 adjoined) is called an almost GCD (AGCD) monoid if for x, y in H , there exists a natural number n = n(x, y) so that x n and y n have an LCM, that is, x n H ∩ y n H is principal. We relate AGCD monoids to the recently introduced inside factorial monoids (there is a subset Q of H so that the submonoid F of H generated by Q and the units of H is factorial and some power of each element of H is in F ). For example, we show that an inside factorial monoid H is an AGCD monoid if and only if the elements of Q are primary in H , or equivalently, H is weakly Krull, distinct elements of Q are v -coprime in H , or the radical of each element of Q is a maximal t -ideal of H . Conditions are given for an AGCD monoid to be inside factorial and the results are put in the context of integral domains. 2000 Mathematics Subject Classification: 13A05, 13F05, 20M14.

The Structure of Right C-rpp Semigroups

We study the strongly rpp-semigroups whose set of idempotents forms a right regular band and the relation ${\cal L}^* \vee {\cal R}$ is a congruence. This kind of strongly rpp-semigroups was called by Y.Q. Guo the right C-rpp semigroups and he characterized them as semilattice of direct products of left cancellative monoid with right zero semigroup. The aim of this paper is to improve upon Guo’s result, giving a tighter description of the semilattice decomposition involving a system of connecting maps. By using our result, we can always construct a right C-rpp semigroup by glueing up the given ingredients by the connecting maps. An example of an infinite right C-rpp semigroup is constructed.

The structure of superabundant semigroups

A structure theorem for superabundant semigroups in terms of semilattices of normalized Rees matrix semigroups over some cancellative monoids is obtained. This result not only provides a construction method for superabundant semigroups but also generalizes the well-known result of Petrich on completely regular semigroups. Some results obtained by Fountain on abundant semigroups are also extended and strengthened.