Commutative Cancellative Monoid Research Articles

On Certain Properties of Square-free and Radical Elements and Factorizations in Commutative Cancellative Monoids

On Certain Properties of Square-free and Radical Elements and Factorizations in Commutative Cancellative Monoids

Graded almost pseudo-valuation domains

Let Γ be a nonzero commutative cancellative monoid (written additively), be a Γ-graded integral domain with for all , and H be the set of nonzero homogeneous elements of R. A homogeneous ideal P of R will be said to be strongly homogeneous primary if implies or for some integer , for every homogeneous elements x, y of RH . We say that R is a graded almost pseudo-valuation domain (gr-APVD) if each homogeneous prime ideal of R is strongly homogeneous primary. In this paper, we study some ring-theoretic properties of gr-APVDs and graded integral domains R such that is a gr-APVD for all homogeneous maximal ideals (resp., homogeneous maximal t-ideals) P of R.

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

Abstract Our main result is the following: let $X$ be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface $X^{\prime}$ with automorphism group isomorphic to the automorphism group of $X$ if and only if $X$ is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety $X$ of any dimension. Recall that normalized additive group actions on $X$ are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of $X$. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

Complemented MacNeille completions and algebras of fractions

We introduce (ℓ-)bimonoids as ordered algebras consisting of two compatible monoidal structures on a partially ordered (lattice-ordered) set. Bimonoids form an appropriate framework for the study of a general notion of complementation, which subsumes both Boolean complements in bounded distributive lattices and multiplicative inverses in monoids. The central question of the paper is whether and how bimonoids can be embedded into complemented bimonoids, generalizing the embedding of cancellative commutative monoids into their groups of fractions and of bounded distributive lattices into their free Boolean extensions. We prove that each commutative (ℓ-)bimonoid embeds into a complete complemented commutative ℓ-bimonoid in a doubly dense way reminiscent of the Dedekind–MacNeille completion. Moreover, this complemented completion, which is term equivalent to a commutative involutive residuated lattice, sometimes contains a tighter complemented envelope analogous to the group of fractions. In the case of cancellative commutative monoids this algebra of fractions is precisely the familiar group of fractions, while in the case of Brouwerian (Heyting) algebras it is a (bounded) idempotent involutive commutative residuated lattice. This construction of the algebra of fractions in fact yields a categorical equivalence between varieties of integral and of involutive residuated structures which subsumes as special cases the known equivalences between Abelian ℓ-groups and their negative cones, and between Sugihara monoids and their negative cones.

On the primality and elasticity of algebraic valuations of cyclic free semirings

A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. Under certain mild conditions on a positive algebraic number [Formula: see text], the additive monoid [Formula: see text] of the evaluation semiring [Formula: see text] is atomic. The atomic structure of both the additive and the multiplicative monoids of [Formula: see text] has been the subject of several recent papers. Here we focus on the monoids [Formula: see text], and we study its omega-primality and elasticity, aiming to better understand some fundamental questions about their atomic decompositions. We prove that when [Formula: see text] is less than [Formula: see text], the atoms of [Formula: see text] are as far from being prime as they can possibly be. Then we establish some results about the elasticity of [Formula: see text], including that when [Formula: see text] is rational, the elasticity of [Formula: see text] is full (this was previously conjectured by Chapman, Gotti and Gotti).

On the additive structure of algebraic valuations of polynomial semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α] of the semiring of polynomials N0[X] using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when N0[α] is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid N0[α], the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for N0[α] to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of α. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.

On transfer Krull monoids

Let H be a cancellative commutative monoid, let mathcal {A}(H) be the set of atoms of H and let widetilde{H} be the root closure of H. Then H is called transfer Krull if there exists a transfer homomorphism from H into a Krull monoid. It is well known that both half-factorial monoids and Krull monoids are transfer Krull monoids. In spite of many examples and counterexamples of transfer Krull monoids (that are neither Krull nor half-factorial), transfer Krull monoids have not been studied systematically (so far) as objects on their own. The main goal of the present paper is to attempt the first in-depth study of transfer Krull monoids. We investigate how the root closure of a monoid can affect the transfer Krull property and under what circumstances transfer Krull monoids have to be half-factorial or Krull. In particular, we show that if widetilde{H} is a DVM, then H is transfer Krull if and only if Hsubseteq widetilde{H} is inert. Moreover, we prove that if widetilde{H} is factorial, then H is transfer Krull if and only if mathcal {A}(widetilde{H})={uvarepsilon mid uin mathcal {A}(H),varepsilon in widetilde{H}^{times }}. We also show that if widetilde{H} is half-factorial, then H is transfer Krull if and only if mathcal {A}(H)subseteq mathcal {A}(widetilde{H}). Finally, we point out that characterizing the transfer Krull property is more intricate for monoids whose root closure is Krull. This is done by providing a series of counterexamples involving reduced affine monoids.

On Undecidability of Finite Subsets Theory for Torsion Abelian Groups

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic.

On length densities

Abstract For a commutative cancellative monoid M, we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ⁡ ( x ) {\operatorname{LD}(x)} , and the entire monoid M, denoted LD ⁡ ( M ) {\operatorname{LD}(M)} . This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ⁢ ( x ) {L(x)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} . We consider some general properties of LD ⁡ ( x ) {\operatorname{LD}(x)} and LD ⁡ ( M ) {\operatorname{LD}(M)} and give a wide variety of examples using numerical semigroups, Puiseux monoids, and Krull monoids. While we give an example of a monoid M with irrational length density, we show that if M is finitely generated, then LD ⁡ ( M ) {\operatorname{LD}(M)} is rational and there is a nonunit element x ∈ M {x\in M} with LD ⁡ ( M ) = LD ⁡ ( x ) {\operatorname{LD}(M)=\operatorname{LD}(x)} (such a monoid is said to have accepted length density). While it is well known that the much studied asymptotic versions of L ⁢ ( x ) {L(x)} , ℓ ⁢ ( x ) {\ell(x)} , and ρ ⁢ ( x ) {\rho(x)} (denoted L ¯ ⁢ ( x ) {\overline{L}(x)} , ℓ ¯ ⁢ ( x ) {\overline{\ell}(x)} , and ρ ¯ ⁢ ( x ) {\overline{\rho}(x)} ) always exist, we show the somewhat surprising result that LD ¯ ⁢ ( x ) = lim n → ∞ ⁡ LD ⁡ ( x n ) {\overline{\operatorname{LD}}(x)=\lim_{n\rightarrow\infty}\operatorname{LD}(x^% {n})} may not exist. We also give some finiteness conditions on M that force the existence of LD ¯ ⁢ ( x ) {\overline{\operatorname{LD}}(x)} .

On graded pseudo-valuation domains

Let Γ be a torsionless commutative cancellative monoid and be a Γ-graded integral domain. In this paper, we provide necessary and sufficient conditions for R to be a graded pseudo-valuation domain. In particular, we determine when the semigroup ring over an integral domain D is a graded pseudo-valuation domain. Also some connections of graded pseudo-valuation domains and UMT-domains are studied.

On Undecidability of Subset Theory for Some Monoids

Early we (with B. N. Karlov) have proved the following claim for the infinite cyclic monoid ℋ. Let exp ℋ be an algebra of finite subsets of ℋ with the same operation, exp ℋ must be a monoid again. So the theory of exp ℋ is equivalent to elementary arithmetic. Thus, the theory of the monoid exp ℋ is undecidable. Here we consider an arbitrary commutative cancellative monoid ℋ with an element of infinite order, and generalize the previous claims to the corresponding monoid exp ℋ.

Factorizations in upper triangular matrices over information semialgebras

An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element factors into atoms, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a bound for the number of atoms in each of its factorizations. These two properties have been systematically studied since they were introduced by Anderson, Anderson, and Zafrullah in 1990. Noetherian domains satisfy the BFP, while Dedekind domains satisfy the FFP. It is well known that, for commutative cancellative monoids (and, in particular, for multiplicative monoids of integral domains), FFP ⇒ BFP ⇒ ACCP ⇒ atomic. For n≥2, we show that each of these four properties transfers back and forth between an information semialgebra S (certain commutative cancellative semiring) and the multiplicative monoid Tn(S)• consisting of n×n upper triangular matrices over S. We also show that a similar transfer behavior takes place if one replaces Tn(S)• by its submonoid Un(S) consisting of upper triangular matrices with units along their main diagonals. As a consequence, we find that the atomic chain FFP ⇒ BFP ⇒ ACCP ⇒ atomic also holds for the two classes comprising the noncommutative monoids Tn(S)• and Un(S). Finally, we construct various rational information semialgebras to verify that, in general, none of the established implications is reversible.

Divisibility properties of twisted semigroup rings

Let R be an integral domain, Γ be a nonzero torsion-free commutative cancellative monoid, t be a twist function of Γ on R, be the semigroup ring of Γ over R, and be the twisted semigroup ring of Γ over R with respect to t. In this paper, we show that is a GCD-domain if and only if R is a GCD-domain and Γ is a GCD-semigroup. Hence, is a GCD-domain if and only if is a GCD-domain, while need not be a UFD even though is a UFD. We show that if then is a UFD if and only if R is a UFD and Γ is a UFS. We also show that if G is a torsion-free abelian group satisfying the ascending chain condition on its cyclic subgroups, then R is a UFD if and only if is a UFD.

Graded Prüfer domains with Clifford homogeneous class semigroups

Let be an integral domain graded by an arbitrary torsionless commutative cancellative monoid Γ. We say that R is a graded Prüfer domain if each nonzero finitely generated homogeneous ideal of R is invertible. It is well known that if D is a Prüfer domain, then every nonzero locally principal ideal of D is invertible if and only if D is of finite character, if and only if D is a Clifford regular domain. In this paper, we generalize this result to graded Prüfer domains. That is, among other things, we prove that the following statements are equivalent for a graded Prüfer domain R; (i) each nonzero h-locally principal homogeneous ideal of R is invertible, (ii) each nonzero nonunit homogeneous element of R is contained in only finitely many maximal homogeneous ideals of R, and (iii) R is Clifford homogeneous regular. Let be the semigroup ring of Γ over an integral domain D and w be the so-called w-operation on . We also study when is Clifford w-regular.

On the atomicity of monoid algebras

Let M be a commutative cancellative monoid, and let R be an integral domain. The question of whether the monoid ring R[x;M] is atomic provided that both M and R are atomic dates back to the 1980s. In 1993, Roitman gave a negative answer to the question for M=N0: he constructed an atomic integral domain R such that the polynomial ring R[x] is not atomic. However, the question of whether a monoid algebra F[x;M] over a field F is atomic provided that M is atomic has been open since then. Here we offer a negative answer to this question. First, we exhibit for any infinite cardinal κ a torsion-free atomic monoid M of rank κ satisfying that the monoid domain R[x;M] is not atomic for any integral domain R. Then for every n≥2 and for each field F of finite characteristic we find a torsion-free atomic monoid M of rank n such that F[x;M] is not atomic. Finally, we construct a torsion-free atomic monoid M of rank 1 such that Z2[x;M] is not atomic.

On properties of square-free elements in commutative cancellative monoids

We discuss various square-free factorizations in monoids in the context of: atomicity, ascending chain condition for principal ideals, decomposition, and a greatest common divisor property. Moreover, we obtain a full characterization of submonoids of factorial monoids in which all square-free elements of a submonoid are square-free in a monoid. We also present a factorial property implying that all atoms of a submonoid are square-free in a monoid.

Semigroup rings as weakly factorial domains, II

Let [Formula: see text] be an integral domain, [Formula: see text] be a nonzero torsionless commutative cancellative monoid with quotient group [Formula: see text], and [Formula: see text] be the semigroup ring of [Formula: see text] over [Formula: see text]. In this paper, among other things, we show that if [Formula: see text] (respectively, [Formula: see text], then [Formula: see text] is a weakly factorial domain if and only if [Formula: see text] is a weakly factorial GCD-domain, [Formula: see text] is a weakly factorial GCD-semigroup, and [Formula: see text] is of type [Formula: see text] (respectively, [Formula: see text] except [Formula: see text]).

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

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.

Unimodular rows over monoid rings

For a commutative Noetherian ring R of dimension d and a commutative cancellative monoid M, the elementary action on unimodular n-rows over the monoid ring R[M] is transitive for n≥max⁡(d+2,3). The starting point is the case of polynomial rings, considered by A. Suslin in the 1970s. The main result completes a project, initiated in the early 1990s, and suggests a new direction in the study of K-theory of monoid rings.

On divisor-closed submonoids and minimal distances in finitely generated monoids

We study the lattice of divisor-closed submonoids of finitely generated cancellative commutative monoids. In case the monoid is an affine semigroup, we give a geometrical characterization of such submonoids in terms of its cone. Finally, we use our results to give an algorithm for computing Δ⁎(H), the set of minimal distances of H.