Monoid Homomorphism Research Articles

RADICALS OF SKEW GENERALIZED POWER SERIES RINGS

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this note for a (S, ω)-Armendariz ring R we study some properties of skew generalized power series ring R[[S, ω]]. In particular, among other results, we show that for a S-compatible (S, ω)-Armendariz ring R, α(R[[S, ω]]) = α(R)[[S, ω]] = Ni ℓ*(R)[[S, ω]], where α is a radical in a class of radicals which includes the Wedderburn, lower nil, Levitzky and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZCn, zip and 2-primal property, transfer between R and the skew generalized power series ring R[[S, ω]], in case R is S-compatible (S, ω)-Armendariz.

TRIANGULAR MATRIX REPRESENTATION OF SKEW GENERALIZED POWER SERIES RINGS

Let R be a ring, (S, ≤) a strictly ordered monoid and ω : S → End (R) a monoid homomorphism. In this paper, we study the triangular matrix representation of skew generalized power series ring R[[S, ω]] which is a compact generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomials rings, (skew) Laurent power series rings, (skew) group rings, (skew) monoid rings, Mal'cev–Neumann rings and generalized power series rings. We investigate that if R is S-compatible and (S, ω)-Armendariz, then the skew generalized power series ring has same triangulating dimension as R. Furthermore, if R is a PWP ring, then skew generalized power series is also PWP ring.

UNIFORM AND COUNIFORM DIMENSION OF GENERALIZED INVERSE POLYNOMIAL MODULES

Let M be a right R-module, (S, <TEX>${\leq}$</TEX>) a strictly totally ordered monoid which is also artinian and <TEX>${\omega}:S{\rightarrow}Aut(R)$</TEX> a monoid homomorphism, and let <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> denote the generalized inverse polynomial module over the skew generalized power series ring [[<TEX>$R^{S,{\leq}},{\omega}$</TEX>]]. In this paper, we prove that <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> has the same uniform dimension as its coefficient module <TEX>$M_R$</TEX>, and that if, in addition, R is a right perfect ring and S is a chain monoid, then <TEX>$[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$</TEX> has the same couniform dimension as its coefficient module <TEX>$M_R$</TEX>.

On Weak Annihilator Ideals of Skew Monoid Rings

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.

LEFT APP-RINGS OF SKEW GENERALIZED POWER SERIES

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any a ∈ R. Let R be a ring, (S, ≤) be a commutative strictly ordered monoid and ω: S → End (R) be a monoid homomorphism. The skew generalized power series ring [[RS, ≤, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings and Malcev–Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring [[RS, ≤, ω]]. It is shown that if (S, ≤) is a commutative strictly totally ordered monoid, ω: S→ Aut (R) a monoid homomorphism and R a ring satisfying the descending chain condition on right annihilators, then [[RS, ≤, ω]] is left APP if and only if for any S-indexed subset A of R, the ideal lR(∑a ∈ A ∑s ∈ S Rωs (a)) is right s-unital.

REVERSIBLE SKEW GENERALIZED POWER SERIES RINGS

AbstractIn this note we show that there exist a semiprime ring R, a strictly ordered artinian, narrow, unique product monoid (S,≤) and a monoid homomorphism ω:S⟶End(R) such that the skew generalized power series ring R[[S,ω]] is semicommutative but R[[S,ω]] is not reversible. This answers a question posed in Marks et al. [‘A unified approach to various generalizations of Armendariz rings’, Bull. Aust. Math. Soc.81 (2010), 361–397].

Weak dimension and right distributivity of skew generalized power series rings

Let R be a ring, S a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of skew polynomial rings, skew power series rings, skew Laurent polynomial rings, skew group rings, and Mal'cev-Neumann Laurent series rings. In the case where S is positively ordered we give sufficient and necessary conditions for the skew generalized power series ring R[[S, ω]] to have weak dimension less than or equal to one. In particular, for such an S we show that the ring R[[S, ω]] is right duo of weak dimension at most one precisely when the lattice of right ideals of the ring R[[S, ω]] is distributive and ω(s) is injective for every s ∈ S.

A UNIFIED APPROACH TO VARIOUS GENERALIZATIONS OF ARMENDARIZ RINGS

AbstractLet R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

ON ANNIHILATOR IDEALS OF SKEW MONOID RINGS

AbstractA ring R is called a left APP-ring if the left annihilator lR(Ra) is pure as a left ideal of R for every a ∈ R; R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring and M an ordered monoid. Assume that there is a monoid homomorphism φ: M ⟶ Aut(R). We give a necessary and sufficient condition for the skew monoid ring (induced by φ) to be left APP (left principally quasi-Baer, quasi-Baer, respectively).

The ascending chain condition for principal left or right ideals of skew generalized power series rings

Let R be a ring, S a monoid and ω : S → End ( R ) a monoid homomorphism. In this paper we prove that if the monoid S is strictly totally ordered or S is commutative torsion-free cancellative semisubtotally ordered, then the ring R 〚 S , ω 〛 of skew generalized power series with coefficients in R and exponents in S is a domain satisfying the ascending chain condition on principal left (resp. right) ideals if and only if R is a domain, R and S satisfy the ascending chain condition on principal left (resp. right) ideals and each ω ( s ) is injective (resp. is injective and preserves nonunits of R). As an immediate consequence we obtain characterizations of power series rings, Laurent series rings, skew power series rings, skew Laurent series rings and generalized power series rings that are domains satisfying the ascending chain condition on principal left (or right) ideals. We construct examples of skew generalized power series domains for which the ascending chain conditions on principal one-sided ideals are not symmetric.

Algebraic fusion of functions with an accumulating parameter and its improvement

AbstractThis paper develops a new framework for fusion that is designed for eliminating the intermediate data structures involved in the composition of functions that have one accumulating parameter. The new fusion framework comprises two steps:algebraic fusionand its subsequentimprovementprocess. The key idea in our development is to regard functions with an accumulating parameter as functions that operate over the monoid ofdata contexts. Algebraic fusion composes each such function with a monoid homomorphism that is derived from the definition of the consumer function to obtain a higher-order function that computes over the monoid of endofunctions. The transformation result may be further refined by an improvement process, which replaces the operation over the monoid of endofunctions (i.e., function closures) with another monoid operation over a monoid structure other than function closures.Using our framework, one can formulate a particular solution to the fusion problem by devising appropriate monoids and monoid homomorphisms. This provides a unified exposition of a variety of fusion methods that have been developed so far in different formalisms. Furthermore, the cleaner formulation makes it possible to argue about some delicate issues on a firm mathematical basis. We demonstrate that algebraic fusion and improvement in the world of complete pointed partial orders (CPOs) and continuous functions can correctly fuse functions that operate on partial and infinite data structures. We also show that subtle differences in termination behaviours of transformed programmes caused by certain different fusion methods can be cleanly explained by corresponding improvement processes that have different underlying monoid structures.

Uniserial rings of skew generalized power series

Let R be a ring, S a strictly ordered monoid and ω : S → End ( R ) a monoid homomorphism. In this paper we obtain some necessary conditions for the skew generalized power series ring R 〚 S , ω 〛 to be right (respectively left) uniserial, and we prove that these conditions are also sufficient when the monoid S is commutative or totally ordered.

Minimal But Inefficient Presentations for Self Semidirect Products of the Free Abelian Monoid on Two Generators

Let A and K both be copies of the free Abelian monoid on two generators. For any connecting monoid homomorphism θ: A → End(K), let M = K ⋊θ A be the corresponding monoid semidirect product. In Çevik (2005), it has been proved necessary and sufficient conditions for the efficiency of a standard presentation for M in terms of the matrix representation for θ. As a main result of this article, we give sufficient conditions for this presentation of M to be minimal but not efficient. To do that we will use the same method as given in Çevik (2003b).

Algebraic fusion of functions with an accumulating parameter and its improvement

We present a unifying solution to the problem of fusion of functions, where both the producer function and the consumer function have one accumulating parameter. The key idea in this development is to formulate the producer function as a function which computes over a monoid of data contexts . Upon this formulation, we develop a fusion method called algebraic fusion based on the elementary theory of universal algebra and monoids. The producer function is fused with a monoid homomorphism that is derived from the definition of the consumer function, and is turned into a higher-order function f that computes over the monoid of endofunctions.We then introduce a general concept called improvement , in order to reduce the cost of computing over the monoid of endofunctions (i. e., function closures). An improvement of the function f via a monoid homomorphism h is a function g that satisfies f =h ºg. This provides a principled way of finding a first-order function representing a solution to the fusion problem. It also presents a clean and unifying account for varying fusion methods that have been proposed so far. Furthermore, we show that our method extends to support partial and infinite data structures, by means of an appropriate monoid structure.

On Descent Theory for Monoid Actions

We give some sufficient conditions for a monoid homomorphism to be an effective descent morphism.

Every finite semigroup is embeddable in a finite relatively free semigroup

The title result is proved by a Murskii-type embedding. Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ d = ξ 2 d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup. It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either { e} or all of S.

The homomorphism problem for trace monoids

It is proved to be decidable, for any given finite subset F of X ∗ , dependency alphabet ( Y, τ) and mapping ϕ : F → Y ∗ , whether or not ϕ can be extended to a monoid homomorphism ϕ ̃ : F ∗ → M(Y,τ) . This contrasts with the undecidability of the symmetric problem, when we consider the dependency alphabet ( X, τ) instead. Some other particular cases of homomorphism and isomorphism problems among trace monoids are also discussed.

Semigroup homomorphisms and fuzzy automata

A generalized Ω-fuzzy automaton over a complete residuated lattice Ω and a monoid (M,*) and with a set S of states is introduced as a monoid homomorphism F:(M,*)→(?,∘), where (?,∘) is a monoid of Ω-fuzzy sets in a set S×S. An extension principle depending of proper filters Φ in Ω is introduced which then enables to introduce morphisms between generalized Ω-fuzzy automata and to introduce the category ℱΦ of these automata. It is proved that categories of classical fuzzy automata, non-deterministic automata and some other systems are equivalent to subcategories of ℱΦ for a suitable filter Φ.

The homomorphism problem for the free monoid

It is proved to be decidable, for any given finite subset F of X ∗ and mapping ϕ :F→X ∗ , whether or not ϕ can be extended to an (injective) monoid homomorphism ϕ ̄ :F ∗→X ∗ . As a corollary, an alternative algorithm for the isomorphism problem for the free monoid is also provided: for any given finite subsets F, G of X ∗ , it is decidable whether or not F ∗≃G ∗ .

On Singular Braids

In Vassiliev theory, there is a natural monoid homomorphism from n-strand singular braids to the group algebra of n-strand braid group. J. Birman conjectured that this monoid homomorphism is injective. We show that the monoid homomorphism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are distinguishable by Vassiliev braid invariants.