Principal ideals of finitely generated commutative monoids

We study the semigroups isomorphic to principal ideals of finitely generated commutative monoids. We define the concept of finite presentation for this kind of semigroups. Furthermore, we show how to obtain information on these semigroups from their presentations.     

A decision procedure on partially commutative free monoids

In this paper we evaluate the complexity of an algorithm for deciding whether a partially commutative free monoid has an infinite number of square-free elements.     

The equation x 2 y 2 = g in partially commutative groups

A partially commutative group is a group defined by generators and relations so that all defining relations are of the form: the commutator of two generators is equal to the identity element. We consider an algorithm for checking whether a given group element is a product of two squares. This generalizes a result of Wicks for free groups.     

Partially commutative inverse monoids

Free partially commutative inverse monoids are investigated. As in the case of free partially commutative monoids or groups (trace monoids or graph groups), free partially commutative inverse monoids are defined as quotients of free inverse monoids modulo a partially defined commutation relation on the generators. A quasi linear time algorithm for the word problem is presented. More precisely, we give an algorithm for a RAM. -completeness of the submonoid membership problem (also known as the generalized word problem) and the membership problem for rational sets is shown. Moreover, free partially commutative inverse monoids modulo a finite idempotent presentation are studied. It turns out that the word problem is decidable if and only if the complement of the partial commutation relation is transitive. The work on this paper has been supported by the DFG research project GELO (Graphen mit entscheidbaren Logiken).     

Parabolic and quasiparabolic subgroups of free partially commutative groups

Let Γ be a finite graph and G be the corresponding free partially commutative group. In this paper we study subgroups generated by vertices of the graph Γ, which we call canonical parabolic subgroups. A natural extension of the definition leads to canonical quasiparabolic subgroups. It is shown that the centralisers of subsets of G are the conjugates of canonical quasiparabolic centralisers satisfying certain graph theoretic conditions.     

for commutative rings with identity

Let , , , , be the usual operators on classes of rings: and for isomorphic and homomorphic images of rings and , , respectively for subrings, direct, and subdirect products of rings. If is a class of commutative rings with identity (and in general of any kind of algebraic structures), then the class is known to be the variety generated by the class . Although the class is in general a proper subclass of the class for many familiar varieties . Our goal is to give an example of a class of commutative rings with identity such that . As a consequence we will describe the structure of two partially ordered monoids of operators.

     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

Automorphisms of the endomorphism semigroup of a free commutative dimonoid

We determine all isomorphisms between the endomorphism semigroups of free commutative dimonoids and prove that all automorphisms of the endomorphism semigroup of a free commutative dimonoid are quasi-inner. In particular, we answer a question of B. I. Plotkin.     

Pitfalls in public key cryptosystems based on free partially commutative monoids and groups

At INDOCRYPT 2003 Abisha, Thomas, and Subramanian proposed two public key schemes based on word problems in free partially commutative monoids and groups. We show that both proposals are vulnerable to chosen ciphertext attacks, and thus in the present form must be considered as insecure.     

Symmetries of the Partial Order of Traces

This paper deals with the automorphism group of the partial order of finite traces. We show that any group can arise as such an automorphism group if we allow arbitrary large dependence alphabets. Restricting to finite dependence alphabets, the automorphism groups are profinite and possess only finitely many simple decomposition factors. Finally, we show that the partial order associated with the Rado graph as dependence alphabet does not give rise to a homogeneous domain thereby answering an open question from Boldi, P., Cardone, F. and Sabadini, N. (1993).     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

Extensions of partially ordered partial abelian monoids

The notion of a partially ordered partial abelian monoid is introduced and extensions of partially ordered abelian monoids by partially ordered abelian groups are studied. Conditions for the extensions to exist are found. The cases when both the above mentioned structures have the Riesz decomposition property, or are lattice ordered, are treated. Some applications to effect algebras and MV-algebras are shown.     

Constructing Minimal Projective Resolutions

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.     

Axiomatizability and completeness of some classes of partially ordered polygons

We study partially ordered monoids over which a class of free (over sets and over posets), projective, and (strongly, weakly) flat partially ordered polygons is axiomatizable, complete, or model complete. Similar issues for polygons were dealt with in papers by V. Gould and A. Stepanova. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-2810.2008.1). Translated from Algebra i Logika, Vol. 48, No. 1, pp. 90–121, January–February, 2009.     

Local homology and dimensional full-valuedness

A natural criterion for dimensional full-valuedness of locally compact spaces with finitely generated local homology is given.     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.