scholarly journals Coherency and Constructions for Monoids

2020 ◽  
Vol 71 (4) ◽  
pp. 1461-1488
Author(s):  
Yang Dandan ◽  
Victoria Gould ◽  
Miklós Hartmann ◽  
Nik Ruškuc ◽  
Rida-E Zenab
Keyword(s):  
Finiteness Condition ◽  
Direct Products ◽  
Finitely Generated ◽  
Rees Matrix Semigroups ◽  
The Relationship ◽  
Brandt Semigroups ◽  
Finitely Presented ◽  
Matrix Semigroups

Abstract A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck–Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of S to be finitely generated.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

PRESENTATIONS FOR SEMIGROUPS AND SEMIGROUPOIDS

2005 ◽  
Vol 15 (02) ◽  
pp. 291-308 ◽  
Author(s):  
MARK KAMBITES
Keyword(s):  
Finite Extension ◽  
Zero Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Combinatorial Properties ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
The Relationship ◽  
Matrix Semigroups

We consider the relationship between the combinatorial properties of semigroupoids in general and semigroups in particular. We show that a semigroupoid is finitely generated [finitely presentable] exactly if the corresponding categorical-at-zero semigroup is finitely generated [respectively, finitely presentable]. This allows us to extend some of the main results of [17], to show that finite generation and presentability are preserved under finite extension of semigroupoids and the taking of cofinite subsemigroupoids. We apply this result to extend the results of [6], giving characterizations of finite generation and finite presentability in Rees matrix semigroups over semigroupoids.

Reflections on some aspects of infinite groups

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
Dennis Spellman
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Limit Groups ◽  
Finitely Presented Groups ◽  
The Relationship ◽  
Finitely Presented

AbstractIn this paper we survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.

scholarly journals Flows on Classes of Regular Semigroups and Cauchy Categories

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Suha Ahmed Wazzan
Keyword(s):  
General Structure ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Regular Semigroups ◽  
Rees Matrix Semigroups ◽  
Completely Simple Semigroups ◽  
Special Case ◽  
Brandt Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

We consider the structure of the flow monoid for some classes of regular semigroups (which are special case of flows on categories) and for Cauchy categories. In detail, we characterize flows for Rees matrix semigroups, rectangular bands, and full transformation semigroups and also describe the Cauchy categories for some classes of regular semigroups such as completely simple semigroups, Brandt semigroups, and rectangular bands. In fact, we obtain a general structure for the flow monoids on Cauchy categories.

scholarly journals Rees matrix semigroups

1990 ◽  
Vol 33 (1) ◽  
pp. 23-37 ◽  
Author(s):  
Mark V. Lawson
Keyword(s):  
Rees Matrix Semigroups ◽  
Algebraic Properties ◽  
Abstract Class ◽  
The Relationship ◽  
Matrix Semigroups

In this paper we provide a new, abstract characterisation of classical Rees matrix semigroups over monoids with zero. The corresponding abstract class of semigroups is obtained by abstracting a number of algebraic properties from completely 0-simple semigroups: in particular, the relationship between arbitrary elements and idempotents.

scholarly journals Yet Another Solution to the Burnside Problem for Matrix Semigroups

2012 ◽  
Vol 55 (1) ◽  
pp. 188-192 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Finiteness Condition ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Matrix Semigroups

AbstractWe use the kernel category to give a finiteness condition for semigroups. As a consequence we provide yet another proof that finitely generated periodic semigroups of matrices are finite.

scholarly journals Embedding any countable semigroup in a 2-generated bisimple monoid

1984 ◽  
Vol 25 (2) ◽  
pp. 153-161 ◽  
Author(s):  
Karl Byleen
Keyword(s):  
Constructive Proof ◽  
Finitely Generated ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups

G. B. Preston [10] proved that any semigroup can be embedded in a bisimple monoid. If S is a countable semigroup, his constructive proof yields a bisimple monoid which is also countable, but not necessarily finitely generated. The main result of this paper is that any countable semigroup can be embedded in a 2-generated bisimple monoid.J. M. Howie [6] proved that any semigroup can be embedded in an idempotentgenerated semigroup. F. Pastijn [9] showed that any semigroup can be embedded in a bisimple idempotent-generated semigroup, and that any countable semigroup can be embedded in a semigroup which is generated by 3 idempotents. Easy proofs of these results using Rees matrix semigroups over a semigroup were given by the author [3]. In this paper, as a corollary to our main result, we deduce that any countable semigroup can be embedded in a bisimple semigroup which is generated by 3 idempotents.

Homological Finite Derivation Type

2003 ◽  
Vol 13 (03) ◽  
pp. 341-359 ◽  
Author(s):  
Juan M. Alonso ◽  
Susan M. Hermiller
Keyword(s):  
Free Monoid ◽  
Higher Dimensions ◽  
Finiteness Condition ◽  
Finitely Generated ◽  
Monoid Ring ◽  
Finite Derivation Type ◽  
Finitely Presented ◽  
Homological Type

In 1987, Squier defined the notion of finite derivation type for a finitely presented monoid. To do this, he associated a 2-complex to the presentation. The monoid then has finite derivation type if, modulo the action of the free monoid ring, the 1-dimensional homotopy of this complex is finitely generated. Cremanns and Otto showed that finite derivation type implies the homological finiteness condition left FP3, and when the monoid is a group, these two properties are equivalent. In this paper we define a new version of finite derivation type, based on homological information, together with an extension of this finite derivation type to higher dimensions, and show connections to homological type FPnfor both monoids and groups.

FINITE PRESENTABILITY OF SEMIGROUP CONSTRUCTIONS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 19-31 ◽  
Author(s):  
ISABEL M. ARAÚJO
Keyword(s):  
Wreath Products ◽  
Direct Products ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroups

We survey some recent results concerning finite presentability of several semigroup constructions. Namely we study direct products, wreath products, Rees matrix semigroups, Bruck–Reilly extensions, general products and strong semilattices of semigroups.

scholarly journals RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT

2012 ◽  
Vol 14 (03) ◽  
pp. 1250017 ◽  
Author(s):  
LEONARDO CABRER ◽  
DANIELE MUNDICI
Keyword(s):  
Abelian Group ◽  
Algebraic Topology ◽  
Lattice Structure ◽  
Order Unit ◽  
Finitely Generated ◽  
Lattice Order ◽  
Translation Invariant ◽  
Projective Lattice ◽  
Rational Polyhedra ◽  
Finitely Presented

An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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