Generators and relations of direct products of semigroups

Robertson, E. F.; Ruškuc, Nik; Wiegold, James

doi:10.1090/s0002-9947-98-02074-1

Cited by 37 publications

(23 citation statements)

References 22 publications

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“…(2) Both S and T are infinite and S 2 = S and T 2 = T . As was proved in [32], S and T admit finite generating sets A and B satisfying the additional conditions that A ⊆ A 2 , B ⊆ B 2 , and A × B is a finite generating set for S × T . Let (a, b) ∈ A × B.…”

Section: Constructionsmentioning

confidence: 83%

“…The situation with direct products of semigroups has some special features that do not arise for groups, because a direct product of finitely generated semigroups is not necessarily itself finitely generated. Robertson et al [32] characterized direct products of semigroups that are finitely generated: S × T is finitely generated if and only if both S and T are finitely generated and:…”

Section: Constructionsmentioning

confidence: 99%

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Growths of Endomorphisms of Finitely Generated Semigroups

Cain¹,

Maltcev

2016

J. Aust. Math. Soc.

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This paper studies the growths of endomorphisms of finitely generated semigroups. The growth is a certain dynamical characteristic describing how iterations of the endomorphism ‘stretch’ balls in the Cayley graph of the semigroup. We make a detailed study of the relation of the growth of an endomorphism of a finitely generated semigroup and the growth of the restrictions of the endomorphism to finitely generated invariant subsemigroups. We also study the possible values endomorphism growths can attain. We show the role of linear algebra in calculating the growths of endomorphisms of homogeneous semigroups. Proofs are a mixture of syntactic algebraic rewriting techniques and analytical tricks. We state various problems and suggestions for future research.

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Section: Constructionsmentioning

confidence: 83%

Section: Constructionsmentioning

confidence: 99%

Growths of Endomorphisms of Finitely Generated Semigroups

Cain¹,

Maltcev

2016

J. Aust. Math. Soc.

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“…In our examination of the finite generation of the Schutzenberger product we will use connections with the direct product (properties (S4) and (S5)). We say that an element x of a semigroup S is indecomposable if there do not exist y, z € S such that x = yz-Then, by [8], the direct product 5 x T is finitely generated if and only if S and T are finitely generated and if one is infinite then the other contains no indecomposable elements.…”

Section: Finite Generationmentioning

confidence: 99%

On finite generation and presentability of Schützenberger products

Gallagher¹,

Ruškucs²

2007

J. Aust. Math. Soc.

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The finite generation and presentation of Schutzenberger products of semigroups are investigated. A general necessary and sufficient condition is established for finite generation. The Schutzenberger product of two groups is finitely presented as an inverse semigroup if and only if the groups are finitely presented, but is not finitely presented as a semigroup unless both groups are finite.2000 Mathematics subject classification: primary 20M05; secondary 20M18.

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“…In , we investigated how properties such as being finitely generated, finitely presented, residually finite behave under direct products. There we have seen that the properties of

A \times B

closely follow those of the factors

A

and

B

, and that it is relatively hard to find counterexamples to the ‘expected’ preservation result:

A \times B has property P if and only if A and B have property P .

In fact, when

scriptP

is finite generation, holds in all congruence permutable varieties and all varieties with idempotent operations [, Theorems 2.2, 2.5], although not for semigroups . When

scriptP

is finite presentability, however, the situation is more complicated.…”

Section: Introductionmentioning

confidence: 99%

Generating subdirect products

Mayr

Ruškuc

2019

J. London Math. Soc.

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We study conditions under which subdirect products of various types of algebraic structures are finitely generated or finitely presented. In the case of two factors, we prove general results for arbitrary congruence permutable varieties, which generalize previously known results for groups, and which apply to modules, rings, K‐algebras and loops. For instance, if C is a fiber product of A and B over a common quotient D, and if A, B and D are finitely presented, then C is finitely generated. For subdirect products of more than two factors, we establish a general connection with projections on pairs of factors and higher commutators. More detailed results are provided for groups, loops, rings and K‐algebras. In particular, let C be a subdirect product of K‐algebras A1,⋯,An for a Noetherian ring K such that the projection of C onto any Ai×Aj has finite co‐rank in Ai×Aj. Then, C is finitely generated (respectively, finitely presented) if and only if all Ai are finitely generated (respectively, finitely presented). Finally, examples of semigroups and lattices are provided which indicate further complications as one ventures beyond congruence permutable varieties.

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Generators and relations of direct products of semigroups

Cited by 37 publications

References 22 publications

Growths of Endomorphisms of Finitely Generated Semigroups

Growths of Endomorphisms of Finitely Generated Semigroups

On finite generation and presentability of Schützenberger products

Generating subdirect products

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