Free Monoids are Coherent

Gould, Victoria; Hartmann, Miklós; Ruškuc, Nik

doi:10.1017/s0013091516000079

Cited by 6 publications

(16 citation statements)

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“…Its predecessors were concerned with free monoids in certain varieties of monoids and unary monoids. In particular, [8] showed that all free monoids are coherent, building upon the earlier observation in [5] that free commutative monoids are coherent, and resolving an open question from that paper. This theme is continued in [7], where it was shown that any free left ample monoid is coherent, while free inverse monoids and free ample monoids of rank > 1 are not.…”

Section: Introductionmentioning

confidence: 64%

“…It is known that there are right coherent monoids that are not weakly right noetherian, for example, any free monoid of rank greater than one [8]. We exhibit two further examples at the end of Section 3, both of them regular, in order to demonstrate the independence of various conditions we consider.…”

Section: Introductionmentioning

confidence: 95%

“…Having a model companion ensures that the first order theory is well behaved, in particular, it is amenable to the application of concepts of stability [3,4,12]. This paper is the third in a recent series [8,7] investigating coherency of monoids. Its predecessors were concerned with free monoids in certain varieties of monoids and unary monoids.…”

Section: Introductionmentioning

confidence: 99%

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Coherency and constructions for monoids

Yang

Gould

Hartmann

et al. 2019

Preprint

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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.

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Section: Introductionmentioning

confidence: 64%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Coherency and constructions for monoids

Yang

Gould

Hartmann

et al. 2019

Preprint

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“…In Sect. 3 we show how it is connected to other finiteness conditions that have been studied recently, such as that of being right coherent [10,11] or right Noetherian [22].…”

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confidence: 68%

Right ideal Howson semigroups

Carson

Gould

2020

Semigroup Forum

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We define a semigroup S to be right ideal Howson if the intersection of any two finitely generated right ideals, or, equivalently, any two principal right ideals, is again finitely generated. We give many examples of such semigroups, including right coherent monoids, finitely aligned semigroups, and inverse semigroups. We investigate the closure of the class of right ideal Howson semigroups under algebraic constructions. For any n 2 N 0 we give a presentation of a right ideal Howson semigroup possessing an intersection of principal right ideals that requires exactly n generators that is, in a particular sense, universal. We give analogous presentations for commutative, and for commutative cancellative, (right) ideal Howson semigroups. Keywords Semigroup Á Monoid Á One-sided ideal Á Presentations 1 Introduction An algebra exhibits the Howson property if the intersection of two finitely generated subalgebras is also finitely generated. This term is in honour of the author of [15], who showed that the intersection of finitely generated subgroups of free groups is finitely generated. There have been a number of investigations of the Howson property for other classes of algebras. In particular, the Howson property for inverse semigroups has been studied by several authors such as Jones and Trotter [16, 17], Lawson and Vdovina [20] and Silva and Soares [25]. By contrast to the situation for Communicated by Mark V. Lawson. The first author acknowledges the support of EPSRC in the form of a Ph.D. studentship.

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“…In Section 6 we argue that the class of right coherent monoids is closed under retract. As a consequence of this, we have an alternative (albeit rather longer) proof to that of [13] that free monoids are coherent. Finally, in Section 7, we show that FIM( ), FLA( ) and FAM( ) are not coherent (for | | 2).…”

Section: Introductionmentioning

confidence: 76%

Coherency, free inverse monoids and related free algebras

Gould¹,

Hartmann²

2016

Math. Proc. Camb. Phil. Soc.

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A monoid S is right coherent if every finitely generated subact of every finitely presented right S-act is finitely presented. This is the non-additive notion corresponding to that for a ring R stating that every finitely generated submodule of every finitely presented right R-module is finitely presented. For monoids (and rings) right coherency is an important finitary property which determines, amongst other things, the existence of a model companion of the class of right S-acts (right R-modules) and hence that the class of existentially closed right S-acts (right R-modules) is axiomatisable.Choo, Lam and Luft have shown that free rings are right (and left) coherent; the authors, together with Ruškuc, have shown that (free) groups, free commutative monoids and free monoids, have the same properties. It is then natural to ask whether other free algebras in varieties of monoids, possibly with an augmented signature, are right coherent. We demonstrate that free inverse monoids are not.Munn described the free inverse monoid FIM(Ω) on Ω as consisting of birooted finite connected subgraphs of the Cayley graph of the free group on Ω. Sitting within FIM(Ω) we have free algebras in other varieties and quasi-varieties, in particular the free left ample monoid FLA(Ω) and the free ample monoid FAM(Ω). The former is the free algebra in the variety of unary monoids corresponding to partial maps with distinguished domain; the latter is the two-sided dual. For example, FLA(Ω) is obtained from FIM(Ω) by considering only subgraphs with vertices labelled by elements of the free monoid on Ω.The main objective of the paper is to show that FLA(Ω) is right coherent. Furthermore, by making use of the same techniques we show that FIM(Ω), FLA(Ω) and FAM(Ω) satisfy (R), (r), (L) and (l), related conditions arising from the axiomatisability of certain classes of right S-acts and of left S-acts.

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Free Monoids are Coherent

Cited by 6 publications

References 5 publications

Coherency and constructions for monoids

Coherency and constructions for monoids

Right ideal Howson semigroups

Coherency, free inverse monoids and related free algebras

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