Malcev presentations for subsemigroups of direct products of ...

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of two free semigroups, but Theorem shows that such direct products are notin general Malcev coherent: the direct product of two free semigroups of rankat least 2 is not Malcev coherent.The Malcev coherence of nilpotent groups was established by [CRRa, Theorem]. Polycyclic groups form a more general class than finitely generatednilpotent groups whilst retaining the property of coherence [Sim, §§ .–.].One therefore naturally asks whether polycyclic groups are Malcev coherent.From Theorem and a result of Rosenblatt [Ros], one obtains a negative answer:polycyclic groups are not in general Malcev coherent.The direct product of a free group and an abelian group is coherent. (Aslight modification of the reasoning of [Mil] proves this assertion and, moregenerally, establishes the coherence of the direct product of a free group and apolycyclic group.) Both virtually free groups [CRRb, Theorem ] and abeliangroups are known to be Malcev coherent. [Every finitely generated subsemigroupof an abelian group — and more generally every finitely generated commutativesemigroup — admits a finite ‘ordinary’ presentation as a consequenceof Rédei’s Theorem [Réd].] Most of this paper is dedicated to the proof of itssecond main result, Theorem , which asserts that every direct product of avirtually free group and an abelian group is Malcev coherent.[This paper is based on Chapter of the author’s Ph.D. thesis [Cai].] Following [ECH + ], the notation used in this paper distinguishesa word from the element of the semigroup or group it represents. Let A be analphabet representing a set of generators for a semigroup S. For any word w ∈A + , denote by w the element of S represented by w. Similarly, if A representsa set of generators for a group G, let w be the element of G represented byw ∈ (A ∪ A −1 ) ∗ . In both cases, for any set of words W, W is the set of allelements represented by at least one word in W.The theory of Malcev presentations was introduced by Spehner [Spe], thoughthey are based on Malcev’s necessary and sufficient condition for the embeddabilityof a semigroup in a group [Mal, Mal]. (Details of the embeddabilitycondition can also be found in [CP, Chapter ].) Although this sectioncontains the definitions and results about Malcev presentations required for therest of the paper, the reader is assumed to be familiar with the basic theory of[ordinary] semigroup presentations. For a fuller exposition of the foundationsof the theory of Malcev presentations, see [Cai, Chapter ]. . Let S be any semigroup. A congruence σ on S is a Malcevcongruence if S/σ is embeddable in a group.If {σ i : i ∈ I} is a set of Malcev congruences on S, then σ = ∩ i∈I σ i is alsoa Malcev congruence on S. This is true because S/σ i embeds in a group G i foreach i ∈ I, so S/σ embeds in ∏ i∈I S/σ i, which in turn embeds in ∏ i∈I G i. Thefollowing definition therefore makes sense. . Let A + be a free semigroup; let ρ ⊆ A + × A + be any binaryrelation on A + . Denote by ρ M the smallest Malcev congruence containing ρ —namely,ρ M = ∩ {σ : σ ⊇ ρ, σ is a Malcev congruence on A+ } .
Then SgM⟨A | ρ⟩ is a Malcev presentation for [any semigroup isomorphic to]A + /ρ M . If both A and ρ are finite, the the Malcev presentation SgM⟨A | ρ⟩ issaid to be finite.Let SgM⟨A | R⟩ be a Malcev presentation for a semigroup S. If S has a finiteMalcev presentation, then it admits one of the form SgM⟨A | Q⟩, where Q is afinite subset of R.The notation SgM⟨A | ρ⟩ distinguishes the Malcev presentation with generatorsA and defining relations ρ from the ordinary semigroup presentationSg⟨A | ρ⟩, which defines A + /ρ # . (Recall that ρ # denotes the smallest congruencecontaining ρ.) Similarly, Gp⟨A|ρ⟩ denotes the group presentation with the sameset of generators and defining relations. Let X be a subset of a group G. Denoteby Sg⟨X⟩ the subsemigroup generated by X, by Mon⟨X⟩ the submonoid generatedby X, and by Gp⟨X⟩ the subgroup generated by X. The free group withbasis A is denoted FG(A).Fix A + and ρ as in the Definition and let S = A + /ρ M . Let A L , A R be twosets in bijection with A under the mappings a ↦→ a L , a ↦→ a R , respectively, withA, A L , A R being pairwise disjoint. Extend the mappings a ↦→ a L , a ↦→ a R toanti-isomorphisms from A ∗ to (A L ) ∗ and (A R ) ∗ , respectively. Letτ = ρ ∪ { (bb R a, a), (abb R , a), (b L ba, a), (ab L b, a) : a ∈ A ∪ A L ∪ A R , b ∈ A } .Let G = Sg⟨A ∪ A L ∪ A R | τ⟩. The semigroup G is actually the universal groupof S, with G ≃ Gp⟨A | ρ⟩. (See [CP, Chapter ] for background informationon universal groups.) Therefore a Malcev presentation for a semigroup is effectivelya presentation for its universal group, with all defining relations confinedto A + × A + . It can be shown thator equivalently thatρ M = τ # ∩ (A + × A + ),ρ M = {(u, v) : u, v ∈ A + represent the same element of Gp⟨A | ρ⟩}.()Therefore, two words u, v ∈ A + represent the same element of S if and only ifthere is a sequenceu = u 0 → u 1 → . . . → u n = v,with n ⩾ 0, where, for each i ∈ {0, . . . , n − 1}, there exist p i , q i , q ′ i , r i ∈ (A ∪A L ∪ A R ) ∗ such that u i = p i q i r i , u i+1 = p i q ′ i r i, and (q i , q ′ i ) ∈ τ or (q′ i , q i) ∈ τ.In fact, it can be shown that u, v ∈ A + represent the same element of S if anyonly if there exists such a sequence with p i ∈ (A ∪ A L ) ∗ and r i ∈ (A ∪ A R ) ∗ .This restriction on the letters that can appear in p i and r i simply means that nochanges can occur to the left of an a L or to the right of an a R . Such a sequenceis called a Malcev ρ-chain (or simply a Malcev chain) from u to v. If (u, v) ∈ ρ M ,then (u, v) is said to be a Malcev consequence of ρ. The first part of the present section is devoted to a proof that thedirect product of two free semigroups of rank at least 2 is not Malcev coherent.This contrasts with Spehner’s proof of the Malcev coherent of free monoids
Page 1: Malcev presentations for subsemigro
Page 5 and 6: The first component of v 2 must beg
Page 7 and 8: Every finitely generated nilpotent
Page 9 and 10: andp ′′ (u, v) = p ′ ((c 1 ,
Page 11 and 12: . SupposeThen, by (), the relations
Page 13 and 14: where x, α i , w, β i , y ′ , y
Page 15 and 16: Therefore σδ ′′ = δ ′ . 13
Page 17 and 18: and this completes the proof. 16Let
Page 19 and 20: . Is every direct product of a free

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