Extension and Amalgamation in Monoids and Semigroups

Renshaw, James

doi:10.1112/plms/s3-52.1.119

Cited by 21 publications

(36 citation statements)

References 7 publications

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“…Introduction. The author [9,11,13] introduced the concept of stability for acts over monoids and showed some strong connections with flatness of monoid acts, with the embeddability of monoid amalgams and with the structure of amalgamated free products of monoids. Our main aim in this paper is to develop these connections and introduce some related properties of act monomorphisms.…”

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confidence: 99%

Stability and flatness in acts over monoids

Renshaw

2002

Colloq. Math.

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Abstract. Our aim in this paper is to study the concept of stability for acts over monoids and in the process develop connections with flatness properties of acts and with some of the current techniques and construction used in the homological classification of monoids. We also present new proofs of some results relating to torsion free acts over monoids and to the embeddability of semigroup amalgams.1. Introduction. The author [9,11,13] introduced the concept of stability for acts over monoids and showed some strong connections with flatness of monoid acts, with the embeddability of monoid amalgams and with the structure of amalgamated free products of monoids. Our main aim in this paper is to develop these connections and introduce some related properties of act monomorphisms. We also demonstrate that some of the current results and techniques from the homological classification of monoids are related to these properties.Some connections between stability, flatness and amalgamation were presented by the author in [11] and [13]. We examine in more detail the connections between stability and flatness in Section 3. The results in this section then act as a "template" for Section 4 where we define a hierarchy of properties related to stability and also to the notion of unitary submonoid. Howie [4] has provided connections between the unitary properties and amalgamation whilst Bulman-Fleming [1] summarises various connections with one of the related properties that we introduce and the homological classification of monoids. Section 5 briefly looks at when monomorphisms into various types of indecomposable act are stable. In Section 6 we demonstrate some connections between stability and the various flatness concepts used in the homological classification theory before proving some interesting consequences in Section 7.

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confidence: 99%

Stability and flatness in acts over monoids

Renshaw

2002

Colloq. Math.

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“…Since ^, = MaK, by multiplying the equations (8) on the left by ax from the right, we get x -xax = xxax = x2ax = ■■■ = xiax = Xi+Xat+Xax, so that x = Xi(axai+Xax), while, by Result 3(i), akai+xak -bkai+xbk for all 1 < k < i. So from (8) we obtain a scheme of length < m joining (x, \®y) to (x', l®y') as follows: By the inductive assumption, x ® y = x' ® y'. Case 2.…”

Section: Proof Of the Main Theoremmentioning

confidence: 95%

“…Our method is to appeal the criterion for an amalgamation base given in [9], which is a modified version of Renshaw's Theorem [8,Theorem 6.11].…”

Section: Main Theorem a Normal Band Has Both (Rep) And (Rep)op If Anmentioning

confidence: 99%

“…On the other hand, if fax = «^fll and bx = a2, then x -xaxxx(bxax) = (x2a2)ax = x2ax. So we can remove xx, w2®y2 from scheme (8). Hence (9) may be assumed.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…There exists some 1 < i < n such that all ak, bk (1 < k < i) belong to fax but axfai+x >f bi+x. By applying Lemma 1 (ii) to (8) we have (axbi+xax)y(= (bxbi+xbi)(w2 ®y2) = ■■■ = (bibi+xbi)(wi+x ®yi+x) = (ai+xbi+xai+x)(Wi+x ®yi+x) = ai+i(wi+x ®yi+x)) = bi+x(wi+2®yi+2).…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

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