On the Second Integral Homology of Bruck-Reilly Extensions of Monoids

Year 2024, Volume: 24 Issue: 2, 260 - 265, 29.04.2024

Melek Yağcı

https://doi.org/10.35414/akufemubid.1343106

Abstract

Let M be a monoid and let θ be an endomorphism on M. Then the set N^0×M×N^0 where N^0 is the set of non-negative integers, is a monoid together with the binary operation

(m,a,n)(p,b,q)=(m-n+r,(aθ^(r-n) ) (bθ^(r-p) ),q-p+r)

where r=max{n,p} and θ^0 is the identity map on M, which is called the Bruck-Reilly extension of M determined by θ and denoted by BR(M,θ). In this paper, we show that the second integral homology of Bruck-Reilly extension of a finite monoid M is

H_2 (BR(M,θ))=H_2 (M)×Z^k

for some k∈N.

Keywords

Monoid, Bruck-Reilly extensions, Second Integral Homology, Presentation

Monoidlerin Bruck-Reilly Genişlemelerinin İkinci Tamsayı Homolojisi

Abstract

M bir monoid ve θ, M üzerinde bir endomorfizm olsun. N^0 negatif olmayan tamsayıların kümesi, r=max{n,p} ve θ^0, M üzerinde birim dönüşüm olmak üzere N^0×M×N^0 kümesi

(m,a,n)(p,b,q)=(m-n+r,(aθ^(r-n) ) (bθ^(r-p) ),q-p+r)

ikili işlemi ile birlikte bir monoid tanımlar. Bu monoide θ nin belirlediği M nin Bruck-Reilly genişlemesi denir ve BR(M,θ) ile gösterilir. Bu çalışmada, bir sonlu M monoidinin Bruck-Reilly genişlemesinin ikinci tamsayı homolojisinin, öyle bir k∈N için

H_2 (BR(M,θ))=H_2 (M)×Z^k

olduğu gösterilmiştir.

Keywords

Monoid, İkinci tamsayı homolojisi, Takdim, Bruck-Reilly Genişlemesi

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