# The Abelian Kernel of an Inverse Semigroup

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background and Notation

#### 2.1. Basic Results on Semigroups

**Lemma**

**1.**

**Lemma**

**2.**

- 1.
- A subsemigroup T of an inverse semigroup S is an inverse subsemigroup if $a\in T$ implies ${a}^{-1}\in T$.
- 2.
- If $s,t$ are elements of an inverse semigroup S, we have that ${\left({s}^{-1}\right)}^{-1}=s$ and ${\left(st\right)}^{-1}={t}^{-1}{s}^{-1}$.
- 3.
- S is an inverse semigroup if, and only if, S is regular and its idempotents commute. As a consequence, if S is an inverse semigroup, $E\left(S\right)$ is a subsemigroup of S.

**Theorem**

**1.**

#### 2.2. Relational Morphisms and Kernels

**Proposition**

**1.**

- 1.
- For every $i\in \mathsf{\Lambda}$, $\tau (i,1,i)=:{K}_{i}\le H$
- 2.
- Given $(i,g,j)\in J$, for every $x\in \tau (i,g,j)$, $\tau (i,g,j)={K}_{i}x=x{K}_{j}$.
- 3.
- For every $s,t\in J$, $\left|\tau \right(s\left)\right|=\left|\tau \right(t\left)\right|$ and $\tau \left(s\right)\tau \left(t\right)=\tau \left(st\right)$, in case that $st\ne 0$.

**Proof.**

**Theorem**

**2.**

**Proposition**

**2**

**Lemma**

**3.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**3.**

**Proof.**

## 3. Minimal Pairs

**Lemma**

**4.**

- 1.
- For every $(i,g,j),({i}^{\prime},{g}^{\prime},{j}^{\prime})\in {\mathcal{M}}^{0}(G,\mathsf{\Lambda},\mathsf{\Lambda},{I}_{\mathsf{\Lambda}})$, $0\ne (i,g,j)({i}^{\prime},{g}^{\prime},{j}^{\prime})$ if, and only if, $j={i}^{\prime}$. Moreover, ${(i,g,j)}^{-1}=(j,{g}^{-1},i)$.
- 2.
- $E\left({J}^{0}\right)=\left\{0\right\}\cup \{(i,1,i):i\in \mathsf{\Lambda}\}$ and ${S}_{(i,1,i)}={\left({J}^{0}\right)}_{(i,1,i)}=(i,G,i)$.
- 3.
- For every $s\in S$ and $x\in {J}^{0}$, $sx,xs\in {J}^{0}$.

**Proposition**

**4.**

**Proof.**

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**5.**

**Proof.**

## 4. The Abelian Kernel of an Inverse Semigroup

**Lemma**

**6.**

- 1.
- For every $i,j\in \mathsf{\Lambda}$, $\tau (i,1,i)=\tau (j,1,j)=:H$ is a subgroup of A. Then, for every $(i,g,j)\in J$, $\tau (i,g,j)=x+H$, for each $x\in \tau (i,g,j)$.
- 2.
- For every element $s\in S$, the equality $\tau (i,g,j)=\tau ({i}^{\prime},{g}^{\prime},{j}^{\prime})$ holds for all ${M}_{s}(i,j)=g\ne 0$, ${M}_{s}({i}^{\prime},{j}^{\prime})={g}^{\prime}\ne 0$, where ${M}_{s}$ is the matrix of projections of s.
- 3.
- There exists a relational morphism $\overline{\tau}:$ S A/H such that $|\overline{\tau}\left(s\right)|=1$, for every $0\ne s$, and ${\overline{\tau}}^{-1}\left(H\right)\cap J={\tau}^{-1}\left(0\right)\cap J$.

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Rhodes, J.; Tilson, B.R. Improved lower bounds for the complexity of finite semigroups. J. Pure Appl. Algebra
**1972**, 2, 13–71. [Google Scholar] [CrossRef] [Green Version] - Ash, C.J. Inevitable graphs: A proof of the type II conjecture and some related decision procedures. Int. J. Algebra Comput.
**1991**, 1, 127–146. [Google Scholar] [CrossRef] - Ribes, L.; Zalesskiǐ, P.A. The pro-p topology of a free group and algorithmic problems in semigroups. Int. J. Algebra Comput.
**1994**, 4, 359–374. [Google Scholar] [CrossRef] - Delgado, M. Abelian pointlikes of a semigroup. Semigroup Forum
**1998**, 56, 339–361. [Google Scholar] [CrossRef] - Steinberg, B. Monoid kernels and profinite topologies on the free Abelian group. Bull. Austral. Math. Soc.
**1999**, 60, 391–402. [Google Scholar] [CrossRef] [Green Version] - Almeida, J.; Shahzamanian, M.H.; Steinberg, B. The pro-nilpotent group topology on a free group. J. Algebra
**2017**, 480, 332–345. [Google Scholar] [CrossRef] [Green Version] - Ballester-Bolinches, A.; Pérez-Calabuig, V. Abelian kernels, profinite topologies and partial automorphisms. Unpublished work.
- Clifford, A.H.; Preston, G.B. The Algebraic Theory of Semigroups; American Mathematical Society: Providence, RI, USA, 1961; Volume I. [Google Scholar]
- Rhodes, J.; Steinberg, B. The q-Theory of Finite Semigroups; Springer Monographs in Mathematics; Springer: New York, NY, USA, 2009. [Google Scholar]
- Rees, D. On semi-groups. Math. Proc. Camb. Philos. Soc.
**1940**, 36, 387–400. [Google Scholar] [CrossRef] - Ballester-Bolinches, A.; Ezquerro, L.M. Classes of Finite Groups; Mathematics and Its Applications; Springer: New York, NY, USA, 2006; Volume 584. [Google Scholar]
- Steinberg, B. Inevitable graphs and profinite topologies: some solutions to algorithmic problems in monoid and automata theory, stemming from group theory. Int. J. Algebra Comput.
**2001**, 11, 25–71. [Google Scholar] [CrossRef]

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Ballester-Bolinches, A.; Pérez-Calabuig, V.
The Abelian Kernel of an Inverse Semigroup. *Mathematics* **2020**, *8*, 1219.
https://doi.org/10.3390/math8081219

**AMA Style**

Ballester-Bolinches A, Pérez-Calabuig V.
The Abelian Kernel of an Inverse Semigroup. *Mathematics*. 2020; 8(8):1219.
https://doi.org/10.3390/math8081219

**Chicago/Turabian Style**

Ballester-Bolinches, A., and V. Pérez-Calabuig.
2020. "The Abelian Kernel of an Inverse Semigroup" *Mathematics* 8, no. 8: 1219.
https://doi.org/10.3390/math8081219