On the Monoid of Unital Endomorphisms of a Boolean Ring

Al Subaiei, Bana; Jarboui, Noômen

doi:10.3390/axioms10040305

Open AccessArticle

On the Monoid of Unital Endomorphisms of a Boolean Ring

by

Bana Al Subaiei

^1,*,†

and

Noômen Jarboui

^2,†

¹

Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

²

Département de Mathématiques, Faculté des Sciences de Sfax, Université de Sfax, P.O. Box 1171, Route de Soukra, Sfax 3038, Tunisia

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Axioms 2021, 10(4), 305; https://doi.org/10.3390/axioms10040305

Submission received: 10 October 2021 / Revised: 6 November 2021 / Accepted: 12 November 2021 / Published: 14 November 2021

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Abstract

Let X be a nonempty set and

P (X)

the power set of X. The aim of this paper is to provide an explicit description of the monoid

{End}_{1_{P (X)}} (P (X))

of unital ring endomorphisms of the Boolean ring

P (X)

and the automorphism group

Aut (P (X))

when X is finite. Among other facts, it is shown that if X has cardinality

n \geq 1

, then

{End}_{1_{P (X)}} (P (X)) ≅ T_{n}^{o p}

, where

T_{n}

is the full transformation monoid on the set X and

Aut (P (X)) ≅ S_{n}

.

Keywords: Boolean ring; power set; prime ideal; maximal ideal; ring endomorphism

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MDPI and ACS Style

Al Subaiei, B.; Jarboui, N. On the Monoid of Unital Endomorphisms of a Boolean Ring. Axioms 2021, 10, 305. https://doi.org/10.3390/axioms10040305

AMA Style

Al Subaiei B, Jarboui N. On the Monoid of Unital Endomorphisms of a Boolean Ring. Axioms. 2021; 10(4):305. https://doi.org/10.3390/axioms10040305

Chicago/Turabian Style

Al Subaiei, Bana, and Noômen Jarboui. 2021. "On the Monoid of Unital Endomorphisms of a Boolean Ring" Axioms 10, no. 4: 305. https://doi.org/10.3390/axioms10040305

APA Style

Al Subaiei, B., & Jarboui, N. (2021). On the Monoid of Unital Endomorphisms of a Boolean Ring. Axioms, 10(4), 305. https://doi.org/10.3390/axioms10040305

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On the Monoid of Unital Endomorphisms of a Boolean Ring

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