On the spectra of commutative semigroups
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On the spectra of commutative semigroups Huanrong Wu1 · Qingguo Li1 Received: 29 January 2019 / Accepted: 22 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This study aims to investigate the Zariski topology on the prime ideals of a commutative semigroup S, denoted by Spec(S). First, we show that a topological space X is homeomorphic to Spec(S) for some commutative semigroup S if and only if X is an SS-space that can be described purely in topological terms. Next, we show that an adjunction exists between the category of commutative semigroups and that of SSspaces. We further show that the category of commutative idempotent semigroups is dually equivalent to that of SS-spaces. Keywords Semigroup · Zariski topology · Spectral space · Topological representation
1 Introduction Spectral theory in rings, initiated by Hochster [7], offers a new approach and tool to link algebra, topology and geometry theories. The result by Hochster shows that a topological space X is homeomorphic to Spec(R), the collection of prime ideals of some commutative ring R endowed with the Zariski topology, if and only if X is a spectral space. Since then, a great quantity of works about spectral theory emerge in rings, semirings, hemirings, semigroups and monoids [11, 13, 15, 16]. Recently, Finocchiaro et.al. [5] brought back the concept of semigroup prime from semigroups to rings; that is, a semigroup prime of a ring R is also a prime ideal of a semigroup (R, ⋅) . The authors studied the Zariski topology on semigroup primes of a ring R. They explored the intrinsic link between the Zariski topology on semigroup primes of a ring R and spectral spaces. This inspires us to study the
Communicated by Marcel Jackson. Supported by National Natural Science Foundation of China (No. 11771134). * Qingguo Li [email protected] 1
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
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Zariski topology on prime ideals of a semigroup solely. Moreover, Ray [17] discussed the spectrum of a monoid and showed that it is a spectral space. As such, we explore the topological algebra on commutative semigroups via spectral spaces. We show that the collection of all prime ideals of a commutative semigroup endowed with the Zariski topology is homeomorphic to the spectrum of a ring (i.e., it is a spectral space). Furthermore, a topological space X is homeomorphic to Spec(S) for some commutative semigroup S if and only if X is an SS-space that can be described purely in topological terms. Moreover, we show that an adjunction exists between the category of commutative semigroups and that of SS-spaces. Finally, we prove that the category of commutative idempotent semigroups is dually equivalent to that of SS-spaces. This shows that an SS-space is a topological representation for commutative idempotent semigroups. The remainder of this paper is organized as follows. In Sect. 2, we shall first briefly recall some related basic definitions and facts about
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