On The Spectrum of Countable MV-algebras
In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra $Free_1$ of McNaughton functions from $[0,1]$ to $[0,1]$, the continuous, piecewise l...
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| Format: | Article |
| Language: | English |
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Islamic Azad University, Bandar Abbas Branch
2023-11-01
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| Series: | Transactions on Fuzzy Sets and Systems |
| Subjects: | |
| Online Access: | https://tfss.journals.iau.ir/article_706608_43e7251b9559c462e5024531565ff316.pdf |
| _version_ | 1827773075342491648 |
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| author | Giacomo Lenzi |
| author_facet | Giacomo Lenzi |
| author_sort | Giacomo Lenzi |
| collection | DOAJ |
| description | In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra $Free_1$ of McNaughton functions from $[0,1]$ to $[0,1]$, the continuous, piecewise linear functions with integer coefficients. The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit. Also, we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic.As an intermediate step we consider the MV-algebra $A_1$ of continuous, piecewise linear functions with rational coefficients. It is known that $A_1$ contains $Free_1$, and that $A_1$ and $Free_1$ are equispectral. However, $A_1$ is in some sense easy to work with than $Free_1$. Now, $A_1$ is still countable. To build an equispectral uncountable MV-algebra $A_2$, we consider certain ``almost rational'' functions on $[0,1]$, which are rational in every initial segment of $[0,1]$, but which can have an irrational limit in $1$.We exploit heavily, via Mundici equivalence, the properties of divisible lattice ordered abelian groups, which have an additional structure of vector spaces over the rational field. |
| first_indexed | 2024-03-11T13:16:52Z |
| format | Article |
| id | doaj.art-bc7ff7b2ab704451a5d108f17fe2cc5b |
| institution | Directory Open Access Journal |
| issn | 2821-0131 |
| language | English |
| last_indexed | 2024-03-11T13:16:52Z |
| publishDate | 2023-11-01 |
| publisher | Islamic Azad University, Bandar Abbas Branch |
| record_format | Article |
| series | Transactions on Fuzzy Sets and Systems |
| spelling | doaj.art-bc7ff7b2ab704451a5d108f17fe2cc5b2023-11-03T11:15:29ZengIslamic Azad University, Bandar Abbas BranchTransactions on Fuzzy Sets and Systems2821-01312023-11-012218419310.30495/tfss.2023.1991906.1082706608On The Spectrum of Countable MV-algebrasGiacomo Lenzi0Department of Mathematics, University of Salerno, Fisciano (SA), Italy.In this paper we consider MV-algebras and their prime spectrum. We show that there is an uncountable MV-algebra that has the same spectrum as the free MV-algebra over one element, that is, the MV-algebra $Free_1$ of McNaughton functions from $[0,1]$ to $[0,1]$, the continuous, piecewise linear functions with integer coefficients. The construction is heavily based on Mundici equivalence between MV-algebras and lattice ordered abelian groups with the strong unit. Also, we heavily use the fact that two MV-algebras have the same spectrum if and only if their lattice of principal ideals is isomorphic.As an intermediate step we consider the MV-algebra $A_1$ of continuous, piecewise linear functions with rational coefficients. It is known that $A_1$ contains $Free_1$, and that $A_1$ and $Free_1$ are equispectral. However, $A_1$ is in some sense easy to work with than $Free_1$. Now, $A_1$ is still countable. To build an equispectral uncountable MV-algebra $A_2$, we consider certain ``almost rational'' functions on $[0,1]$, which are rational in every initial segment of $[0,1]$, but which can have an irrational limit in $1$.We exploit heavily, via Mundici equivalence, the properties of divisible lattice ordered abelian groups, which have an additional structure of vector spaces over the rational field.https://tfss.journals.iau.ir/article_706608_43e7251b9559c462e5024531565ff316.pdfmv-algebrasprime spectrumlattice ordered abelian groups |
| spellingShingle | Giacomo Lenzi On The Spectrum of Countable MV-algebras Transactions on Fuzzy Sets and Systems mv-algebras prime spectrum lattice ordered abelian groups |
| title | On The Spectrum of Countable MV-algebras |
| title_full | On The Spectrum of Countable MV-algebras |
| title_fullStr | On The Spectrum of Countable MV-algebras |
| title_full_unstemmed | On The Spectrum of Countable MV-algebras |
| title_short | On The Spectrum of Countable MV-algebras |
| title_sort | on the spectrum of countable mv algebras |
| topic | mv-algebras prime spectrum lattice ordered abelian groups |
| url | https://tfss.journals.iau.ir/article_706608_43e7251b9559c462e5024531565ff316.pdf |
| work_keys_str_mv | AT giacomolenzi onthespectrumofcountablemvalgebras |