‎On The Spectrum of Countable MV-algebras

Transactions on Fuzzy Sets and Systems

مقاله 11، دوره 2، شماره 2، بهمن 2023، صفحه 184-193   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2023.1991906.1082

نویسنده

Giacomo Lenzi^*

Department of Mathematics, University of Salerno, Fisciano (SA), Italy.

چکیده

‎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.

کلیدواژه‌ها

MV-algebras‎؛ Prime spectrum‎؛ Lattice ordered abelian groups‎

مراجع

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