Strong semisimplicity in MV-algebras

BUSANICHE, MANUELA

Córdoba

Congreso; Congreso Latinoamericano de Matemáticos; 2012

Unión Matemática de América Latina y el Caribe-

We are interesting in the study of finitely generated theories in ${\L}$ukasiewicz infinite-valued propositional calculus ${\L}_{\infty}$. Recall the correspondence between theories with $n$-variables formulas in ${\L}_{\infty}$ and ideals in the free MV-algebra with $n$ generators. Since the free MV-algebra with $n$-generators is an algebra of piecewise linear continuous functions from the $n$-cube into $[0,1]$, there is nowadays a trend of using tools from polyhedral topology to obtain important results about MV-algebras, thus for ${\L}_{\infty}$ (see \cite{Mun}).We study strong semisimplicity of finitely generated MV-algebras using the mentioned tools. An MV-algebra $A$ is called strongly semisimple  if for any $a\in A$ the intersection of all maximal ideals $J$ such that $a\in J$ is the ideal generated by $a$.%In logical terms that translate into those finitely generated theories $\Theta$ such that for each  formula $\theta$, the intersection of all maximal subtheories of $\Theta$ containing $\theta$ is the subtheory generated by $\theta.$ We show an MV-algebra that is not strongly semisimple and we see that every singly generated MV-algebra is strongly semisimple.