Convergences in perfect BL-algebras

The aim of the paper is to investigate some concepts of convergence in the class of perfect BL-algebras. Similarity convergence was developed by G. Georgescu and A. Popescu in the case of the residuated lattices, while the convergence with a fixed regulator was studied by Cernák for lattice-ordered groups and MV-algebras and by the author for residuated lattices. In this paper we study the similarity convergence and the convergence with a fixed regulator for the perfect BL-algebras. The main result is the construction of Cauchy completion of a perfect BL-algebra.Peer Reviewe

On generators of commutative semifields

We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of generators and show that it grows linearly with the depth of an associated rooted forest.Comment: 12 page

Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.Comment: 54 page