arXiv subjects

Leonardo M. Cabrer

Publications and source records attributed to Leonardo M. Cabrer.

5 recordsLinked to original sources

arXiv2020

Piggybacking over unbounded distributive lattices

This paper fills a gap in the literature on natural duality theory. It concerns dual representations of categories of distributive-lattice-based algebras in which the lattice reducts are not assumed to have bounds. The development of theory to parallel what is known for the exhaustively-studied bounded case was initially driven by need. This arose in connection with a major investigation of Sugihara algebras and Sugihara monoids. The theorems in this paper apply in a systematic way to a range of examples:varieties of Sugihara type; other classes of algebras previously treated ad hoc; and further classes as required.

math.RA↗

arXiv2018

Checking Admissibility Using Natural Dualities

This paper presents a new method for obtaining small algebras to check the admissibility-equivalently, validity in free algebras-of quasi-identities in a finitely generated quasivariety. Unlike a previous algebraic approach of Metcalfe and Rothlisberger that is feasible only when the relevant free algebra is not too large, this method exploits natural dualities for quasivarieties to work with structures of smaller cardinality and surjective rather than injective morphisms. A number of case studies are described here that could not be be solved using the algebraic approach, including (quasi)varieties of MS-algebras, double Stone algebras, and involutive Stone algebras.

math.LO↗

arXiv2016

MV-algebras, infinite dimensional polyhedra, and natural dualities

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.

math.RA↗

arXiv2013

Gödel algebras: interactive dualities and their applications

We present a technique for deriving certain new natural dualities for any variety of algebras generated by a finite Heyting chain. The dualities we construct are tailored to admit a transparent translation to the more pictorial Priestley/Esakia duality and back again. This enables us to combine the two approaches and so to capitalise on the virtues of both, in particular the categorical good behaviour of a natural duality: we thereby demonstrate the fullness, or not, of each of our dualities; we obtain new results on amalgamation; and we also provide a simple treatment of coproducts.

math.RA↗

arXiv2013

MV-algebras freely generated by finite Kleene algebras

If V and W are varieties of algebras such that any V-algebra A has a reduct U(A) in W, there is a forgetful functor U: V->W that acts by A |-> U(A) on objects, and identically on homomorphisms. This functor U always has a left adjoint F: W->V by general considerations. One calls F(B) the V-algebra freely generated by the W-algebra B. Two problems arise naturally in this broad setting. The description problem is to describe the structure of the V-algebra F(B) as explicitly as possible in terms of the structure of the W-algebra B. The recognition problem is to find conditions on the structure of a given V-algebra A that are necessary and sufficient for the existence of a W-algebra B such that F(B) is isomorphic to A. Building on and extending previous work on MV-algebras freely generated by finite distributive lattices, in this paper we provide solutions to the description and recognition problems in case V is the variety of MV-algebras, W is the variety of Kleene algebras, and B is finitely generated--equivalently, finite. The proofs rely heavily on the Davey-Werner natural duality for Kleene algebras, on the representation of finitely presented MV-algebras by compact rational polyhedra, and on the theory of bases of MV-algebras.

math.LO↗