The Reticulation of a Universal Algebra

The reticulation of an algebra $A$ is a bounded distributive lattice ${\cal L}(A)$ whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of $A$, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra $A$ from a semi-degenerate congruence-modular variety ${\cal C}$ in the case when the commutator of $A$, applied to compact congruences of $A$, produces compact congruences, in particular when ${\cal C}$ has principal commutators; furthermore, it turns out that weaker conditions than the fact that $A$ belongs to a congruence-modular variety are sufficient for $A$ to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from ${\cal C}$ is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and ${\cal C}$, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.Comment: 29 page

‎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 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

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl