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

Functorial Properties of the Reticulation of a Universal Algebra

The reticulation of an algebra A is a bounded distributive lattice whose prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic to the prime spectrum of congruences of A, with its own Stone topology. The reticulation allows algebraic and topological properties to be transferred between the algebra A and this bounded distributive lattice, a transfer which is facilitated if we can define a reticulation functor from a variety containing A to the variety of (bounded) distributive lattices. In this paper, we continue the study of the reticulation of a universal algebra initiated in [27], where we have used the notion of prime congruence introduced through the term condition commutator, for the purpose of creating a common setting for the study of the reticulation, applicable both to classical algebraic structures and to the algebras of logics. We characterize morphisms which admit an image through th

Reticulation of Quasi-commutative Algebras

The commutator operation in a congruence-modular variety $\mathcal{V}$ allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993). The reticulation of an algebra $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (endowed with the topology defined by Agliano). In a recent paper, C. Mure\c{s}an and the author defined the reticulation for the algebras $A$ in a semidegenerate congruence-modular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ is closed under commutators. This theory does not cover the Belluce reticulation for non-commutative rings. In this paper we shall introduce the quasi-commutative algebras in a semidegenerate congruence-modular variety $\mathcal{V}$ as a generalization of the Belluce quasi-commutative rings. We define and study a notion of reticulation for the quasi-commutative algebras such that the Belluce reticulation for the quasi-commutative rings can be obtained as a particular case. We prove a characterization theorem for the quasi-commutative algebras and some transfer properties by means of the reticulationComment: arXiv admin note: text overlap with arXiv:2205.0217

A representation theorem for integral rigs and its applications to residuated lattices

We prove that every integral rig in Sets is (functorially) the rig of global sections of a sheaf of really local integral rigs. We also show that this representation result may be lifted to residuated integral rigs and then restricted to varieties of these. In particular, as a corollary, we obtain a representation theorem for pre-linear residuated join-semilattices in terms of totally ordered fibers. The restriction of this result to the level of MV-algebras coincides with the Dubuc-Poveda representation theorem.Comment: Manuscript submitted for publicatio

Semidegenerate Congruence-modular Algebras Admitting a Reticulation

The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construction of the lattice L(R) was generalized by Belluce for each unital ring R and the reticulation was defined by axioms. In a recent paper we generalized the Belluce construction for algebras in a semidegenerate congruence-modular variety V. For any algebra A ∈ V we defined a bounded distributive lattice L(A), but in general the prime spectrum Spec(A) of A is not homeomorphic with the prime spectrum SpecId (L(A)). We introduced the quasi-commutative algebras in the variety V (as a generalization of Belluce’s quasi-commutative rings) and proved that for any algebra A ∈ V, the spectra Spec(A) and SpecId (L(A)) are homeomorphic. In this paper we define the reticulation A ∈ V by four axioms and prove that any two reticulations of A are isomorphic lattices. By using the uniqueness of reticulation and other results from the mentioned paper, we obtain a characterization theorem for the algebras A ∈ V that admit a reticulation: A is quasi-commutative if and only if A admits a reticulation. This result is a universal algebra generalization of the following Belluce theorem: a ring R is quasi-commutative if and only if R admits a reticulation. Another subject treated in this paper is the spectral closure of the prime spectrum Spec(A) of an algebra A ∈ V, a notion that generalizes the Belluce spectral closure of the prime spectrum of a ring