arXiv subjects

Claudia Mureşan

Publications and source records attributed to Claudia Mureşan.

At least 19 recordsLinked to original sources

arXiv2024

Congruence Extensions in Congruence-modular Varieties

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semi-degenerate congruence-modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive w.r.t. arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence-modular varieties.

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arXiv2022

Subreducts and Subvarieties of PBZ*--lattices

PBZ*-lattices are bounded lattice-ordered structures endowed with two complements, called Kleene and Brouwer; by definition, they are the paraorthomodular Brouwer-Zadeh lattices in which the pairs of elements with their Kleene complements satisfy the Strong De Morgan condition. These algebras arise in the study of Quantum Logics and they form a variety PBZL* which includes orthomodular lattices with an extended signature (with the two complements coinciding), as well as antiortholattices (whose Brouwer complements are trivial). The former turn out to have directly irreducible lattice reducts and, under distributivity, no nontrivial elements with bounded lattice complements. We establish a lattice isomorphism between the lattice of subvarieties of the variety SAOL generated by the antiortholattices with the Strong De Morgan property and the ordinal sum of the three-element chain with the lattice of subvarieties of the variety PKA of pseudo-Kleene algebras, which also gives us axiomatizations for all subvarieties of SAOL from those of the subvarieties of PKA and proves that the variety PKA is generated by the class of the bounded involution lattice reducts of the members of SAOL and thus of those of any subvariety of PBZL* that includes SAOL, hence neither of these classes is a variety. We also obtain an infinity of pairwise disjoint infinite ascending chains of varieties of PBZ*-lattices, out of which one is formed of subvarieties of SAOL and another one from subvarieties of the variety of distributive PBZ*-lattices.

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arXiv2021

On Nontrivial Weak Dicomplementations and the Lattice Congruences that Preserve Them

We study the existence of nontrivial and of representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices, then use this study to determine the finite (dual) weakly complemented lattices with the largest numbers of congruences, along with the structures of their congruence lattices. It turns out that, if $n\geq 7$ is a natural number, then the four largest numbers of congruences of the $n$--element (dual) weakly complemented lattices are: $2^{n-2}+1$, $2^{n-3}+1$, $5\cdot 2^{n-6}+1$ and $2^{n-4}+1$. For smaller numbers of elements, several intermediate numbers of congruences appear between the elements of this sequence. After determining these numbers, along with the structures of the (dual) weakly complemented lattices having these numbers of congruences, we derive a similar result for weakly dicomplemented lattices.

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arXiv2020

Stone Commutator Lattices and Baer Rings

In this paper, we transfer Davey`s characterization for $κ$--Stone bounded distributive lattices to lattices with certain kinds of quotients, in particular to commutator lattices with certain properties, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of these lattices. We then apply these results to certain congruence lattices, in particular to those of semiprime members of semi--degenerate congruence--modular varieties, and use this particular case to transfer Davey`s Theorem to commutative unitary rings.

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arXiv2019

Functorial Properties of the Reticulation of a Universal Algebra

The {\em 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 bounded distributive lattices, a transfer which is facilitated if we can define a {\em 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 \cite{retic}, where we have used the notion of a prime congruence introduced through the term condition commutator. We characterize morphisms which admit an image through the reticulation and investigate the kinds of varieties that admit reticulation functors; we prove that these include semi--degenerate congruence--distributive varieties with the Compact Intersection Property and semi--degenerate congruence--distributive varieties with congruence intersection terms, as well as generalizations of these, and additional varietal properties ensure that the reticulation functors preserve the injectivity of morphisms. We also study the property of morphisms of having an image through the reticulation in relation to another property, involving the complemented elements of congruence lattices, exemplify the transfer of properties through the reticulation with conditions Going Up, Going Down, Lying Over and the Congruence Boolean Lifting Property, and illustrate the applicability of such a transfer by using it to derive results for certain types of varieties from properties of bounded distributive lattices.

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arXiv2019

On PBZ*-lattices

We continue our investigation of paraorthomodular BZ*-lattices (PBZ*-lattices), started in \cite{GLP1+,PBZ2,rgcmfp,pbzsums,pbz5}. We shed further light on the structure of the subvariety lattice of the variety $\mathbb{PBZL}^{\ast }$ of PBZ*-lattices; in particular, we provide axiomatic bases for some of its members. Further, we show that some distributive subvarieties of $\mathbb{PBZL}^{\ast }$ are term-equivalent to well-known varieties of expanded Kleene lattices or of nonclassical modal algebras. By so doing, we somehow help the reader to locate PBZ*-lattices on the atlas of algebraic structures for nonclassical logics.

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arXiv2019

A Note on Congruences of Infinite Bounded Involution Lattices

We prove that an infinite (bounded) involution lattice and even pseudo--Kleene algebra can have any number of congruences between $2$ and its number of elements or equalling its number of subsets, regardless of whether it has as many ideals as elements or as many ideals as subsets; consequently, the same holds for antiortholattices. Under the Generalized Continuum Hypothesis, this means that an infinite (bounded) involution lattice, pseudo--Kleene algebra or antiortholattice can have any number of congruences between $2$ and its number of subsets, regardless of its number of ideals.

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arXiv2019

PBZ*-Lattices: Structure Theory and Subvarieties

We investigate the structure theory of the variety of \emph{PBZ*-lattices} and some of its proper subvarieties. These lattices with additional structure originate in the foundations of quantum mechanics and can be viewed as a common generalisation of orthomodular lattices and Kleene algebras expanded by an extra unary operation. We lay down the basics of the theories of ideals and of central elements in PBZ*-lattices, we prove some structure theorems, and we explore some connections with the theories of subtractive and binary discriminator varieties.

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arXiv2019

Cancelling Congruences of Lattices, While Keeping Their Numbers of Filters and Ideals

In this paper, we study the congruences, prime filters and prime ideals of horizontal sums of bounded lattices, then, through a construction based on horizontal sums and without enforcing the Continuum Hypothesis, we are modifying an example from \cite{eucard} into a solution to the problem we have proposed in the same article: finding a lattice with the cardinalities of the sets of filters, ideals and congruences pairwise distinct.

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arXiv2018

Ordinal and Horizontal Sums Constructing PBZ*-lattices

PBZ*-lattices are algebraic structures related to quantum logics, which consist of bounded lattices endowed with two kinds of complements, named {\em Kleene} and {\em Brouwer}, such that the Kleene complement satisfies a weakening of the orthomodularity condition and the De Morgan laws, while the Brouwer complement only needs to satisfy the De Morgan laws for the pairs of elements with their Kleene complements. PBZ*-lattices form a variety $\mathbb{PBZL}^{\ast }$, which includes the variety $\mathbb{OML}$ of orthomodular lattices (considered with an extended signature, by letting their two complements coincide) and the variety $V(\mathbb{AOL})$ generated by the class $\mathbb{AOL}$ of antiortholattices. We investigate the congruences of antiortholattices, in particular of those obtained through certain ordinal sums and of those whose Brower complements satisfy the De Morgan laws, infer characterizations for their subdirect irreducibility and prove that even the lattice reducts of antiortholattices are directly irreducible. Since the two complements act the same on the lattice bounds in all PBZ*-lattices, we can define the horizontal sum of any nontrivial PBZ*-lattices, obtained by glueing them at their smallest and at their largest elements; a horizontal sum of two nontrivial PBZ*-lattices is a PBZ*-lattice exactly when at least one of its summands is an orthomodular lattice. We investigate the algebraic structures and the congruence lattices of these horizontal sums, then the varieties they generate. We obtain a relative axiomatization of the variety $V(\mathbb{OML}\boxplus \mathbb{AOL})$ generated by the horizontal sums of nontrivial orthomodular lattices with nontrivial antiortholattices w.r.t. $\mathbb{PBZL}^{\ast }$, as well as a relative axiomatization of the join of varieties $\mathbb{OML}\vee V(\mathbb{AOL})$ w.r.t. $V(\mathbb{OML}\boxplus \mathbb{AOL})$.

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arXiv2018

Some Extremal Values of the Number of Congruences of a Finite Lattice

We study the smallest, as well as the largest numbers of congruences of lattices of an arbitrary finite cardinality $n$. Continuing the work of Freese and Cz\' edli, we prove that the third, fourth and fifth largest numbers of congruences of an $n$--element lattice are: $5\cdot 2^{n-5}$ if $n\geq 5$, respectively $2^{n-3}$ and $7\cdot 2^{n-6}$ if $n\geq 6$. We also determine the structures of the $n$--element lattices having $5\cdot 2^{n-5}$, respectively $2^{n-3}$ congruences, along with the structures of their congruence lattices.

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arXiv2017

On principal congruences and the number of congruences of a lattice with more ideals than filters

Let $λ$ and $κ$ be cardinal numbers such that $κ$ is infinite and either $2\leq λ\leq κ$, or $λ=2^κ$. We prove that there exists a lattice $L$ with exactly $λ$ many congruences, $2^κ$ many ideals, but only $κ$ many filters. Furthermore, if $λ\geq 2$ is an integer of the form $2^m\cdot 3^n$, then we can choose $L$ to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this $L$ is even relatively complemented for $λ=2$. Related to some earlier results of George Grätzer and the first author, we also prove that if $P$ is a bounded ordered set (in other words, a bounded poset) with at least two elements, $G$ is a group, and $κ$ is an infinite cardinal such that $κ\geq |P|$ and $κ\geq |G|$, then there exists a lattice $L$ of cardinality $κ$ such that (i) the principal congruences of $L$ form an ordered set isomorphic to $P$, (ii) the automorphism group of $L$ is isomorphic to $G$, (iii) $L$ has $2^κ$ many ideals, but (iv) $L$ has only $κ$ many filters.

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arXiv2017

Transferring Davey`s Theorem on Annihilators in Bounded Distributive Lattices to Modular Congruence Lattices and Rings

Congruence lattices of semiprime algebras from semi--degenerate congruence--modular varieties fulfill the equivalences from B. A. Davey`s well--known characterization theorem for $m$--Stone bounded distributive lattices, moreover, changing the cardinalities in those equivalent conditions does not change their validity. I prove this by transferring Davey`s Theorem from bounded distributive lattices to such congruence lattices through a certain lattice morphism and using the fact that the codomain of that morphism is a frame. Furthermore, these equivalent conditions are preserved by finite direct products of such algebras, and similar equivalences are fulfilled by the elements of semiprime commutative unitary rings and, dualized, by the elements of complete residuated lattices.

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arXiv2017

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.

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arXiv2016

Going Up and Lying Over in Congruence--modular Algebras

In this paper, we extend properties Going Up and Lying Over from ring theory to the general setting of congruence--modular equational classes, using the notion of prime congruence defined through the commutator. We show how these two properties relate to each other, prove that they are preserved by finite direct products and quotients and provide algebraic and topological characterizations for them. We also point out many kinds of varieties in which these properties always hold.

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arXiv2016

Taking Prime, Maximal and Two--class Congruences Through Morphisms

In this paper we study prime, maximal and two--class congruences from the point of view of the relationships between them in various kinds of universal algebras, as well as their direct and inverse images through morphisms. This research has also produced a set of interesting results concerning the prime and the maximal congruences of several kinds of lattices.

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arXiv2015

Factor Congruence Lifting Property

In previous work, we have introduced and studied a lifting property in congruence--distributive universal algebras which we have defined based on the Boolean congruences of such algebras, and which we have called the Congruence Boolean Lifting Property. In a similar way, a lifting property based on factor congruences can be defined in congruence--distributive algebras; in this paper we introduce this property, which we have called the Factor Congruence Lifting Property, and study it, partly in relation to the Congruence Boolean Lifting Property, and to other lifting properties in particular classes of algebras.

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arXiv2015

Congruence Boolean Lifting Property

We introduce and study the Congruence Boolean Lifting Property (CBLP) for congruence--distributive universal algebras, as well as a property related to CBLP, which we have called $(\star )$. CBLP extends the so--called Boolean Lifting Properties (BLP) from MV--algebras, BL--algebras and residuated lattices, but differs from the BLP when particularized to bounded distributive lattices. Important classes of universal algebras, such as discriminator varieties, fulfill the CBLP. The main results of the present paper include a characterization theorem for congruence--distributive algebras with CBLP and a structure theorem for semilocal arithmetical algebras with CBLP. When we particularize the CBLP to the class of residuated lattices and to that of bounded distributive lattices and we study its relation to other Boolean Lifting Properties for these algebras, interesting results concerning the image of the reticulation functor between these classes are revealed.

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