Claudia Muresan

University of Bucharest | Unibuc · Faculty of Mathematics and Computer Science

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 for...

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 con...

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...

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. Furthermore, when they have at most as many congruences as elements, t...

We continue our investigation of paraorthomodular BZ*-lattices (PBZ\(^{*}\)–lattices), started in Giuntini et al. (2016, 2017, 2018, 2020), Mureşan (2019). We shed further light on the structure of the subvariety lattice of the variety \(\mathbb {PBZL}^{\mathbb {*}}\) of PBZ\(^{*}\)–lattices; in particular, we provide axiomatic bases for some of it...

PBZ\(^{*}\)–lattices are lattices with additional operations that arise in the context of the unsharp approach to quantum logic. They include orthomodular lattices and Kleene algebras with an extra unary operation. We study in the framework of PBZ\(^{*}\)–lattices two constructions—the ordinal sum construction and the horizontal sum construction—th...

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.

We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czédli’s recent work, we prove that the third, fourth and fifth largest numbers of congruences of an n–element lattice are: 5 ⋅ 2n− 5 if n ≥ 5, 2n− 3 and 7 ⋅ 2n− 6 if n ≥ 6, respec...

We investigate the structure theory of the variety of 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...

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 dist...

We study the existence of nontrivial (dual) weak complementations and the representable (dual) weak complementations, along with the lattice congruences that preserve them, in different constructions of bounded lattices.

We continue the study of the reticulation of a universal algebra initiated in \cite{retic}, characterizing morphisms which admit an image through the reticulation and investigating the kinds of varieties that admit reticulation functors.

PBZ*--lattices are bounded lattice--ordered structures arising in the study of quantum logics, which include orthomodular lattices, as well as antiortholattices. Antiortholattices turn out not only to be directly irreducible, but also to have directly irreducible lattice reducts. Their presence in varieties of PBZ*--lattices determines the lengths...

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 Morg...

We prove that, under the Generalized Continuum Hypothesis, an infinite bounded involution lattice can have any number of (involution preserving lattice) congruences between $2$ and its number of subsets, regardless of its number of ideals.

The reticulation of an algebra A is a bounded distributive lattice 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 C in the case when the...

We investigate the structure theory of some subvarieties of the variety of \emph{PBZ*-lattices}. 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 explore the connections between...

In this paper, we are transferring Davey's characterization of $m$-Stone bounded distributive lattices to commutator lattices, and obtain related results on prime, radical, complemented and compact elements, annihilators and congruences of commutator lattices.

In this paper we determine the largest and second largest possible numbers of congruences of finite pseudo-Kleene algebras, along with the structures of the pseudo-Kleene algebras with these numbers of congruences, which also show the largest and second largest possible numbers of congruences of finite antiortholattices with $0$ meet-irreducible, a...

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 $...

Modifying a lattice through horizontal sums, as to cancel its congruences, without significant effect on its number of filters or ideals, and using such a construction to obtain a lattice with the cardinalities of the sets of congruences, filters and ideals pairwise distinct, more precisely an infinite simple lattice with as many filters as element...

Let $\lambda$ and $\kappa$ be cardinal numbers such that $\kappa$ is infinite and either $2\leq \lambda\leq \kappa$, or $\lambda=2^\kappa$. We prove that there exists a lattice $L$ with exactly $\lambda$ many congruences, $2^\kappa$ many ideals, but only $\kappa$ many filters. Furthermore, if $\lambda\geq 2$ is an integer of the form $2^m\cdot 3^n$...

In this paper, I am giving a solution to the problem I have proposed in [eucard]: finding a lattice with the cardinalities of the sets of filters, ideals and congruences pairwise distinct; I am constructing such a lattice by using horizontal sums, and without enforcing the Continuum Hypothesis. This research has also produced a set of results on (p...

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...

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...

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 al...

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...

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--d...

We define lifting properties for universal algebras, which we study in this general context and then particularize to various such properties in certain classes of algebras. Next we focus on residuated lattices, in which we investigate lifting properties for Boolean and idempotent elements modulo arbitrary, as well as specific kinds of filters. We...

In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congruences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting properti...

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 particulari...

In this paper, we introduce the lifting properties for the Boolean elements of bounded distributive lattices with respect to the congru-ences, filters and ideals, we establish how they relate to each other and to significant algebraic properties, and we determine important classes of bounded distributive lattices which satisfy these lifting propert...

In this paper we define the Boolean Lifting Property (BLP) for residuated lattices to be the property that all Boolean elements can be lifted modulo every filter, and study residuated lattices with BLP. Boolean algebras, chains, local and hyperarchimedean residuated lattices have BLP. BLP behaves interestingly in direct products and involutive resi...

In this paper we present some applications of the reticulation of a residuated lattice, in the form of a transfer of properties between the category of bounded distributive lattices and that of residuated lattices through the reticulation functor. The results we are presenting are related to co-Stone algebras; among other applications, we transfer...

Bosbach states represent a way of probabilisticly evaluating the formulas from various (commutative or non-commutative) many-valued logics. They are defined on the algebras corresponding to these logics with values in $[0,1]$. Starting from the observation that in the definition of Bosbach states there intervenes the standard MV-algebra structure o...

In this paper we define, inspired by ring theory, the class of maximal residuated lattices with lifting Boolean center and prove a structure theorem for them: any maximal residuated lattice with lifting Boolean center is isomorphic to a finite direct product of local residuated lattices.

In this paper we present some applications of the reticulation of a residuated lattice, in the form of a transfer of properties between the category of bounded distributive lattices and that of residuated lattices through the reticulation functor. The results we are presenting are related to co-Stone algebras; among other applications, we transfer...

In this article we prove a set of preservation properties of the reticulation functor for residuated lattices (for instance preservation of subalgebras, finite direct products, inductive limits, Boolean powers) and we transfer certain properties between bounded distributive lattices and residuated lattices through the reticulation, focusing on Ston...

In this paper we define the reticulation of a residuated lattice, prove that it has "good properties", present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.

In this paper we study the dense elements and the radical of a residuated lattice, residuated lattices with lifting Boolean center, simple, local, semilocal and quasi-local residuated lattices. BL-algebras have lifting Boolean center; moreover, Glivenko residuated lattices which fulfill a certain equation (that is satisfied by BL-algebras) have lif...

In this paper we define the reticulation of a residuated lattice, prove that it has "good properties", present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.

In this paper we define the reticulation of a residuated lattice, prove that it has "good properties", present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.