Variety Of Lattices Research Articles

Remarks on an algebraic semantics for paraconsistent Nelson's logic

In the paper Busaniche and Cignoli (2009) we presented a quasivariety of commutative residuated lattices, called NPc-lattices, that serves as an algebraic semantics for paraconsistent Nelson's logic. In the present paper we show that NPc-lattices form a subvariety of the variety of commutative residuated lattices, we study congruences of NPc-lattices and some subvarieties of NPc-lattices.

Correlated phases of bosons in tilted frustrated lattices

We study the ``tilting'' of Mott insulators of bosons into metastable states. These are described by Hamiltonians acting on resonant subspaces and have rich possibilities for correlated phases with nontrivial entanglement of pseudospin degrees of freedom encoded in the boson density. We extend a previous study [S. Sachdev, K. Sengupta, and S. M. Girvin, Phys. Rev. B 66, 075128 (2002)] of cubic lattices to a variety of lattices and tilt directions in two dimensions: square, decorated square, triangular, and kagome. For certain configurations three-body interactions are necessary to ensure that the energy of the effective resonant subspace is bounded from below. We find quantum phases with Ising density wave order and with superfluidity transverse to the tilt direction, and a quantum liquid state with no broken symmetry. The existence of the quantum liquid state is shown by an exact solution for a particular correlated boson model. We also find cases for which the resonant subspace is described by effective quantum dimer models.

Lee classes for pseudocomplemented semilattices, revisited

We give syntactic and semantic characterisations of the Lee classes of the varieties of pseudocomplemented semilattices and pseudocomplemented distributive lattices. The main result clarifies earlier work of K. B. Lee and of the second author.

Study of the properties of 2D photonic crystal structures as a function of the air-filling fraction and refractive index contrast

We present a study of the primary band gaps of a variety of two-dimensional photonic crystal lattices over the air-filling fraction (AFF) parameter space for different refractive index contrasts. Our results exhibit a variety of unexpected trends and may be useful for better designs of photonic crystal devices.

Congruence openings of additive Green’s relations on a semiring

In a series of papers, Green’s relations on the additive and multiplicative reducts of a semiring proved to be a very useful tool in the study of semirings. However, in the vast majority of cases, Green’s relations are not congruences, and we show that in such cases it is much more convenient to use the congruence openings of Green’s relations, instead of the Green’s relations themselves. By means of these congruence openings we define and study several very interesting operators on the lattices of varieties of semirings and additively idempotent semirings, and, in particular, we establish order embeddings of the lattice of varieties of additively idempotent semirings into the direct products of the lattices of open (resp. closed) varieties with respect to two opening (resp. closure) operators on this lattice that we introduced.

The subregular variety to the variety of special lattices

We recall the basic geometric properties of the full lattice variety, the projective variety parametrizing special lattices over Witt vectors which was introduced in Haboush (2005) [6]. It is an analog in unequal characteristic, of a certain Schubert variety in the affine Grassmannian for SL n , and it is normal and a locally complete intersection (Haboush and Sano, submitted for publication [7], Sano (2004) [15]). In this paper, I prove that the complement of its smooth locus, the subregular variety in it, is also normal and a locally complete intersection. The result is analogous to the geometry of the subregular subvariety of the nilpotent cone.

Demazure resolutions as varieties of lattices with infinitesimal structure

Let k be a field of positive characteristic. We construct, for each dominant cocharacter λ of the standard maximal torus in Sl n , a closed subvariety D ( λ ) of the multigraded Hilbert scheme of an affine space over k , such that the k -valued points of D ( λ ) can be interpreted as lattices in k ( ( z ) ) n endowed with infinitesimal structure. The variety D ( λ ) carries a natural Sl n ( k 〚 z 〛 ) -action. Moreover, for any λ we construct an Sl n ( k 〚 z 〛 ) -equivariant universal homeomorphism from D ( λ ) to a Demazure resolution of the Schubert variety S ( λ ) associated with λ in the affine Grassmannian. Lattices in D ( λ ) have non-trivial infinitesimal structure if and only if they lie over the boundary of the big cell of S ( λ ) .

Energy transfer mechanism for downconversion in the (Pr3+,Yb3+) couple

Downconversion of one visible photon into two infrared photons has been reported for the lanthanide ion couple (${\text{Pr}}^{3+}$, ${\text{Yb}}^{3+}$) in a variety of host lattices. The mechanism responsible for downconversion is controversial and has been reported to be either a two-step energy transfer process (via two first-order transfer steps, the first being cross relaxation) or cooperative energy transfer from ${\text{Pr}}^{3+}$ to two ${\text{Yb}}^{3+}$ ions (a second-order process). Here we report experiments on downconversion for the (${\text{Pr}}^{3+}$, ${\text{Yb}}^{3+}$) in ${\text{LiYF}}_{4}$. Luminescence decay curves of the ${\text{Pr}}^{3+}$ emission are recorded as a function of the ${\text{Yb}}^{3+}$ concentration and analyzed using Monte Carlo simulations for both cooperative energy transfer and energy transfer through cross relaxation. We obtain a good agreement between experiment and simulations for energy transfer by cross relaxation but not for cooperative energy transfer. The observation that cross relaxation is more efficient than cooperative energy transfer is consistent with Judd-Ofelt calculations for the transition probabilities involved in the two energy transfer processes and the lower probability for the second-order cooperative transfer.

Generalized ordinal sums and translations

We extend the lattice embedding of the axiomatic extensions of the positive fragment of intuitionistic logic into the axiomatic extensions of intuitionistic logic to the setting of substructural logics. Our approach is algebraic and uses residuated lattices, the algebraic models for substructural logics. We generalize the notion of the ordinal sum of two residuated lattices and use it to obtain embeddings between subvariety lattices of certain residuated lattice varieties. As a special case we obtain the above mentioned embedding of the subvariety lattice of Brouwerian algebras into an interval of the subvariety lattice of Heyting algebras. We describe the embeddings both in model theoretic terms, focusing on the subdirectly irreducible algebras, and in syntactic terms, by showing how to translate the equational bases of the varieties.

Continuum many top varieties of GMV-algebras and unital ℓ-groups

We prove that there are continuum many top varieties of GMV-algebras and examine the countable lattice of varieties generated by periodic primitive unital l-groups with the unit a rational translation.

Minimal varieties of m-groups

We study the lattice of varieties of m-groups. The main result is a description of the covers of the smallest nontrivial variety of m-groups.

Mapping Extension of an Algebra

A new construction of algebras called a mapping extension of an algebra is here introduced. The construction yields a generalization of some classical constructions such as the nilpotent extension of an algebra, inflation of a semigroup but also the square extension construction introduced recently for idempotent groupoids. The mapping extension construction is defined for algebras of any fixed type, however nullary operation symbols are here not admitted. It is based on the notion of a retraction and some system of mappings. A mapping extension of a given algebra is constructed as a counterimage algebra by a specially defined retraction. Varieties of algebras satisfying an identity φ(x) ≈ x for a term φ not being a variable (such as varieties of lattices, Boolean algebras, groups and rings) are especially interesting because for such a variety [Formula: see text], all mapping extensions by φ of algebras from [Formula: see text] form an equational class. In the last section, combinatorial properties of the mapping extension construction are considered.

Complexity of quasivariety lattices for varieties of differential groupoids

In 1987, Romanowska and Roszkowska found an explicit description of the lattice of varieties of differential groupoids. In 2008, the author proved that the lattice of quasivarieties of differential groupoids is \( \mathcal{Q} \)-universal. In the present article, we find an example of a minimal \( \mathcal{Q} \)-universal variety of differential groupoids.

Commutative integral bounded residuated lattices with an added involution

A symmetric residuated lattice is an algebra A = ( A , ∨ , ∧ , ∗ , → , ∼ , 1 , 0 ) such that ( A , ∨ , ∧ , ∗ , → , 1 , 0 ) is a commutative integral bounded residuated lattice and the equations ∼ ∼ x = x and ∼ ( x ∨ y ) = ∼ x ∧ ∼ y are satisfied. The aim of the paper is to investigate the properties of the unary operation ε defined by the prescription ε x = ∼ x → 0 . We give necessary and sufficient conditions for ε being an interior operator. Since these conditions are rather restrictive (for instance, on a symmetric Heyting algebra ε is an interior operator if and only the equation ( x → 0 ) ∨ ( ( x → 0 ) → 0 ) = 1 is satisfied) we consider when an iteration of ε is an interior operator. In particular we consider the chain of varieties of symmetric residuated lattices such that the n iteration of ε is a boolean interior operator. For instance, we show that these varieties are semisimple. When n = 1 , we obtain the variety of symmetric stonean residuated lattices. We also characterize the subvarieties admitting representations as subdirect products of chains. These results generalize and in many cases also simplify, results existing in the literature.

Ordered groups with a conucleus

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between Π MTL -algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a “quantale like” category of algebras.

Experimental demonstration of painting arbitrary and dynamic potentials for Bose–Einstein condensates

There is a pressing need for robust and straightforward methods to create potentials for trapping Bose–Einstein condensates (BECs) that are simultaneously dynamic, fully arbitrary and sufficiently stable to not heat the ultracold gas. We show here how to accomplish these goals, using a rapidly moving laser beam that ‘paints’ a time-averaged optical dipole potential in which we create BECs in a variety of geometries, including toroids, ring lattices and square lattices. Matter wave interference patterns confirm that the trapped gas is a condensate. As a simple illustration of dynamics, we show that the technique can transform a toroidal condensate into a ring lattice and back into a toroid. The technique is general and should work with any sufficiently polarizable low-energy particles.

Iterative separation in distributive congruence lattices

Abstract In [PLOŠČICA, M.: Separation in distributive congruence lattices, Algebra Universalis 49 (2003), 1–12] we defined separable sets in algebraic lattices and showed a close connection between the types of non-separable sets in congruence lattices of algebras in a finitely generated congruence distributive variety and the structure of subdirectly irreducible algebras in . Now we generalize these results using the concept of separable mappings (defined on some trees) and apply them to some lattice varieties.

CRITICAL POINTS OF PAIRS OF VARIETIES OF ALGEBRAS

For a class [Formula: see text] of algebras, denote by Conc[Formula: see text] the class of all (∨, 0)-semilattices isomorphic to the semilattice ConcA of all compact congruences of A, for some A in [Formula: see text]. For classes [Formula: see text] and [Formula: see text] of algebras, we denote by [Formula: see text] the smallest cardinality of a (∨, 0)-semilattices in Conc[Formula: see text] which is not in Conc[Formula: see text] if it exists, ∞ otherwise. We prove a general theorem, with categorical flavor, that implies that for all finitely generated congruence-distributive varieties [Formula: see text] and [Formula: see text], [Formula: see text] is either finite, or ℵn for some natural number n, or ∞. We also find two finitely generated modular lattice varieties [Formula: see text] and [Formula: see text] such that [Formula: see text], thus answering a question by J. Tůma and F. Wehrung.

On lattice properties of epigroups

The structure of an epigroup with upper semimodular subepigroup lattice is described modulo groups. A special case is distinguished when this lattice belongs to a quasivariety contained in the variety of all modular lattices. It is also shown that certain properties of epigroups are invariant under taking a lattice isomorphic epigroup; this takes place, in particular, for epigroups decomposable into a semilattice of archimedean epigroups and for some types of archimedean epigroups.

THE VARIETY OF RINGS WITH INVOLUTION SATISFYING x7≈ x

We give a complete description of the lattice of varieties of rings with involution satisfying x7≈ x by identity bases. There are 90 such varieties. If we substitute in each ring of such a variety the operations by term operations of the same arity we obtain a so-called class of derived rings. We discuss the case when the class of derived rings belongs to the original variety. In particular, we describe the class of derived rings for the variety of rings generated by the two-element Galois-field.