Structure of lattices of varieties and lattices of quasivarieties: similarity and difference. III

We give representations for lattices of varieties and lattices of quasivarieties in terms of inverse limits of lattices satisfying a number of additional conditions. Specifically, it is proved that, for any locally finite variety (quasivariety) of algebras V, L _v(V)[resp., L _q(V)] is isomorphic to an inverse limit of a family of finite join semidistributive at 0 (resp., finite lower bounded) lattices. A similar statement is shown to hold for lattices of pseudo-quasivarieties. Various applications are offered; in particular, we solve the problem of Lampe on comparing lattices of varieties with lattices of locally finite ones. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 646-666, November-December, 1995.     

The Lattice of Completions of an Ordered Set

For any ordered set P, the join dense completions of P form a complete lattice K(P) with least element O(P), the lattice of order ideals of P, and greatest element M(P), the Dedekind–MacNeille completion P. The lattice K(P) is isomorphic to an ideal of the lattice of all closure operators on the lattice O(P). Thus it inherits some local structural properties which hold in the lattice of closure operators on any complete lattice. In particular, if K(P) is finite, then it is an upper semimodular lattice and an upper bounded homomorphic image of a free lattice, and hence meet semidistributive.     

Lattice properties preserved under free products

We argue for a number of interesting properties of lattices that are preserved under free products, viz., (1) residual finiteness; (2) approximability by (finite) bounded (upper bounded, lower bounded) lattices; (3) boundedness (upper boundedness, lower boundedness). In memory of Victor of Gorbunov Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 66–73, January–February, 2000. Original article submitted March 10, 1999.     

Semidistributive and distributively decomposable rings

A module is said to be distributive if the lattice of all its submodules is distributive. A module is called semidistributive if it is a direct sum of distributive modules. Right semidistributive rings, as well as distributively decomposable rings, are investigated. Translated fromMatematicheskie Zemetki, Vol. 65, No. 2, pp. 307–313, February, 1999.     

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice. Received February 9, 2000; accepted in final form November 23, 2000.     

Generalizing semidistributivity

A latticeL is called congruence normal if it can be generated by doubling of convex sets starting with the one-element lattice. In the special case of intervals, the lattice is called bounded. It has been proven thatL is bounded if and only ifL is congruence normal and semidistributive.In this paper we study the connection between certain classes of convex sets and generalized semidistributive laws. These so-called doubling classes are pseudovarieties which can be described by implications as well as by forbiden substructures. In the end, we examine the structure of the lattice of all doubling classes.     

Cyclic terms for SD⋁ varieties revisited

We present a direct proof showing that every finite algebra generating a congruence join semidistributive variety has a cyclic term.     

Congruence properties of lattices of quasivarieties

The congruence properties close to being lower boundedness in the sense of McKenzie are treated. In particular, the affirmative answer is obtained to a known question as to whether finite lattices of quasivarieties are lower bounded in the case where quasivarieties are congruence-Noetherian and locally finite. Namely, we state that for every congruence-Noetherian or finitely generated locally finite quasivariety K, the lattice L_q(K) possesses the Day-Pudlak-Tuma property. But if a quasivariety is locally finite without the condition of being finitely generated), that lattice satisfies only the Pudlak-Tuma property. Translated fromAlgebra i Logika, Vol. 36, No. 6, pp. 605–620, Noember, 1997.     

The join of varieties with nilpotent intersection

It is proved that the join of finitely based varieties with nilpotent intersection has a finite basis of identities. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 228–242, March–April, 1996.     

The general gould type integral with respect to a multisubmeasure

In two earlier papers [GAVRILUŢ, A.: A Gould type integral with respect to a multisubmeasure, Math. Slovaca 58 (2008), 1–20] and [Gavriluţ, A.: On some properties of the Gould type integral with respect to a multisubmeasure, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 52 (2006), 177–194], we defined and studied a Gould type integral for a real valued, bounded function with respect to a multisubmeasure having finite variation. In this paper, we introduce and study the properties of a Gould type integral in the general setting: the function may be unbounded and the variation of the multisubmeasure may be infinite.     

The canonical join complex for biclosed sets

The canonical join complex of a semidistributive lattice is a simplicial complex whose faces are canonical join representations of elements of the semidistributive lattice. We give a combinatorial classification of the faces of the canonical join complex of the lattice of biclosed sets of segments supported by a tree, as introduced by the third author and McConville. We also use our classification to describe the elements of the shard intersection order of the lattice of biclosed sets. As a consequence, we prove that this shard intersection order is a lattice.     

On the isoperimetric problem in the Heisenberg group ℍ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

It has been recently conjectured that, in the context of the Heisenberg group ℍⁿ endowed with its Carnot–Carathéodory metric and Haar measure, the isoperimetric sets (i.e., minimizers of the ℍ-perimeter among sets of constant Haar measure) could coincide with the solutions to a “restricted” isoperimetric problem within the class of sets having finite perimeter, smooth boundary, and cylindrical symmetry. In this paper, we derive new properties of these restricted isoperimetric sets, which we call Heisenberg bubbles. In particular, we show that their boundary has constant mean ℍ-curvature and, quite surprisingly, that it is foliated by the family of minimal geodesics connecting two special points. In view of a possible strategy for proving that Heisenberg bubbles are actually isoperimetric among the whole class of measurable subsets of ℍⁿ, we turn our attention to the relationship between volume, perimeter, and ε-enlargements. In particular, we prove a Brunn–Minkowski inequality with topological exponent as well as the fact that the ℍ-perimeter of a bounded, open set F⊂ℍⁿ of class C² can be computed via a generalized Minkowski content, defined by means of any bounded set whose horizontal projection is the 2n-dimensional unit disc. Some consequences of these properties are discussed. Mathematics Subject Classification (2000) 28A75, 22E25, 49Q20     

Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks

We study the Cahn-Hilliard equation in a bounded smooth domain without any symmetry assumptions. We prove that for any fixed positive integer K there exist interior K–spike solutions whose peaks have maximal possible distance from the boundary and from one another. This implies that for any bounded and smooth domain there exist interior K–peak solutions. The central ingredient of our analysis is the novel derivation and exploitation of a reduction of the energy to finite dimensions (Lemma 5.5) with variables which are closely related to the location of the peaks. We do not assume nondegeneracy of the points of maximal distance to the boundary but can do with a global condition instead which in many cases is weaker. Received March 5, 1999 / Accepted June 11, 1999     

Dynamical combinatorics and torsion classes

For finite semidistributive lattices the map κ gives a bijection between the sets of completely join-irreducible elements and completely meet-irreducible elements.Here we study the κ-map in the context of torsion classes. It is well-known that the lattice of torsion classes for an artin algebra is semidistributive, but in general it is far from finite. We show the κ-map is well-defined on the set of completely join-irreducible elements, even when the lattice of torsion classes is infinite. We then extend κ to a map on torsion classes which have canonical join representations given by the special torsion classes associated to the minimal extending modules introduced by the first and third authors and A. Carroll in 2019.For hereditary algebras, we show that the extended κ-map on torsion classes is essentially the same as Ringel's ?-map on wide subcategories. Also in the hereditary case, we relate the square of κ to the Auslander-Reiten translation.     

Asymptotics of Weil�CPetersson Geodesics II: Bounded Geometry and Unbounded Entropy

We use ending laminations for Weil–Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil–Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil–Petersson geodesics. As an application, we show theWeil–Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.     

Stability Analysis in Continuous and Discrete Time, using the Cayley Transform

For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman distance. This is extended to a class of Desch–Schappacher perturbations.     

On the semidistributivity of elements in weak congruence lattices of algebras and groups

Weak congruence lattices and semidistributive congruence lattices are both recent topics in universal algebra. This motivates the main result of the present paper, which asserts that a finite group G is a Dedekind group if and only if the diagonal relation is a join-semidistributive element in the lattice of weak congruences of G. A variant in terms of subgroups rather than weak congruences is also given. It is pointed out that no similar result is valid for rings. An open problem and some results on the join-semidistributivity of weak congruence lattices are also included. This research of the second and third authors was partially supported by Serbian Ministry of Science and Environment, Grant No. 144011 and by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, grant ”Lattice methods and applications”.     

Lattices embeddable in subsemigroup lattices. I. Semilattices

V. B. Repnitskii showed that any lattice embeds in some subsemilattice lattice. In his proof, use was made of a result by D. Bredikhin and B. Schein, stating that any lattice embeds in the suborder lattice of a suitable partial order. We present a direct proof of Repnitskii’s result, which is independent of Bredikhin—Schein’s, giving the answer to a question posed by L. N. Shevrin and A. J. Ovsyannikov. We also show that a finite lattice is lower bounded iff it is isomorphic to the lattice of subsemilattices of a finite semilattice that are closed under a distributive quasiorder. Supported by INTAS grant No. 03-51-4110; RF Ministry of Education grant No. E02-1.0-32; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2112.2003.1; a grant from the Russian Science Support Foundation; SB RAS Young Researchers Support project No. 11. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 215–230, March–April, 2006.     

Existence of non-trivial harmonic functions on Cartan–Hadamard manifolds of unbounded curvature

The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem. We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.