Graph isomorphic discrete partially ordered sets and semimodularity

In [RATANAPRASERT, C.—DAVEY, B.: Semimodular lattices with isomorphic graphs, Order 4 (1987), 1–13], the authors found conditions under which an isomorphism of graphs of discrete lattices transfers semimodularity. The main theorem of the present paper generalizes their result for discrete partially ordered sets.     

Hereditarily strong lattices

Lower (or: dually) semimodular lattices behave in certain situations like consistent upper semimodular lattices. A common property of both classes of lattices is that they are hereditarily strong in the sense that every interval is a strong sublattice (cf. Section 2). We relate the properties “hereditarily strong” and “consistent” as well as their duals to lower semimodularity and to modularity (Section 3). Finally we characterize modularity for consistent lower semimodular lattices by means of forbidden sublattices (Section 4).     

Lattices of lattice homomorphisms

IfL andM are lattices, Hom (L, M) denotes the set of homomorphisms ofL intoM with the pointwise partial order.L is calledcatalytic if Hom (L, M) is a lattice for every latticeM. Among other results, it is shown that every retract of a lattice completely freely generated by a partially ordered set is catalytic, and that every catalytic lattice is semidistributive and satisfies Whitman's condition (W). Presented by R. McKenzie     

On a problem involving bounded epimorphisms of lattices

A latticeL satisfies thebounded epimorphism condition if wheneverM is a lattce and ϕ:M →L is a bounded epimorphism, there exists a homomorphismι:L →M such that ιϕ=id _L . we show that the class of finite lattices satisfying the bounded epimorphism condition is properly contained in the class of finite lattices satisfying Whitman's condition (W). We also introduce a property defined for finite lattices that is sufficient to imply the bounded epimorphism condition. Presented by B. Jónsson.     

Congruences in partial abelian semigroups

A partial abelian semigroup (PAS) is a structure , where is a partial binary operation on L with domain , which is commutative and associative (whenever the corresponding elements exist). A class of congruences on partial abelian semigroups are studied such that the corresponding quotient is again a PAS. If M is a subset of a PAS L, we say that are perspective with respect to M, if there is such that and A subset M is called weakly algebraic if perspectivity with respect to M is a congruence. Some conditions are shown under which a congruence coincides with a perspectivity with respect to an appropriate set M. Especially, conditions under which the corresponding quotient is a D-poset are found. It is also shown that every congruence of MV-algebras and orthomodular lattices is given by a perspectivity with respect to an appropriate set M. Received July 17, 1995; accepted in final form September 16, 1996.     

Some Applications of Rademacher Sequences in Banach Lattices

We give several applications of Rademacher sequences in abstract Banach lattices. We characterise those Banach lattices with an atomic dual in terms of weak* sequential convergence. We give an alternative treatment of results of Rosenthal, generalising a classical result of Pitt, on the compactness of operators from L^p into L^q. Finally we generalise earlier work of ours by showing that, amongst Banach lattices F with an order continuous norm, those having the property that the linear span of the positive compact operators fromE into F is complete under the regular norm for all Banach lattices E are precisely the atomic lattices.     

On Projecting Symmetries by Unbranched Regular Coverings of Riemann Surfaces

A symmetry of a Riemann surface X of genus g is an antiholomorphic involution σ of X. It is a classical result of Harnack that the set of fixed points of σ consists of k closed Jordan curves, called ovals, for some k, 0 ≤ k ≤ g + 1; when k = g or k = g+1 we say, following Natanzon [8], that σ is an (M – 1)- or an M-symmetry, respectively. Given a Riemann surface X with an M-symmetry, a Riemann surface Y and a regular covering p: X → Y, we prove that Y admits either an M- or an (M – 1)-symmetry and whenever p is unbranched we describe the groups of covering transformations of p. In the case that X is hyperelliptic we calculate as well the number of unbranched regular coverings p: X → Y in which X has an M-symmetry. The first two authors are supported by MTM2005-01637, the third by SAB2005-0049.     

Primes,irreducibles and extremal lattices

This paper studies certain types of join and meet-irreducibles called coprimes and primes. These elements can be used to characterize certain types of lattices. For example, a lattice is distributive if and only if every join-irreducible is coprime. Similarly, a lattice is meet-pseudocomplemented if and only if each atom is coprime. Furthermore, these elements naturally decompose lattices into sublattices so that often properties of the original lattice can be deduced from properties of the sublattice. Not every lattice has primes and coprimes. This paper shows that lattices which are long enough must have primes and coprimes and that these elements and the resulting decompositions can be used to study such lattices.The length of every finite lattice is bounded above by the minimum of the number of meet-irreducibles (meet-rank) and the number of join-irreducibles (join-rank) that it has. This paper studies lattices for which length=join-rank or length=meet-rank. These are called p-extremal lattices and they have interesting decompositions and properties. For example, ranked, p-extremal lattices are either lower locally distributive (join-rank=length), upper locally distributive (meet-rank=length) or distributive (join-rank=meet-rank=length). In the absence of the Jordan-Dedekind chain condition, p-extremal lattices still have many interesting properties. Of special interest are the lattices that satisfy both equalities. Such lattices are called extremal; this class includes distributive lattices and the associativity lattices of Tamari. Even though they have interesting decompositions, extremal lattices cannot be characterized algebraically since any finite lattice can be embedded as a subinterval into an extremal lattice. This paper shows how prime and coprime elements, and the poset of irreducibles can be used to analyze p-extremal and other types of lattices.The results presented in this paper are used to deduce many key properties of the Tamari lattices. These lattices behave much like distributive lattices even though they violate the Jordan-Dedekind chain condition very strongly having maximal chains that vary in length from N-1 to N(N-1)/2 where N is a parameter used in the construction of these lattices.     

Continuity of Actions of Groups and Semigroups on Banach Spaces

It is shown that if a locally compact group acts isometricallyon a Banach space X leaving a closed subspace M invariant, andif the induced actions on M and X/M are strongly continuous,then the action on X is strongly continuous. Since this maybe of interest for one-parameter semigroups, similar resultsare proved for actions of suitable topological semigroups. Othergeneralizations are given for (suitable) non-isometric actions,non-locally compact groups, and non-Banach spaces; corollariesconcerning 1-cocycles and uniformly continuous actions are given.     

Sublattices of complete lattices with continuity conditions

Various embedding problems of lattices into complete lattices are solved. We prove that for any join-semilattice S with the minimal join-cover refinement property, the ideal lattice Id S of S is both algebraic and dually algebraic. Furthermore, if there are no infinite D-sequences in J(S), then Id S can be embedded into a direct product of finite lower bounded lattices. We also find a system of infinitary identities that characterize sublattices of complete, lower continuous, and join-semidistributive lattices. These conditions are satisfied by any (not necessarily finitely generated) lower bounded lattice and by any locally finite, join-semidistributive lattice. Furthermore, they imply M. Erné’s dual staircase distributivity.On the other hand, we prove that the subspace lattice of any infinite-dimensional vector space cannot be embedded into any ℵ₀-complete, ℵ₀-upper continuous, and ℵ₀-lower continuous lattice. A similar result holds for the lattice of all order-convex subsets of any infinite chain.Dedicated to the memory of Ivan RivalReceived April 4, 2003; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

On Strongly Essential Submodules

The submodules with the property of the title (N ? M is strongly essential in M if ∏ _I N is essential in ∏ _I M for any index set I) are introduced and fully investigated.

It is shown that for each submodule N of M there exists a subset T ? M such that N + T is strongly essential submodule of M and (N:T) = Ann(T), T  ∩  N = 0. Basic properties of these objects and several examples are given and the counterparts of the related concepts to essential submodules are also introduced and studied. It is shown that each maximal left ideal of a left fully bounded ring is either a summand or strongly essential. Rings over which no module has a proper strongly essential submodule are characterized. It is also shown that the left Loewy rings are the only rings over which the essential submodules and strongly essential submodules of any left module coincide. Finally, a new characterization of left FBN rings is observed.     

Lattices of Intermediate Subfactors

LetNbe an irreducible subfactor of a typeII₁factorM. If the Jones index [M:N] is finite, then the set at(NM) of the intermediate subfactors for the inclusionNMforms afinitelattice. The commuting and co-commuting square conditions for intermediate subfactors are related to the modular identity in the lattice at(NN). In particular, simplicity of a finite groupGis characterized in terms of commuting square conditions of intermediate subfactors forNM=NG. We investigate the question of which finite lattices can be realized as intermediate subfactor lattices.     

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.     

On regularity of sup-preserving maps: generalizing Zareckiĭ’s theorem

A sup-preserving map f between complete lattices L and M is regular if there exists a sup-preserving map g from M to L such that fgf=f. In the class of completely distributive lattices, this paper demonstrates a necessary and sufficient condition for f to be regular. When L=M is a power set, our theorem reduces to the well known Zareckiĭ’s theorem which characterizes regular elements in the semigroup of all binary relations on a set. Another application of our result is a generalization of Zareckiĭ’s theorem for quantale-valued relations.     

Properties (u) and (V*) of Pelczynski in symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V ^*) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.     

Direct Sum of Distributive Lattices on the Perfect Matchings of a Plane Bipartite Graph

Let G be a plane bipartite graph and M(G){\cal M}(G) the set of perfect matchings of G. A property that the Z-transformation digraph of perfect matchings of G is acyclic implies a partially ordered relation on M(G){\cal M}(G). It was shown that M(G){\cal M}(G) is a distributive lattice if G is (weakly) elementary. Based on the unit decomposition of alternating cycle systems, in this article we show that the poset M(G){\cal M}(G) is direct sum of finite distributive lattices if G is non-weakly elementary; Further, if G is elementary, then the height of distributive lattice M(G){\cal M}(G) equals the diameter of Z-transformation graph, and both quantities have a sharp upper bound é\fracn(n+2)4ù\lceil\frac{n(n+2)}{4}\rceil, where n denotes the number of inner faces of G.