Notes on sectionally complemented lattices. II

Summary G. Grätzer and H. Lakser proved in 1986 that for the finite distributive lattices D and E, with |D| > 1, and for the {0, 1}-homomorphism φ of D into E, there exists a finite lattice L and an ideal I of L such that D ≡ Con L, E ≡ Con I, and φ is represented by the restriction map. In their recent survey of finite congruence lattices, G. Grätzer and E. T. Schmidt ask whether this result can be improved by requiring that L be sectionally complemented. In this note, we provide an affirmative answer. The key to the solution is to generalize the 1960 sectional complement (see Part I) from finite orders to finite preorders.     

Notes on sectionally complemented lattices. III

Summary In a recent survey article, G. Grätzer and E. T. Schmidt raise the problem when is the ideal lattice of a sectionally complemented chopped lattice sectionally complemented. The only general result is a 1999 lemma of theirs, stating that if the finite chopped lattice is the union of two ideals that intersect in a two-element ideal U, then the ideal lattice of M is sectionally complemented. In this paper, we present examples showing that in many ways their result is optimal. A typical result is the following: For any finite sectionally complemented lattice U with more than two elements, there exists a finite sectionally complemented chopped lattice M that is (i) the union of two ideals intersecting in the ideal U; (ii) the ideal lattice of M is not sectionally complemented.     

Notes on sectionally complemented lattices. IV How far does the Atom Lemma go?

There are two results in the literature that prove that the ideal lattice of a finite, sectionally complemented, chopped lattice is again sectionally complemented. The first is in the 1962 paper of G. Grätzer and E. T. Schmidt, where the ideal lattice is viewed as a closure space to prove that it is sectionally complemented; we call the sectional complement constructed then the 1960 sectional complement. The second is the Atom Lemma from a 1999 paper of the same authors that states that if a finite, sectionally complemented, chopped lattice is made up of two lattices overlapping in an atom and a zero, then the ideal lattice is sectionally complemented. In this paper, we show that the method of proving the Atom Lemma also applies to the 1962 result. In fact, we get a stronger statement, in that we get many sectional complements and they are rather close to the componentwise sectional complement.     

Congruence-preserving extensions of finite lattices to sectionally complemented lattices

In 1962, the authors proved that every finite distributive lattice can be represented as the congruence lattice of a finite sectionally complemented lattice. In 1992, M. Tischendorf verified that every finite lattice has a congruence-preserving extension to an atomistic lattice. In this paper, we bring these two results together. We prove that every finite lattice has a congruence-preserving extension to a finite sectionally complemented lattice.

     

On the endomorphism monoids of (uniquely) complemented lattices

Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.

     

Two extension theorems. Modular functions on complemented lattices

We prove an extension theorem for modular functions on arbitrary lattices and an extension theorem for measures on orthomodular lattices. The first is used to obtain a representation of modular vector-valued functions defined on complemented lattices by measures on Boolean algebras. With the aid of this representation theorem we transfer control measure theorems, Vitali-Hahn-Saks and Nikodým theorems and the Liapunoff theorem about the range of measures to the setting of modular functions on complemented lattices.     

Layer-projective lattices. I

The class of layer-projective lattices is singled out. For example, it contains the lattices of subgroups of finite Abelianp-groups, finite modular lattices of centralizers that are indecomposable into a finite sum, and lattices of subspaces of a finite-dimensional linear space over a finite field that are invariant with respect to a linear operator with zero eigenvalues. In the class of layer-projective lattices, the notion of type (of a lattice) is naturally introduced and the isomorphism problem for lattices of the same type is posed. This problem is positively solved for some special types of layer-projective lattices. The main method is the layer-wise lifting of the coordinates. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 170–182, February, 1998.     

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.     

Gamma invariants for dense lattices

A subclass of the class of dense lattices, called strongly dense lattices, is defined and its properties investigated. In particular, an invariant of strongly dense lattices is defined and shown to take all possible values. Connections to module theory are explored, including some constructions of von Neumann regular rings with strongly dense lattices of two-sided ideals. Received September 16, 1997; accepted in final form July 22, 1998.     

Commutators and decompositions of orthomodular lattices

We show that in any complete OML (orthomodular lattice) there exists a commutatorc such that [0,c ] is a Boolean algebra. This fact allows us to prove that a complete OML satisfying the relative centre property is isomorphic to a direct product [0,a] × [0,a ] wherea is a join of two commutators, [0,a] is an OML without Boolean quotient and [0,a ] is a Boolean algebra. The proof uses a new characterization of the relative centre property in complete OMLs. In a final section, we specify the previous direct decomposition in the more particular case of locally modular OMLs.     

On lattices embeddable into subsemigroup lattices. V. Trees

We prove that the class of finite lattices embeddable into the subsemilattice lattices of semilattices which are (n-ary) trees can be axiomatized by identities within the class of finite lattices, whence it forms a pseudovariety.     

On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitski? showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitski? result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.     

Lattices embeddable in subsemigroup lattices. II. Cancellative semigroups

Repnitskii proved that any lattice embeds in a subsemigroup lattice of some commutative, cancellative, idempotent free semigroup with unique roots. In that proof, use is made of a result by Bredikhin and Schein stating that any lattice embeds in a suborder lattice of suitable partial order. Here, we present a direct proof of Repnitskii’s result which is independent of Bredikhin-Schein’s, thus giving the answer to the question posed by Shevrin and Ovsyannikov. 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. 4, pp. 436–446, July–August, 2006.     

Three-pencil lattices on triangulations

In this paper, three-pencil lattices on triangulations are studied. The explicit representation of a lattice, based upon barycentric coordinates, enables us to construct lattice points in a simple and numerically stable way. Further, this representation carries over to triangulations in a natural way. The construction is based upon group action of S ₃ on triangle vertices, and it is shown that the number of degrees of freedom is equal to the number of vertices of the triangulation.      

Efficient algorithms on distributive lattices

We present several efficient algorithms on distributive lattices. They are based on a compact representation of the lattice, called the ideal tree. This allows us to exploit regularities in the structure of distributive lattices. The algorithms include a linear-time algorithm to reconstruct the covering graph of a distributive lattice from its ideal tree, a linear-time incremental algorithm for building the ideal lattice of a poset and a new incremental algorithm for listing the ideals of a poset in a combinatorial Gray code manner (in an code.)     

Notes on locally order-polynomially complete lattices

No Abstract. .Dedicated to the memory of Ivan RivalReceived December 1, 2002; accepted in final form June 16, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Notes on planar semimodular lattices. VI. On the structure theorem of planar semimodular lattices

In a recent paper, G. Czédli and E. T. Schmidt present a structure theorem for planar semimodular lattices. In this note, we present an alternative proof.     

Varieties of semigroups with non-trivial identities on subsemigroup lattices

The present notice is devoted to the characterization up to the group case of varieties of semigroups whose subsemigroup lattices satisfy non-trivial identities. Received November 2, 1999; accepted in final form April 23, 2000.