Recursively inconsistent algorithmic problems on 1-constructivizable relatively complemented distributive lattices

Let be a 1-constructivizable relatively complemented distributive lattice with an infinite set of atoms. We construct a computable family of algorithmic ₂ ^r -problems on each of which is decidable w.r.t. an appropriate constructivization of but any two of which cannot be simultaneously decidable. Thus we solve the problems on a spectra and relations between algorithmic dimensions for the lattices concerned. A criterion stating the existence of a least element in the algebraic reducibility structure is found and algebraic conditions are specified for solving the problem of effective choice of constructivizations, given specifications of problems in .Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 667-680, November-December, 1995.Supported by RFFR grant No. 93-011-16014 and by ISF grant NQ 6000.     

Orthomodular lattices and permutable congruences

If a variety of ortholattices is congruence-permutable, then we prove that it is a variety of orthomodular lattices.Dedicated to the memory of Ivan RivalReceived October 7, 2003; accepted in final form July 12, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Biordered sets and complemented modular lattices

In this paper we give a structure theorem for the biordered set of a strongly regular Baer semigroup. As a result we shall be able to construct the biordered set of the multiplicative semigroup of a regular ring in terms of the complemented modular lattice which is coordinatized by this regular ring. The author's research was done while he was a visiting professor at the University of Nebraska, supported by a Fulbright-Hays Award.     

Finite lattices and congruences. A survey

In the early forties, R.P. Dilworth proved his famous result: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In one of our early papers, we presented the first published proof of this result; in fact we proved: Every finite distributive lattice D can be represented as the congruence lattice of a finite sectionally complemented lattice L.We have been publishing papers on this topic for 45 years. In this survey paper, we are going to review some of our results and a host of related results by others: Making L nice.If being nice is an algebraic property such as being semimodular or sectionally complemented, then we have tried in many instances to prove a stronger form of these results by verifying that every finite lattice has a congruence-preserving extension that is nice. We shall discuss some of the techniques we use to construct nice lattices and congruence-preserving extensions.We shall describe some results on the spectrum of a congruence of a finite sectionally complemented lattice, measuring the sizes of the congruence classes. It turns out that with very few restrictions, these can be as bad as we wish.We shall also review some results on simultaneous representation of two distributive lattices. We conclude with the magic wand construction, which holds out the promise of obtaining results that go beyond what can be achieved with the older techniques.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived November 26, 2002; accepted in final form June 18, 2004.     

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. I

Summary In 1944, R.P. Dilworth proved (unpublished) that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L. In 1960, G. Grätzer and E. T. Schmidt improved this result by constructing a finite sectionally complemented lattice L whose congruence lattice represents D. In L, sectional complements do not have to be unique. The one sectional complement constructed by G. Grätzer and E. T. Schmidt in 1960, we shall call the 1960 sectional complement. This paper examines it in detail. The main result is an algebraic characterization of the 1960 sectional complement.     

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.     

Representing homomorphisms of distributive lattices as restrictions of congruences of rectangular lattices

Let be a {0, 1}-homomorphism of a finite distributive lattice D into the congruence lattice Con L of a rectangular (whence finite, planar, and semimodular) lattice L. We prove that L is a filter of an appropriate rectangular lattice K such that ConK is isomorphic with D and is represented by the restriction map from Con K to Con L. The particular case where is an embedding was proved by E.T. Schmidt. Our result implies that each {0, 1}-lattice homomorphism between two finite distributive lattices can be represented by the restriction of congruences of an appropriate rectangular lattice to a rectangular filter.     

Semigroups whose lattices of right congruences are semiatomic

The main theorems presented here are characterizations of a semigroup with a left identity whose lattice of right congruences is semiatomic. These theorems are preceded by a number of results on minimal right congruences.     

Elementary equivalence and relatively free products of lattices

It is shown that if is a non-trivial variety of lattices, then there existA, B, C ∈ such thatB≡C but notA*B≡A*C. Except in the case when is the variety of all distributive lattices,A can be taken to consist of just one element. For the varietyD of all distributive lattices, it is shown that for anyB, C and any finiteA, B≡C if and only ifA*B≡A*C. The work of the first author was supported in part by NSF Grant GP-29129, and the work of the second author by a grant from the N.R.C. of Canada. Presented by C. C. Chang.     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.