Bigeneration in complete lattices and principal separation in ordered sets

By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called L-generators resp. C-subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) L-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable L, the complete lattices with a smallest join-dense L-subbasis consisting of L-primes are the L-completions of principally separated posets.     

Algebraically complete lattices

We prove that the class of existentially complete lattices is not an elementary class; thus the theory of lattices does not have a model-companion. Finally we observe that there is a locally finite finitely generic lattice.     

Affine complete distributive lattices

We prove a characterization theorem for affine complete distributive lattices. To do so we introduce the notions of almost principal ideal and almost principal.This research was supported by GA SAV Grant 362/93.     

On complete congruence lattices of join-infinite distributive lattices

In [5] G. Gr?tzer and E. T. Schmidt raised the problem of characterizing the complete congruence lattices of complete lattices satisfying the Join-Infinite Distributive Identity (JID) and the Meet-Infinite Distributive Identity (MID) and proved the theorem: Any complete lattice with more than two elements and with a meet-irreducible zero cannot be represented as the lattice of complete congruence relations of a complete lattice satisfying the (JID) and (MID). In this note we generalize this result by showing that the complete congruence lattice of every complete lattice satisfying (JID) and (MID) is a zero-dimensional complete lattice satisfying (JID). Some consequences are discussed. Received March 6, 2000; accepted in final form September 12, 2000.     

Generalized principal lattices and cubic pencils

Given a cubic pencil, an addition of lines can be defined in order to construct generalized principal lattices. In this paper we show the converse: the lines defining a generalized principal lattice belong to the same cubic pencil, which is unique for degrees ≥ 4. Partially supported by the Spanish Research Grant MTM2006-03388, by Gobierno de Aragón and Fondo Social Europeo.     

On order polynomially complete lattices

One of the problems in the theory of order polynomially complete lattices is the question whether an order polynomially complete lattice is necessarily finite. In this note we give a partial answer to this problem by showing: No unbounded lattice is order polynomially complete. From this we deduce that a polynomially complete lattice cannot be countably infinite.Presented by I. Rosenberg.     

Meet decompositions in complete lattices

In the present paper we shall study infinite meet decompositions of an element of a complete lattice. We give here a generalization of some results of papers [2] and [3].     

Some results on almost principal element lattices

In this paper, we explore locally principal element lattices in terms of primary, semiprimary and prime power elements.     

Characterizing fully principal congruence representable distributive lattices

Motivated by a recent paper of G. Grätzer, a finite distributive lattice D is called fully principal congruence representable if for every subset Q of D containing 0, 1, and the set J(D) of nonzero join-irreducible elements of D, there exists a finite lattice L and an isomorphism from the congruence lattice of L onto D such that Q corresponds to the set of principal congruences of L under this isomorphism. A separate paper of the present author contains a necessary condition of full principal congruence representability: D should be planar with at most one join-reducible coatom. Here we prove that this condition is sufficient. Furthermore, even the automorphism group of L can arbitrarily be stipulated in this case. Also, we generalize a recent result of G. Grätzer on principal congruence representable subsets of a distributive lattice whose top element is join-irreducible by proving that the automorphism group of the lattice we construct can be arbitrary.     

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.     

Remarks on affine complete distributive lattices

We characterise the Priestley spaces corresponding to affine complete bounded distributive lattices. Moreover we prove that the class of affine complete bounded distributive lattices is closed under products and free products. We show that every (not necessarily bounded) distributive lattice can be embedded in an affine complete one and that ℚ ∩ [0, 1] is initial in the class of affine complete lattices.     

Strictly locally order affine complete lattices

We call a latticeL strictly locally order-affine complete if, given a finite subsemilatticeS ofL ⁿ, every functionf: S L which preserves congruences and order, is a polynomial function. The main results are the following: (1) all relatively complemented lattices are strictly locally order-affine complete; (2) a finite modular lattice is strictly locally order-affine complete if and only if it is relatively complemented. These results extend and generalize the earlier results of D. Dorninger [2] and R. Wille [9, 10].     

A complete logic forn-permutable congruence lattices

A family of logical systems, which may be regarded as extending equational logic, is studied. The equationsf=g of equational logic are generalized to congruence equivalence formulasf≡g (modx), wheref andg are terms interpreted as elements of an algebraV of some specified type. and termx is interpreted as a member of ann-permutable lattice of congruences forV. Formal concepts of proof and derivability from systems of hypotheses are developed. These proofs, like those of equational logic. require only finite algebraic processes, without manipulation of logical quantifiers or connectives. The logical systems are shown to be correct and complete: a well-formed statement is derivable from a system of hypotheses if and only if it is valid in all models of these hypotheses.     

On nestedly complete topological vector lattices

A topological vector lattice E is called (σ-)nestedly complete if every downward directed net (resp., decreasing sequence) of order intervals in E whose ‘diameters’ tend to zero has a nonempty intersection. Some characterizations of the (σ-)nested completeness are given, and it is shown that if E is metrizable and nestedly complete, so is each of its quotients E/I, where I is a closed ideal in E. Conversely, if a closed ideal I in E is (sequentially) complete and E/I is (σ-)nestedly complete, so is E. However, the nested completeness is not a three-space property: an example is given where both I and E/I are nestedly complete while E is not. It is also shown that the nested completeness and the related notion of nested density come up quite naturally when extending some positive linear operators. Finally, the nested and other completeness type properties of vector lattices C(S) are investigated.     

Some properties of complete congruence lattices

For a complete lattice C, we consider the problem of establishing when the complete lattice of complete congruence relations on C is a complete sublattice of the complete lattices of join- or meet-complete congruence relations on C. We first argue that this problem is not trivial, and then we show that it admits an affirmative answer whenever C is continuous for the join case and, dually, co-continuous for the meet case. As a consequence, we prove that if C is continuous then each principal filter generated by a continuous complete congruence on C is pseudocomplemented. Received January 6, 1998; accepted in final form July 2, 1998.     

Regular coverings in complete modular lattices

Received September 12, 1994; accepted in final form June 27, 1996.