Infinite intervals in free lattices

if u/v is any infinite interval in FL(X), the free lattice with generating set X, and then u/v contains a sublattice isomorphic to FL(Y). This result answers a question posed by Ralph Freese and J. B. Nation in their paper Covers in Free Lattices [2] and the principal techniques used in proving this result all come from that paper. The most difficult step in the argument is to show that there are incomparable elements in any infinite interval.     

Uniform binary geometries

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the upper interval structure determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.     

On Product MV-Algebras

In this paper we apply the notion of the product MV-algebra in accordance with the definition given by B. Riean. We investigate the convex embeddability of an MV-algebra into a product MV-algebra. We found sufficient conditions under which any two direct product decompositions of a product MV-algebra have isomorphic refinements.     

Self complementary topologies and preorders

A topology on a set X is self complementary if there is a homeomorphic copy on the same set that is a complement in the lattice of topologies on X. The problem of characterizing finite self complementary topologies leads us to redefine the problem in terms of preorders (i.e. reflexive, transitive relations). A preorder P on a set X is self complementary if there is an isomorphic copy P of P on X that is arc disjoint to P (except for loops) and with the property that PP is strongly connected. We characterize here self complementary finite partial orders and self complementary finite equivalence relations.     

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.     

Posets isomorphic to their extensions

A standard extension for a poset P is a system Q of lower ends (descending subsets) of P containing all principal ideals of P. An isomorphism between P and Q is called recycling if [Y]Q for all YQ. The existence of such an isomorphism has rather restrictive consequences for the system Q in question. For example, if Q contains all lower ends generated by chains then a recycling isomorphism between P and Q forces Q to be precisely the system of all principal ideals. For certain standard extensions Q, it turns out that every isomorphism between P and Q (if there is any) must be recycling. Our results include the well-known fact that a poset cannot be isomorphic to the system of all lower ends, as well as the fact that a poset is isomorphic to the system of all ideals (i.e., directed lower ends) only if every ideal is principal.     

Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F₂, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004.     

Automorphism groups of comparability and covering graphs

Let G be a group and H a subgroup of G. It is shown that there exists a partially ordered set (X, ) such that G is isomorphic to the group of all automorphisms of the comparability graph of (X, ) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (X, ). There also exists a partially ordered set (Y, ) such that G is isomorphic to the group of all automorphisms of the covering graph of (Y, ) and such that under this isomorphism H is mapped onto the group of all order-automorphisms of (Y, ). In this representation X and Y can be taken to be finite if G is finite and of the same cardinality as G if G is infinite.     

Many varieties of lattices with nonsurjective epimorphisms

We use dominions to show that many varieties of lattices have nonsurjective epimorphisms. The variety D of distributive lattices is treated in detail. We show that the dominion in D of a sublattice is the closure of M under relative complementation in L. This dominion is also the largest sublattice of L in which M is epimorphically embedded. In any variety of lattices larger than D, the dominion of M in L is just M. Received May 1, 2001; accepted in final form October 4, 2005.     

Complementing lattice-ordered groups: The projectable case

A complemented l-group G is one in which to each a G there corresponds a b G so that |a||b|=0, while |a||b| is a unit of G. For projectable l-groups this is so precisely when the group possesses a unit.The article introduces the notion of complementation, and the situation for projectable l-groups is analyzed in some detail; in particular, it is shown that any projectable l-group having a projectable complementation in which it is convex has a unique maximal one of this kind.A portion of this research was carried out while this author was a Stauffer Visiting Professor at the University of Kansas during the year 1986–87. He thanks his colleagues in mathematics at that institution for their hospitality.     

A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.     

Identities of cofinal sublattices

If V is a variety of lattices and L a free lattice in V on uncountably many generators, then any cofinal sublattice of L generates all of V. On the other hand, any modular lattice without chains of order-type +1 has a cofinal distributive sublattice. More generally, if a modular lattice L has a distributive sublattice which is cofinal modulo intervals with ACC, this may be enlarged to a cofinal distributive sublattice. Examples are given showing that these existence results are sharp in several ways. Some similar results and questions on existence of cofinal sublattices with DCC are noted.This work was done while the first author was partly supported by NSF contract MCS 82-02632, and the second author by an NSF Graduate Fellowship.     

On context patterns associated with concept lattices

In this paper, we consider the following reconstruction problem: Given two ordered sets (G, ) and (M, ) representing join- and meet-irreducible elements, respectively together with three relationsJ,, onG×M modelling comparability (gm) and maximal noncomparability with respect tog (gm, butgm*) and with respect tom (gm, butgm*). We determine necessary and sufficient conditions for the existence of a finite latticeL and injections :GJ(L) and :MM(L) such that the given order relations and the abstract relations coincide with the one induced by the latticeL.     

The Shifting Lemma and shifting lattice identities

Gumm [6] used the Shifting Lemma with high success in congruence modular varieties. Later, some analogous diagrammatic statements, including the Triangular Scheme from [1] were also investigated. The present paper deals with the purely lattice theoretic underlying reason for the validity of these lemmas. The shift of a lattice identity, a special Horn sentence, is introduced. To any lattice identity and to any variable y occurring in we introduce a Horn sentence S(, y). When S(, y) happens to be equivalent to , we call it a shift of . When has a shift then it gives rise to diagrammatic statements resembling the Shifting Lemma and the Triangular Scheme. Some known lattice identities will be shown to have a shift while some others have no shift.     

When an algebraic frame is regular

It is shown that an algebraic frame L is regular if and only if its compact elements are complemented. More generally, it is shown that each pseudocomplemented element is regular if and only if each , with c compact, is complemented. With a mild assumption on L, each , with c compact, is regular precisely when for any two minimal primes p and q of L. These results are then interpreted in various frames of subobjects of lattice-ordered groups and f-rings.     

The PT-order and the fixed point property

The PT-order, or passing through order, of a poset P is a quasi order defined on P so that ab holds if and only if every maximal chain of P which passes throug a also passes through b. We show that if P is chain complete, then it contains a subset X which has the properties that (i) each element of X is -maximal, (ii) X is a -antichain, and (iii) X is -dominating; we call such a subset a -good subset of P. A -good subset is a retract of P and any two -good subsets are order isomorphic. It is also shown that if P is chain complete, then it has the fixed point property if and only if a -good subset also has the fixed point property. Since a retract of a chain complete poset is also chain complete, the construction may be iterated transfinitely. This leads to the notion of the core of P (a -good subset of itself) which is the transfinite analogue of the core of a finite poset obtained by dismantling.Research partially supported by grants from the National Natural Science Foundation of China and The Natural Science Foundation of Shaanxi province.Research supported by NSERC grant #69-0982.     

Normal subgroups and elementary theories of lattice-ordered groups

We show that any lattice-ordered group (l-group) G can be l-embedded into continuously many l-groups H _i which are pairwise elementarily inequivalent both as groups and as lattices with constant e. Our groups H _i can be distinguished by group-theoretical first-order properties which are induced by lattice-theoretically nice properties of their normal subgroup lattices. Moreover, they can be taken to be 2-transitive automorphism groups A(S _i ) of infinite linearly ordered sets (S _i , ) such that each group A(S _i ) has only inner automorphisms. We also show that any countable l-group G can be l-embedded into a countable l-group H whose normal subgroup lattice is isomorphic to the lattice of all ideals of the countable dense Boolean algebra B.     

Infinite independent sets in distributive lattices

We show that a poset P contains a subset isomorphic to if and only if the poset J(P) consisting of ideals of P contains a subset isomorphic to the power set of κ. If P is a join-semilattice this amounts to the fact that P contains an independent set of size κ. We show that if κ := ω and P is a distributive lattice, then this amounts to the fact that P contains either or as sublattices, where Γ and Δ are two special meet-semilattices already considered by J. D. Lawson, M. Mislove and H. A. Priestley.Dedicated to the memory of Ivan RivalReceived April 22, 2003; accepted in final form July 11, 2004.This revised version was published online in August 2005 with a corrected cover date.     

Intervals in lattices of quasiorders

We investigate the structure of intervals in the lattice of all closed quasiorders on a compact or discrete space. As a first step, we show that if the intervalI has no infinite chains then the underlying space may be assumed to be finite, and in particular,I must be finite, too. We compute several upper bounds for its size in terms of its heighth, which in turn can be computed easily by means of the least and the greatest element ofI. The cover degreec of the interval (i.e. the maximal number of atoms in a subinterval) is less than 4h. Moreover, ifc4(n–1) thenI contains a Boolean subinterval of size 2 ⁿ , and ifI is geometric then it is already a finite Boolean lattice. While every finite distributive lattice is isomorphic to some interval of quasiorders, we show that a nondistributive finite interval of quasiorders is neither a vertical sum nor a horizontal sum of two lattices, with exception of the pentagon. Many further lattices are excluded from the class of intervals of quasiorders by the fact that no join-irreducible element of such an interval can have two incomparable join-irreducible complements. Up to isomorphism, we determine all quasiorder intervals with less than 9 elements and all quasiorder intervals with two complementary atoms or coatoms.     

On the duality between varieties and algebraic theories

It is proved that for any ultrametric space (X, d), the set L(X) of its closed balls is a lattice . It is complete, atomic, tree-like, and real graduated. For any such lattice , the set A(L) of its atoms can be naturally equipped with an ultrametric . These assignments are inverse of one another: where the first equality means an isometry while the second one is a lattice isomorphism. A similar correspondence established for morphisms, shows that there is an isomorphism of categories. The category ULTRAMETR of ultrametric spaces and non-expanding maps is isomorphic to the category LAT* of complete, atomic, tree-like, real graduated lattices and isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism functor and its relations to the categorical operations and action of other functors. Basic properties of a space (such as completeness, spherical completeness, total boundedness, compactness, etc.) are translated into algebraic properties of the corresponding lattice L(X).