LLAMPS IN SLIM RECTANGULAR PLANAR SEMIMODULARLATTICES
G ´ABOR CZ´EDLI
Dedicated to L´aszl´o K´erchy on his seventieth birthday
Abstract.
A planar (upper) semimodular lattice L is slim if the five-elementnondistributive modular lattice M does not occur among its sublattices. (Pla-nar lattices are finite by definition.) Slim rectangular lattices as particularslim planar semimodular lattices were defined by G. Gr¨atzer and E. Knappin 2007. In 2009, they also proved that the congruence lattices of slim planarsemimodular lattices with at least three elements are the same as those of slimrectangular lattices. In order to provide an effective tool for studying thesecongruence lattices, we introduce the concept of lamps of slim rectangular lat-tices and prove several of their properties. Lamps and several tools based onthem allow us to prove in a new and easy way that the congruence lattices ofslim planar semimodular lattices satisfy the two previously known properties.Also, we use lamps to prove that these congruence lattices satisfy four newproperties including the two-pendant four-crown property and the forbiddenmarriage property . Note on the dedication.
Professor
L´aszl´o K´erchy is the previous editor-in-chiefof Acta Sci. Math. (Szeged). I have been knowing him since 1967, when I enrolledin a high school where he was a second-year student with widely acknowledgedmathematical talent. His influence had played a role that I became a student at theBolyai (Mathematical) Institute in Szeged. Furthermore, he helped me to becomeone of his roommates in Lor´and E¨otv¨os University Dormitory in 1972. Leadingby his example, I could become a member of the Bolyai Institute right after mygraduation. With my gratitude, I dedicate this paper to his seventieth birthday.1.
Introduction
The theory of planar semimodular lattices has been an intensively studied part oflattice theory since Gr¨atzer and Knapp’s pioneering [21]. The key role in the theoryof these lattices is played by slim planar semimodular lattices; their definition ispostponed to Section 2. The importance of slim planar semimodular lattices issurveyed, for example, in Cz´edli and Kurusa [10], Cz´edli and Gr¨atzer [7], Cz´edliand Schmidt [12], and Gr¨atzer and Nation [23].
Slim rectangular lattices , to bedefined later, were introduced in Gr¨atzer and Knapp [22] as particular slim planar
Date : February 28, 2021 Hint: check the author’s website for possible updates.2020
Mathematics Subject Classification.
Key words and phrases.
Rectangular lattice, slim semimodular lattice, multi-fork extension,lattice diagram, edge of normal slope, precipitous edge, lattice congruence, two-pendant four-crown property, lamp, congruence lattice, forbidden marriage property.This research was supported by the National Research, Development and Innovation Fund ofHungary under funding scheme K 134851. a r X i v : . [ m a t h . R A ] F e b G. CZ´EDLI semimodular lattices and they will play a crucial role in our proofs. The studyof congruence lattices
Con L of slim planar semimodular lattices L goes back toGr¨atzer and Knapp [22]. These congruence lattices are finite distributive latticesand, in addition, we know from Cz´edli [4] and Gr¨atzer [16], [17], and [20] that theyhave special properties. Target.
One of our targets is to develop effective tools to derive the above-mentionedspecial properties in a new and easy way and to present four new properties. To doso, we are going to define the lamps of a slim rectangular lattice L so that the setof lamps becomes a poset (partially ordered set) isomorphic to the poset J(Con L )of join-irreducible congruences of L . It remains a problem whether the propertiesrecalled or proved in the present paper and in Cz´edli and Gr¨atzer [8] characterizethe congruence lattices of slim planar semimodular lattices. Outline.
For comparison with our lamps, the rest of the present section mentionsthe known ways to describe J(Con L ) for a finite lattice L . In Section 2, we recallthe concept of slim planar semimodular lattices, that of slim rectangular lattices,and that of their C -diagrams. Also in Section 2, we introduce the concept oflamps of these lattices and prove our (Main) Lemma 2.11. This lemma providesthe main tool for “illuminating” the congruence lattices of slim planar semimodularlattices. In Section 3, further tools are given and several consequences of (the Main)Lemma 2.11 are proved. In particular, this section proves that the congruence lat-tices of slim planar semimodular lattices satisfy both previously known properties,see Corollaries 3.3 and 3.6, and two new properties, see Corollaries 3.4 and 3.5.Section 4 defines the two-pendant four-crown property and the forbidden marriageproperty, and proves that the congruence lattices of slim planar semimodular lat-tices satisfy these two properties, too; see Theorem 4.3. Comparison with earlier approaches to J (Con L ) . Let L be a finite lattice; its congruence lattice is denoted by Con L . Since Con L is distributive, it is determinedby the poset J(Con L ) = (cid:104) J(Con L ); ≤(cid:105) of its nonzero join-irreducible elements .There are three known ways to describe this poset.First, one can use the join dependency relation defined on J ( L ); see Lemma 2.36of the monograph Freese, Jeˇzek, and Nation [14], where this relation is attributedto Day [13].Second, Gr¨atzer [17] takes (and well describes) the prime-perspectivity relationon the set of prime intervals of L . His description becomes more powerful if L happens to be a slim planar semimodular lattice: for such a lattice, Gr¨atzer’sSwing Lemma applies, see [18] and see also Cz´edli, Gr¨atzer, and Lakser [9] andCz´edli and Makay [11].Third, but only for a slim rectangular lattice L , Cz´edli [2] defined a relation onthe set of trajectories of L while Gr¨atzer [18] defined an analogous relation on theset of prime intervals. Although it happened in a different way, Theorem 7.3(ii) ofCz´edli [2] indicates that the dual of the approach based on join dependency relationcould also have been used to derive a more or less similar description of J(Con L ).A relation ρ ⊆ X on a set X is a quasiorder (also called preorder ) if it isreflexive and transitive. Each of the three approaches mentioned above definesonly a quasiorder on a set X in the first step; this set consists of join-irreducibleelements, prime intervals, or trajectories, respectively. In the next step, we have toform the quotient set X/ ( ρ ∩ ρ − ) and equip it with the quotient relation ρ/ ( ρ ∩ ρ − ) AMPS IN SLIM RECTANGULAR LATTICES 3 to obtain (cid:104)
J(Con L ); ≤(cid:105) up to isomorphism. For a slim rectangular lattice L , ourlamps provide a more efficient description of J(Con L ) since we do not have to forma quotient set. 2. From diagrams to lamps
By definition, planar lattices are finite. A slim planar semimodular lattice is aplanar (upper) semimodular lattice L such that one of the following three conditionsholds:(i) M , the five-element nondistributive modular lattice, is not a sublattice of L ,(ii) M is not a cover-preserving sublattice of L ,(iii) J( L ), the set of nonzero join-irreducible elements of L , is the union of twochains;see Gr¨atzer and Knapp [21] and Cz´edli and Schmidt [12], or the book chapter Cz´edliand Gr¨atzer [7] for the equivalence of these three conditions for planar semimodularlattices (but not for other lattices.)Let L be a slim planar semimodular lattice; we always assume that a planardiagram of L is fixed. The left boundary chain and the right boundary chain of L are denoted by C left ( L ) and C right ( L ), respectively. Here and at several otherconcepts occurring later, we heavily rely on the fact that the diagram of L is fixed;indeed, C left ( L ) and C right ( L ) depend on the diagram, not only on L .Following Gr¨atzer and Knapp [22], a slim planar semimodular lattice is calleda slim rectangular lattice if | L | ≥
4, C left ( L ) has exactly one doubly irreducibleelement, lc( L ), C right ( L ) has exactly one doubly irreducible element, rc( L ), andthese two doubly irreducibly elements are complementary, that is, lc( L ) ∨ rc( L ) = 1and lc( L ) ∧ rc( L ) = 0. Here lc( L ) and rc( L ) are called the left corner (element) andthe right corner (element) of the rectangular lattice L . Note that | L | ≥ | L | ≥
3. Note also that the definition of rectangularity doesnot depend on how the diagram is fixed since lc( L ) and rc( L ) are the only doublyirreducible elements of L .Let us emphasize that a slim rectangular lattice is planar and semimodularby definition, whereby the title of the paper is redundant. The purpose of thisredundancy is to give more information about the content of the paper.The (principal) ideals ↓ lc( L ) and ↓ rc( L ) are chains and they are called the bottomleft boundary chain and the bottom right boundary chain , respectively, while thefilters ↑ lc( L ) and ↑ rc( L ) are also chains, the top left boundary chain and top rightboundary chain , respectively. The lower boundary and the upper boundary of L are ↓ lc( L ) ∪ ↓ rc( L ) and ↑ lc( L ) ∪ ↑ rc( L ), respectively. Note also that J( L ) ∪ { } equals the lower boundary ↓ lc( L ) ∪ ↓ rc( L ), but for the set M( L ) of non-unit meet-irreducible elements , we only have that ↑ lc( L ) ∪ ↑ rc( L ) ⊆ M( L ) ∪ { } . For example,the lattices S ( n )7 for n ∈ N + := { , , , . . . } , defined in Cz´edli [2] and presentedhere in Figure 1 for n ≤
4, are slim rectangular lattices.If p and q are elements of a lattice such that p ≺ q , then the prime interval , thatis, the two-element interval [ p, q ] is an edge of the diagram. Following Cz´edli [5],we need the following concepts. Definition 2.1 (Types of diagrams) . The slope of the line {(cid:104) x, x (cid:105) : x ∈ R } andthat of the line {(cid:104) x, − x (cid:105) : x ∈ R } are called normal slopes . This allows us to speak G. CZ´EDLI
Figure 1. S (1)7 , S (2)7 , S (3)7 , and S (4)7 of lines, line segments, and edges of normal slopes. For example, an edge [ p, q ] ofa lattice diagram is of a normal slope iff the angle that this edge makes with ahorizontal line is π/ ◦ ) or 3 π/ ◦ ). If this angle is strictly between π/ π/
4, then the edge is precipitous . For examples, vertical edges are precipitous. Wesay that a diagram of a slim rectangular lattice L belongs to C or, in other words,it is a C -diagram if every edge [ p, q ] such that p ∈ M( L ) \ (C left ( L ) ∪ C right ( L )) isprecipitous and all the other edges are of normal slopes. A C -diagram of L belongsto C or, shortly saying, it is a C -diagram if any two edges on the lower boundaryare of the same geometric length.The diagrams in Figures 1, 3, 7, and 11, L = L in Figure 6, and the diagramsdenoted by L in Figures 2 and 4 are C -diagrams. In addition to these C -diagrams,the diagrams in Figures 5 and 6 are C -diagrams while the diagram in Figure 10can be a part of a C -diagram. In fact, all diagrams of slim rectangular lattices inthis paper are C -diagrams. Note that we believe that only C -diagrams can givesatisfactory insight into the congruence lattices of slim rectangular lattices. Fordiagrams, drawn or not, let us agree in the following. Convention 2.2.
In the rest of the paper, all diagrams of slim rectangular lat-tices are assumed to be C -diagrams. Furthermore, L will always denote a slimrectangular lattice with a fixed C -diagram.The reader may wonder how the new concepts in the following definition obtainedtheir name; the explanation will be given in the paragraph preceding Definition 2.8. Definition 2.3 (Lamps) . Let L be a slim rectangular lattice with a fixed C -diagram.(i) An edge n = [ p, q ] of L is a neon tube if p ∈ M( L ). The elements p and q are the foot , denoted by Foot( n ), and the top of this neon tube. Clearly, aneon tube is determined by its foot.(ii) If an edge [ p, q ] is a neon tube such that p belongs to the boundary of L (equivalently, if ↓ p contains a doubly irreducible element), then [ p, q ] isalso a boundary lamp (with a unique neon tube [ p, q ]). If I = [ p, q ] is aboundary lamp, then p is called the foot of I and is denoted by Foot( I )while Peak( I ) := q is the peak of I . Note the terminological difference:neon tubes have tops but lamps have peaks.(iii) Assume that q ∈ L is the top of a neon tube whose foot is not on theboundary C left ( L ) ∪ C right ( L ) of L , and let β q := (cid:94) { p i : [ p i , q ] is a neon tube and p i / ∈ C left ( L ) ∪ C right ( L ) } . (2.1) AMPS IN SLIM RECTANGULAR LATTICES 5
Then the interval I := [ β q , q ] is an internal lamp of L . The prime intervals[ p i , q ] such that p i ∈ M( L ) but p i / ∈ C left ( L ) ∪ C right ( L ) are the neon tubes of this lamp. If I is an internal lamp, then either I is a neon tube and wesay that I has a unique neon tube, or I has more than one neon tubes.The element q is the peak of the lamp I and it is denoted by Peak( I ) whileFoot( I ) := β q is the foot of I .(iv) The lamps of L are its boundary lamps and its internal lamps. Clearly, forevery lamp I of L ,Foot( I ) = (cid:94) { Foot( n ) : n is a neon tube of I } . (2.2)Since a slim rectangular lattice L has only two doubly irreducible elements, lc( L )and rc( L ), the non-containment in (2.1) is equivalent to the condition that “ ↓ p i doesnot contain a doubly irreducible element”. Therefore, the concept of lamps doesnot depend on the diagram of L . Note that in a reasonable sense, the C diagramof L is unique, and so is its C diagram; see Cz´edli [5]. To help the reader in findingthe lamps in our diagrams, let us agree to the following. Convention 2.4.
In our diagrams of slim rectangular lattices, the foots of lampsare exactly the black-filled elements. (Except possibly for Figure 10, which can bebut need not be the whole lattice in question.) The thick edges are always neontubes (but there can be neon tubes that are not thick edges).Note that in (the slim rectangular lattices of) Figures 1, 2, 4, and 11, the neontubes are exactly the thick edges. In addition to the fact that neon tubes are easy torecognize as edges with bottom elements in M( L ), there is another way to recognizethem even more easily; the following remark follows from definitions. Remark 2.5.
Neon tubes in a C -diagram of a slim rectangular lattice (and so inour figures) are exactly the precipitous edges and the edges on the upper boundary ↑ lc( L ) ∪ ↑ rc( L ). Regions of a slim rectangular lattice L are defined as closed planar polygonssurrounded by some edges of (the fixed diagram) of L ; see Kelly and Rival [25] foran elaborate treatise of these regions, or see Cz´edli and Gr¨atzer [7]. Note that everyinterval of a planar lattice determines a region; possibly of area 0 if the intervalis a chain. The affine plane on which diagrams are drawn is often identified with R via the classical coordinatization. By the full geometric rectangle of a slimrectangular lattice L with a fixed diagram we mean the closed geometric rectanglewhose boundary is the union of all edges belonging to C left ( L ) ∪ C right ( L ). It is arectangle indeed since we allow C diagrams only. Smaller geometric rectangles arealso relevant; this is why we have the second part of the following definition. Definition 2.6 (geometric shapes associated with lamps) . Keeping Convention 2.2in mind, let I = [ p, q ] = [Foot( I ) , Peak( I )] be a lamp of L .(i) The body of I , denoted by Body( I ) = Body([ p, q ]) is the region determinedby [ p, q ]. Note that Body( I ) is a line segment if I has only one neon tube,and (by Remark 2.5) it is a quadrangle of positive area having two precipi-tous upper edges and two lower edges of normal slopes otherwise.(ii) Assume that I is an internal lamp, and define r as the meet of all lowercovers of q . Then the interval [ r, q ] is a region; this region is denoted byCircR( I ) = CircR([ p, q ]) and it is called the circumscribed rectangle of I . G. CZ´EDLI
For example, if I is the only internal lamp of S ( n )7 , then CircR( I ) is the fullgeometric rectangle of S ( n )7 for all n ∈ N + while Body( I ) is the dark-grey area for n ∈ { , , } in Figure 1. To see another example, if E is the lamp with two neontubes labelled by e on the left of Figure 2, then Body( I ) is the dark-grey area andthe vertices of CircR( I ) are x , y , z , and t . Note that by the dual of Cz´edli [3,Proposition 3.13], r in Definition 2.6(ii) can also be defined as the meet of theleftmost lower cover and the rightmost lower cover of q . (2.3) Figure 2.
Lamp( L ) ∼ = J(Con L ), whence Lamp( L ) determines Con L Definition 2.7 (line segments associated with lamps) . Let I := [ p, q ] be a lampof a slim rectangular lattice L with a fixed C -diagram, and let F stand for thefull geometric rectangle of L . Let (cid:104) p x , p y (cid:105) ∈ R and (cid:104) q x , q y (cid:105) ∈ R be the geometricpoints corresponding to p = Foot( I ) and q = Peak( I ). As usual, R + will stand forthe set of non-negative real numbers. We define the following four (geometric) linesegments of normal slopes; see Figure 3 where these line segments are dashed edgesof normal slopes.LRoof( I ) := {(cid:104) ξ, η (cid:105) ∈ F : ( ∃ t ∈ R + ) ( ξ = q x − t and η = q y − t ) } , RRoof( I ) := {(cid:104) ξ, η (cid:105) ∈ F : ( ∃ t ∈ R + ) ( ξ = q x + t and η = q y − t ) } , LFloor( I ) := {(cid:104) ξ, η (cid:105) ∈ F : ( ∃ t ∈ R + ) ( ξ = p x − t and η = p y − t ) } , RFloor( I ) := {(cid:104) ξ, η (cid:105) ∈ F : ( ∃ t ∈ R + ) ( ξ = p x + t and η = p y − t ) } . These line segments are called the left roof , the right roof , the left floor , and the rightfloor of I , respectively. Note that LRoof( I ) and LFloor( I ) lie on the same geometricline if and only if I is a boundary lamp on the left boundary, and analogously forRRoof( I ) and RFloor( I ). We defined the roof of I and the floor of I as follows:Roof( I ) := LRoof( I ) ∪ RRoof( I ) , andFloor( I ) := LFloor( I ) ∪ RFloor( I );they are ∧∧∧∧∧∧∧∧∧ -shaped broken lines (that is, chevrons pointing upwards). AMPS IN SLIM RECTANGULAR LATTICES 7
Figure 3.
Four line segments associated with I In real life, neon tubes and lamps are for illuminating in the sense of emittinglight beams. Our lamps do this only downwards with normal slopes; the photonsthey emit can only go to the directions (cid:104) , − (cid:105) and (cid:104)− , − (cid:105) . Definition 2.8 belowdescribes this more precisely. Note at this point that in addition to brightening withlight, “illuminating” also means intellectual enlightening, that is, making thingsclear for human mind. This explains that lamps and neon tubes occur in ourterminology just introduced: by illuminating a part of a C -diagram in a visualgeometrical way like in physics, lamps also illuminate the congruence structure byenlightening it in intellectual sense. Convention 2.2 is still in effect. Definition 2.8 (Illuminated sets) . Let I := [Foot( I ) , q ] = [Foot( I ) , Peak( I )] be alamp of L . A geometric point (cid:104) x, y (cid:105) of the full geometric rectangle of L is illuminatedby I from the left if the lamp has a neon tube [ p i , q ] such that the edge [ p i , q ] as ageometric line segment has a nonempty intersection with the half-line {(cid:104) x − t, y + t (cid:105) : 0 ≤ t ∈ R } . Similarly, a point (cid:104) x, y (cid:105) of the full geometric rectangle of L is illuminated by I from the right if the half-line {(cid:104) x + t, y + t (cid:105) : 0 ≤ t ∈ R } has a nonempty intersection with at least one of the neon tubes of I . If (cid:104) x, y (cid:105) isilluminated from the left or from the right, then we simply say that this point is illuminated by the lamp I . The set of points illuminated by the lamp I , that ofpoints illuminated by I from the right, and that from the left are denoted byLit( I ) = Lit([Foot( I ) , Peak( I )]) , LeftLit( I ) = LeftLit([Foot( I ) , Peak( I )]) , andRightLit( I ) = RightLit([Foot( I ) , Peak( I )]) (2.4)respectively. The acronym “Lit” and its variants come from “light” rather thanfrom “illuminate”. (The outlook of “Ill” coming from “illuminate” would not be G. CZ´EDLI satisfactory and would heavily depend on the font used.) Let us emphasize that, say,LeftLit( I ) consist of points illuminated from the right ; the notation is explainedby the fact that the geometric points of LeftLit( I ) are on the left of (and downfrom) I . Note that LeftLit( I ) is of positive geometric area if and only if I is not aboundary lamp on the left boundary, and analogously for RightLit( I ). Finally, wealso define Lit + ( I ) := Lit( I ) \ Floor( I ) . (2.5)Note that Lit( I ) is LeftLit( I ) ∪ RightLit( I ). By Definition 2.7 and 2.8,Lit( I ) is (topologically) bordered by Roof( I ), Floor( I ),and appropriate line segments of C left ( L ) and C right ( L ),and so it is bordered by line segments of normal slopes. (2.6)Note also that the intersection LeftLit( I ) ∩ RightLit( I ) can be of positive (geometric)area in the plane and that both LeftLit( I ) and RightLit( I ) are of positive area ifand only if I is an internal lamp. Figure 4.
Illustrating (2.4) and Lamp( L ) ∼ = J(Con L )For example, each of S (1)7 , S (2)7 , S (3)7 , and S (4)7 of Figure 1 has a unique in-ternal lamp, namely, the interval spanned by the black-filled enlarged element inthe middle and 1. The illuminated set of this lamp is the “ ∧∧∧∧∧∧∧∧∧ -shaped” grey-filledhexagon (containing light-grey and dark-grey points). Also, S ( n )7 has exactly twoboundary lamps and the illuminated set of each of these two lamps is the wholegeometric rectangle of S ( n )7 , for every n ∈ N + . If E denotes the lamp with two e -labelled neon tubes on the right of Figure 2, then Lit( I ) is the “ ∧∧∧∧∧∧∧∧∧ -shaped” grey-filledhexagon. In Figure 4, let E and G denote the lamps consisting of the e -labellededges and the g -labelled edges, respectively. On the left of this figure, LeftLit( E )and LeftLit( G ) are the grey-filled trapezoids while the grey-filled trapezoids on theright are RightLit( E ) and RightLit( G ). Let us note that, for any slim rectangularlattice L , two distinct internal lamps can never have the same peak, (2.7)but all the three lamps, two boundary and one internal, have the same peak in S ( n )7 . AMPS IN SLIM RECTANGULAR LATTICES 9
Next, we introduce some relations on the set of lamps. Even though not all ofthese relations are applied in the subsequent sections, they and Lemma 2.11 willhopefully be useful in future investigation of the congruence lattices of slim planarsemimodular lattices; for example, in Cz´edli and Gr¨atzer [8].
Definition 2.9 (Relations defined for lamps) . Let L be a slim rectangular latticewith a fixed C -diagram. The set of lamps of L will be denoted by Lamp( L ). Onthis set, we define eight irreflexive binary relations; seven in geometric ways basedon Definitions 2.6–2.8 and one in a purely algebraic way; these relations will soonbe shown to be the same. For I, J ∈ Lamp( L ),(i) let (cid:104) I, J (cid:105) ∈ ρ Body mean that I (cid:54) = J , Body( I ) ⊆ Lit( J ), and I is an internallamp;(ii) let (cid:104) I, J (cid:105) ∈ ρ CircR mean that I is an internal lamp, CircR( I ) ⊆ Lit( J ), and I (cid:54) = J ;(iii) let (cid:104) I, J (cid:105) ∈ ρ alg mean that Peak( I ) ≤ Peak( J ) but Foot( I ) (cid:54)≤ Foot( J );(iv) let (cid:104) I, J (cid:105) ∈ ρ LRBody mean that I is an internal lamp, I (cid:54) = J , andBody( I ) ⊆ LeftLit( J ) or Body( I ) ⊆ RightLit( J );(v) let (cid:104) I, J (cid:105) ∈ ρ LRCircR mean that I is an internal lamp, I (cid:54) = J , andCircR( I ) ⊆ LeftLit( J ) or CircR( I ) ⊆ RightLit( J );(vi) let (cid:104) I, J (cid:105) ∈ ρ foot mean that I (cid:54) = J , Foot( I ) ∈ Lit( J ), and I is an internallamp;(vii) let (cid:104) I, J (cid:105) ∈ ρ infoot mean that I (cid:54) = J and Foot( I ) is in the geometric (or, inother words, topological) interior of Lit( J ); and, finally,(viii) let (cid:104) I, J (cid:105) ∈ ρ in+foot mean that Foot( I ) ∈ Lit + ( J ). Remark 2.10.
In each of (i), (ii),. . . ,(viii) of Definition 2.9, I is an internal lampand I (cid:54) = J ; this follows easily from other stipulations even where this is not explicitlymentioned.Now we are in the position to formulate the main result of this section. Thecongruence generated by a pair (cid:104) x, y (cid:105) of elements will be denoted by con( x, y ). If p = [ x, y ] is an interval, then we can write con( p ) instead of con( x, y ). Lemma 2.11 (Main Lemma) . If L is a slim rectangular lattice with a fixed C -diagram, then the following four assertions hold. (i) The relations ρ Body , ρ CircR , ρ alg , ρ LRBody , ρ LRCircR , ρ foot , ρ infoot , and ρ in+foot are all the same. (ii) The reflexive transitive closure of ρ alg is a partial ordering of the set Lamp( L ) of all lamps of L ; we denote this reflexive transitive closure by ≤ . (iii) The poset (cid:104)
Lamp( L ); ≤(cid:105) is isomorphic to the poset (cid:104) J(Con L ); ≤(cid:105) of nonzerojoin-irreducible congruences of L with respect to the ordering inherited from Con L , and the map ϕ : Lamp( L ) → J(Con L ) , defined by [ p, q ] (cid:55)→ con( p, q ) ,is an order isomorphism. (iv) If I ≺ J (that is, J covers I ) in Lamp( L ) , then (cid:104) I, J (cid:105) ∈ ρ alg . The proof of this lemma heavily relies upon Cz´edli [2] and [5]. Before present-ing this proof, we need some preparations. First, we need to recall the multiforkstructure theory from Cz´edli [2]. Minimal regions of a planar lattice are called cells .Every slim planar semimodular lattice, in particular, every slim rectangular lattice L is a 4 -cell lattice , that is, its cells are formed by four edges; see Gr¨atzer andKnapp [21]. So the cells of L are of the from [ a ∧ b, a ∨ b ] such that a (cid:107) b and[ a ∧ b, a ∨ b ] = { a ∧ b, a, b, a ∨ b } . This cell is said to be a distributive cell if theprincipal ideal ↓ ( a ∨ b ) is a distributive lattice. Let n ∈ N + . To obtain the multiforkextension ( of rank n ) of L at a distributive -cell J means that we change J to acopy of S ( n )7 and keep adding new elements while going to the southeast and thesouthwest to preserve semimodularity. This is visualized by Figure 5, where L isdrawn on the left, J is the grey-filled 4-cell, n = 3, and the slim rectangular lattice L (cid:48) we obtain by the multifork extension of L at J of rank 3 is L (cid:48) , drawn on theright. The new elements, that is, the elements of L (cid:48) \ L , are the pentagon-shapedones. (For a more detailed definition of a multifork extension, which we do not needhere since Figures 5 and 6 are sufficient for our purposes, the reader can resort toCz´edli [2]. Note that [2] uses the term “ n -fold” rather than “of rank n ”.) Figure 5. L (cid:48) is the multifork extension of rank 3 at the 4-cell J By a grid we mean the direct product of two finite nonsingleton chains or a C -diagram of such a direct product. Note that a slim rectangular lattice is distributiveif and only if it is a grid. Note also that, in a slim rectangular lattice with a fixed C -diagram, a 4-cell I = [ p, q ] is distributive if and only ifevery edge in the ideal ↓ q is of a normal slope; (2.8)the “only if” part of (2.8) is Corollary 6.5 of Cz´edli [5] while the “if” part followseasily by using that if all edges of ↓ q are of normal slopes then ↓ q is a (sublatticeof a) grid. Lemma 2.12 (Theorem 3.7 of Cz´edli [2]) . Each slim rectangular lattice is obtainedfrom a grid by a sequence of multi-fork extensions at distributive -cells, and everylattice obtained in this way is a slim rectangular lattice. Figure 6 illustrates how we can obtain the lattice L defined by Figure 2 fromthe initial grid L in six steps. For i = 1 , , . . . L i is obtained from L i − by amultifork extension at the grey-filled 4-cell of L i − . We still need one importantconcept, which we recall from Cz´edli [2]; it was originally introduced in Cz´edli andSchmidt [12]. AMPS IN SLIM RECTANGULAR LATTICES 11
Figure 6.
Illustrating Lemma 2.12: a sequence of multifork extensions
Definition 2.13 (Trajectories) . Let L be a slim rectangular lattice with a fixed C -diagram. The set of its edges , that is, the set of its prime intervals is denotedby Edge( L ). We say that p , q ∈ Edge( L ) are consecutive edges if they are oppositesides of a 4-cell. Maximal sequences of consecutive edges are called trajectories . Inother words, the blocks of the least equivalence relation on Edge( L ) including theconsecutiveness relation are called trajectories. If a trajectory has an edge on theupper boundary (equivalently, if it has an edge that is the unique neon tube of aboundary lamp), then this trajectory a straight trajectory . Otherwise, it is a hattrajectory . Each trajectory has a unique edge that is a neon tube; this edge is calledthe top edge of the trajectory. The top edge of a trajectory u will be denoted byTopE( u ) while Traj( L ) will stand for the set of trajectories of L .For example, if L is the lattice on the left of Figure 5, then it has eight trajecto-ries. One of the eight trajectories is a hat trajectory and consists of the h -labellededges. There are seven straight trajectories and one of these seven consists of the s -labelled edges. Note that all neon tubes of L are drawn by thick lines. On theright of the same figure, only the neon tubes of the new lamp are thick. Also, L (cid:48) has exactly four hat trajectories and seven straight trajectories. One of the hat tra-jectories consists of the h -labelled edges while the s -labelled edges form a straighttrajectory. Proof of Lemma 2.11.
Let L be a slim rectangular lattice with a fixed C -diagram.To prove part (i), we need some preparations. We know from Lemma 2.12 that L is obtained by a sequence L , L , . . . , L k = L such that L is agrid and L i is a multifork extension of L i − at adistributive 4-cell H i of L i − for i ∈ { , . . . , k } . (2.9)This is illustrated by Figure 6. In Lamp( L ), there are only boundary lamps. Itis clear by definitions that each internal lamp arises from the replacement of thedistributive 4-cell H i of L i − , grey-filled in Figures 5 and 6, by a copy of S ( n i )7 , for some i ∈ { , . . . , k } and n i ∈ N + . Shortly saying,every internal lamp comes to existence from amultifork extension. Furthermore, if a lamp K comes by a multifork extension at a 4-cell H i , thenCircR( I ) is the geometric region determined by H i ; (2.10)the second half of (2.10) follows from (2.8) and Convention 2.2. For example, onthe right of Figure 5, the new lamp what the multifork extension has just broughtis the lamp with pentagon-shaped black-filled foot and the thick edges are its neontubes. Similarly, the new lamp is the one with thick neon tube(s) in each of L , . . . , L = L in Figure 6. Keeping Convention 2.2 in mind, it is clear that if [ p, q ] is thenew lamp that the multifork extension of L i − brings into L i , then [lc( L i )] ∧ p, p ]and [rc( L i )] ∧ p, p ] are chains with all of their edges being of normal slopes. Observethat no geometric line segment that consists of some edgesof L i disappears at further multifork extension steps, (2.11)but there can appear more vertices on it. This fact, Convention 2.2, (2.8), (2.10),and the fact that L i − is a sublattice of its multifork extension L i for all i ∈{ , . . . , k } yield thatif I = [ p, q ] ∈ Lamp( L ), then the lowest point of LRoof( I )and that of LFloor( I ) are lc( L ) ∧ q and lc( L ) ∧ p , respectively,and the intervals [lc( L ) ∧ q, q ] and [lc( L ) ∧ p, p ] are chains.That is, LRoof( I ) and LFloor( I ) correspond to intervals thatare chains. The edges of these chains are of the same normalslope. Similar statements hold for “right” instead of “left”. (2.12)For later reference, note another easy consequence of (2.9) and (2.11), or (2.12):With reference to (2.9), assume that i < j ≤ k and a lamp I ispresent in L i . Then Lit( I ) is the same in L i as in L j and L . (2.13)For an internal lamp I , the leftmost neon tube and the (not necessarily distinct)rightmost neon tube of I are the right upper edge and the left upper edge of a 4-cell,respectively. Combining this fact, (2.10), and Observation 6.8(iii) of Cz´edli [5], weconclude that the two upper (geometric) sides of the circumscribedrectangle CircR( I ) of an internal lamp I are edges (thatis, prime intervals) of L . (2.14)Since LeftLit( J ) ⊆ Lit( J ) and RightLit( J ) ⊆ Lit( J ) for every J ∈ Lamp( L ), ρ LRCircR ⊆ ρ CircR and ρ LRBody ⊆ ρ Body . (2.15)Next, we are going to prove that ρ CircR ⊆ ρ Body and ρ LRCircR ⊆ ρ LRBody . (2.16)To do so, assume that (cid:104) I, J (cid:105) ∈ ρ CircR and (cid:104) I (cid:48) , J (cid:48) (cid:105) ∈ ρ LRCircR . This means thatCircR( I ) ⊆ Lit( J ) and CircR( I (cid:48) ) ⊆ LeftLit( J (cid:48) ) or CircR( I (cid:48) ) ⊆ RightLit( J (cid:48) ), re-spectively. The definition of S ( n )7 and multifork extensions, (2.10), and (2.11) yieldthat Body( I ) ⊆ CircR( I ) and Body( I (cid:48) ) ⊆ CircR( I (cid:48) ). Combining these inclusionswith the earlier ones, we have that Body( I ) ⊆ Lit( J ) and Body( I (cid:48) ) ⊆ LeftLit( J (cid:48) )or Body( I (cid:48) ) ⊆ RightLit( J (cid:48) ). Hence, (cid:104) I, J (cid:105) ∈ ρ Body and (cid:104) I (cid:48) , J (cid:48) (cid:105) ∈ ρ LRBody , provingthe validity of (2.16).
AMPS IN SLIM RECTANGULAR LATTICES 13
We claim that ρ Body ⊆ ρ alg . (2.17)To show this, assume that (cid:104) I, J (cid:105) ∈ ρ Body , that is, Body( I ) ⊆ Lit( J ), I (cid:54) = J , and I is an internal lamp. We know from (2.6) that Lit( J ) is bordered by geometricline segments of normal slopes. Hence, Corollary 6.1 of Cz´edli [5] and Body( I ) ⊆ Lit( J ) yield that Peak( I ) ≤ Peak( J ). Figure 1 and Convention 2.2 imply thatFoot( I ) is not on Floor( J ), the lower geometric boundary of Lit( J ). It is trivial byBody( I ) ⊆ Lit( J ) that Foot( I ) is not (geometrically and strictly) below Floor( J ).Hence, the just mentioned Corollary 6.1 of [5] shows that Foot( I ) (cid:54)≤ Foot( J ). Wehave obtained that (cid:104) I, J (cid:105) ∈ ρ alg . Thus, ρ Body ⊆ ρ alg , proving (2.17). Figure 7.
Proving that ρ foot ⊆ ρ LRCircR
By construction, see Figure 1, CircR( I ) contains Foot( I ) as a geometric pointin its topological interior for every internal lamp I ∈ Lamp( L ). This clearly yieldsthe first inclusion in (2.18) below: ρ CircR ⊆ ρ infoot ⊆ ρ in+foot ⊆ ρ foot ⊆ ρ LRCircR . (2.18)The second and the third inclusions above are trivial. In order to show the fourthinclusion, ρ foot ⊆ ρ LRCircR , assume that (cid:104)
I, J (cid:105) ∈ ρ foot . We know that I (cid:54) = J , I isan internal lamp, and Foot( I ) ∈ Lit( J ). Since Lit( J ) = LeftLit( J ) ∪ RightLit( J ),left-right symmetry allows us to assume that the geometric point Foot( I ) belongsto LeftLit( J ). But, as it is clear from Figure 1 and (2.10), Foot( I ) is in the (topo-logical) interior of CircR( I ). Hence,there is an open set U ⊆ R such that U ⊆ CircR( I ) ∩ LeftLit( J ). (2.19)We claim that CircR( I ) ⊆ LeftLit( J ). Suppose, for a contradiction, that this in-clusion fails. We know from (2.6), (2.10), (2.11), (2.13), and Remark 2.5 thatLeftLit( J ) is surrounded by edges of normal slopes except its top right precipitousside. Also, we obtain from (2.8), (2.10), and (2.14) that CircR( I ) is a rectanglewhose sides are of normal slopes, and the upper two sides are edges (that is, primeintervals). These facts, (2.19), and CircR( I ) (cid:54)⊆ LeftLit( J ) yield that a side ofCircR( I ) crosses a side of LeftLit( J ). But edges do not cross in a planar diagram,whence no side of LeftLit( J ) crosses an upper edge of CircR( I ). So if we visual-ize LeftLit( J ) by light-grey color and CircR( I ) is dark-grey, then none of the sixlittle dark-grey rectangles on the left of Figure 7 can be CircR( I ). Note that, in both parts of this Figure, J is formed by the three thick neon tubes. Since theabove-mentioned six dark-grey rectangles represent generality, the only possibilityis that the top vertex Peak( I ) of CircR( I ) is positioned like the top of the dark-greyrectangle on the right of Figure 7. (There can be lattice elements not indicated inthe diagram, and we took into consideration that the top sides of CircR( J ) arealso edges and cannot be crossed and that (2.7) holds). But then Corollary 6.1 ofCz´edli [5] yields that Peak( J ) ≤ Peak( I ). With reference to (2.9) and (2.10), let i and j denote the subscripts such that I and J came to existence in L i and L j ,respectively. Since none of the lattices S ( n )7 , n ∈ N + , is distributive, j > i . Sothe elements of L i are old when L j is constructed as a multifork extension of L j − .Here by an old element of L j we mean an element of L j − . It is clear by the conceptof multifork extensions and that of LeftLit( J ), (2.8), and (2.10) that no old elementbelongs to the topological interior of Lit( J ). But Foot( I ) ∈ L i ⊆ L j − does belongto this interior, which is a contradiction showing that CircR( I ) ⊆ LeftLit( J ). Thus, (cid:104) I, J (cid:105) ∈ ρ LRCircR , and we obtain that ρ foot ⊆ ρ LRCircR . This completes the proofof (2.18).Next, we are going show that ρ alg ⊆ ρ foot . (2.20)To verify this, assume that (cid:104) I, J (cid:105) ∈ ρ alg . We know that Peak( I ) ≤ Peak( J ) butFoot( I ) (cid:54)≤ Foot( J ). Observe that Foot( I ) ≤ Peak( I ) ≤ Peak( J ). We know thatPeak( J ) and Foot( J ) are the vertices of the ∧∧∧∧∧∧∧∧∧ -shaped Roof( J ) and Floor( J ), re-spectively, and Roof( J ) and Floor( J ) consist of line segments of normal slopes.Hence, it follows from Corollary 6.1 of Cz´edli [5] and Foot( I ) ≤ Peak( J ) thatFoot( I ) is geometrically below or on Roof( J ). On the other hand, Corollary 6.1 of[5] and Foot( I ) (cid:54)≤ Foot( J ) imply that Foot( I ) is neither geometrically below, noron Floor( J ). So Foot( I ) is above Floor( J ). Therefore, Foot( I ) is geometrically be-tween Floor( J ) and Roof( J ). Thus, (2.6) gives that Foot( I ) ∈ Lit( J ). This showsthat (cid:104) I, J (cid:105) ∈ ρ foot , and we have shown the validity of (2.20). Figure 8.
An overview of what has already been provedThe directed graph in Figure 8 visualizes what we have already shown. Each(directed) edge ρ → ρ of the graph means that ρ ⊆ ρ has been formulated in thedisplayed equation given by the label of the edge. The directed graph is stronglyconnected, implying part (i).Next, we turn our attention to parts (ii) and (iii). For a trajectory u of L ,TopE( u ) is a neon tube, so it belongs to a unique lamp; we denote this lamp byLmp( u ). Lamps are special intervals. Hence, in agreement with how the notation AMPS IN SLIM RECTANGULAR LATTICES 15 con() was introduced right before Lemma 2.11, con(Lmp( u )) will denote the con-gruence con(Foot(Lmp( u )) , Peak(Lmp( u ))) generated by the interval Lmp( u ). Weclaim that, for any trajectory u of L ,con(TopE( u )) = con(Lmp( u )) . (2.21)To show (2.21), observe that this assertion is trivial if TopE( u ) is the only neontube of Lmp( u ) since then the same prime interval generates the congruence onboth sides of (2.21).Hence, we can assume that Lmp( u ) has more than one neon tubes; clearly, thenLmp( u ) is an internal lamp. Let p := Foot(Lmp( u )), q := Peak(Lmp( u )), and let[ p , q ] = TopE( u ), [ p , q ], . . . , [ p m , q ] be a list of all neon tubes of Lmp( u ). Withthis notation, (2.21) asserts that con( p , q ) = con( p, q ). Since the congruence blocksof a finite lattice are intervals and p ≤ p ≤ q , the inequality con( p , q ) ≤ con( p, q )is clear. It follows in a straightforward way by inspecting the lattice S ( m )7 , seeFigure 1, or it follows trivially by the Swing Lemma, see Gr¨atzer [18] or Cz´edli,Gr¨atzer and Lakser [9], that con( p , q ) = con( p , q ) = · · · = con( p m , q ). Hence,( p i , q ) ∈ con( p , q ) for i ∈ { , . . . , m } , and we obtain that( p, q ) (2.2) = ( p ∧ · · · ∧ p m , q ∧ · · · ∧ q ) ∈ con( p , q ) . (2.22)This yields the converse inequality con( p , q ) ≥ con( p, q ) and proves (2.21).As usual, the smallest element and largest element of an interval I will often bedenoted by 0 I and 1 I , respectively. Following Cz´edli [2, Definition 4.3], we definetwo relations on Traj( L ). For trajectories u, v ∈ Traj( L ), we let (cid:104) u, v (cid:105) ∈ σ def ⇐⇒ (cid:26) TopE( u ) ≤ TopE( v ) , 0 TopE( u ) (cid:54)≤ TopE( v ) ,and u is a hat trajectory. (2.23)The second relation is τ , the reflexive transitive closure of σ . We also need athird relation on Traj( L ); it is Θ := τ ∩ τ − . The Θ -block of a trajectory u ∈ Traj( L ) will be denoted by u/ Θ . Since τ is a quasiordering , that is, a reflexive andtransitive relation, it is well known that the quotient set Traj( L ) / Θ turns into aposet (cid:104) Traj( L ) / Θ , τ / Θ (cid:105) by defining (cid:104) u/ Θ , v/ Θ (cid:105) ∈ τ / Θ def ⇐⇒ (cid:104) u, v (cid:105) ∈ τ (2.24)for u, v ∈ Traj( L ); see, for example, (4.1) in Cz´edli [2]. We claim that, for any u, v ∈ Traj( L ), (cid:104) u/ Θ , v/ Θ (cid:105) ∈ τ / Θ ⇐⇒ Lmp( u ) ≤ Lmp( v ) . (2.25)As the first step towards (2.25), we show that, for any u, v ∈ Lamp( L ), u/ Θ = v/ Θ ⇐⇒ Lmp( u ) = Lmp( v ) . (2.26)The ⇐ part of (2.26) is quite easy. Assume that Lmp( u ) = Lmp( v ). Then it isclear by Figure 1 and (2.23) that (cid:104) u, v (cid:105) and (cid:104) v, u (cid:105) belong to σ . Hence, both (cid:104) u, v (cid:105) and (cid:104) v, u (cid:105) are in τ , whence u/ Θ = v/ Θ .To show the converse implication, assume that u/ Θ = v/ Θ and, to excludea trivial case, u (cid:54) = v . This assumption gives that (cid:104) u, v (cid:105) ∈ τ , and thus there is a k ∈ N + and there are elements w = u , w , . . . , w k = v such that (cid:104) w i − , w i (cid:105) ∈ σ forall i ∈ { , . . . , k } . By (2.23), u = u is a hat trajectory and 1 TopE( u ) = 1 TopE( w ) ≤ TopE( w ) ≤ · · · ≤ TopE( w k ) = 1 TopE( v ) . That is, 1 TopE( u ) ≤ TopE( v ) . Since (cid:104) v, u (cid:105) is also in τ , we also have that v is a hat trajectory and 1 TopE( v ) ≤ TopE( u ) . Hence, TopE( v ) = 1 TopE( u ) and so Peak(Lmp( u )) = Peak(Lmp( v )). Thus, Lmp( u ) =Lmp( v ) by (2.7), proving (2.26).In the next step towards (2.25), (2.26) allows us to assume that u/ Θ (cid:54) = v/ Θ ,which is equivalent to Lmp( u ) (cid:54) = Lmp( v ). First, we show thatif (cid:104) u, v (cid:105) ∈ σ , then Lmp( u ) ≤ Lmp( v ) . (2.27)Let u = u , . . . , u k and v = v , . . . , v t be the neon tubes of Lmp( u ) and Lmp( v ),respectively. With reference to (2.9) and (2.10), let i and j denote the subscriptssuch that Lmp( u ) and Lmp( v ) came to existence in L i and L j , respectively. Weobtain from (cid:104) u, v (cid:105) ∈ σ that Peak(Lmp( u )) = 1 TopE( u ) ≤ TopE( v ) = Peak(Lmp( v )).By (2.23), u is a hat trajectory, whence Lmp( u ) is an internal lamp and so i ≥ S ( n )7 , n ∈ N + , is distributive, i > j . Since the sequence(2.9) is increasing, we obtain that 0 TopE( v ) ∈ L i − . It is clear from (2.10) and thedescription of multifork extensions (see Figures 5 and 6, and see also Figure 1) thatif, according to (2.9), a lamp K comes to existence in L (cid:96) , x ∈ L (cid:96) − , and Foot( K ) ≤ x , then Peak( K ) ≤ x . (2.28)Suppose for a contradiction that Foot(Lmp( u )) ≤ TopE( v ) . Applying (2.28) with (cid:96) = i and K = I , the already-mentioned 0 TopE( v ) ∈ L i − leads to Peak(Lmp( u )) ≤ TopE( v ) . But then 0 TopE( u ) < TopE( u ) = Peak(Lmp( u )) ≤ TopE( v ) , which is acontradiction since 0 TopE( u ) (cid:54)≤ TopE( v ) by (cid:104) u, v (cid:105) ∈ σ . Hence, Foot(Lmp( u )) (cid:54)≤ TopE( v ) . Combining this with Foot(Lmp( v )) ≤ TopE( v ) , see (2.2), we obtainthat Foot(Lmp( u )) (cid:54)≤ Foot(Lmp( v )). We have already seen that Peak(Lmp( u )) ≤ Peak(Lmp( v )), whereby (cid:104) Lmp( u ) , Lmp( v ) (cid:105) ∈ ρ alg . This yields that Lmp( u ) ≤ Lmp( v ) since “ ≤ ” is the reflexive transitive closure of ρ alg . Thus, we have shownthe validity of (2.27).Next, we assert that, for any u, v ∈ Traj( L ),if (cid:104) Lmp( u ) , Lmp( v ) (cid:105) ∈ ρ alg , then (cid:104) u, v (cid:105) ∈ τ . (2.29)To show this, let v = v, v , . . . , v k be the neon tubes of Lmp( v ). Assume that thepair (cid:104) Lmp( u ) , Lmp( v ) (cid:105) belongs to ρ alg . Then1 TopE( u ) = Peak(Lmp( u )) ≤ Peak(Lmp( v )) =1 TopE( v j ) for all j ∈ { , . . . , k } , (2.30)and we know from Remark 2.10 that Lmp( u ) is an internal lamp. Hence, u is a hat trajectory. On the other hand, 0 TopE( u ) (cid:54)≤ Foot(Lmp( v )) since other-wise Foot(Lmp( u )) ≤ TopE( u ) ≤ Foot(Lmp( v )) would contradict the containment (cid:104) Lmp( u ) , Lmp( v ) (cid:105) ∈ ρ alg . If we had that 0 TopE( u ) ≤ TopE( v j ) for all j ∈ { , . . . , k } ,then we would obtain that0 TopE( u ) ≤ (cid:94) j ∈{ ,...,k } TopE( v j ) (2.2) = Foot(Lmp( v )) , contradicting 0 TopE( u ) (cid:54)≤ Foot(Lmp( v )). Hence, there exists a j ∈ { , . . . , k } suchthat 0 TopE( u ) (cid:54)≤ TopE( v j ) . Thus, using (2.30), (2.23), and the fact that u is a hattrajectory, we obtain that (cid:104) u, v j (cid:105) ∈ σ . Hence, (cid:104) u, v j (cid:105) ∈ τ . Also, Lmp( v j ) = Lmp( v )and (2.26) yield that (cid:104) v j , v (cid:105) ∈ τ . By transitivity, (cid:104) u, v (cid:105) ∈ τ , proving (2.29).By (2.24), what (2.25) asserts is equivalent to the statement that (cid:104) u, v (cid:105) ∈ τ ⇐⇒ Lmp( u ) ≤ Lmp( v ). Since τ on Traj( L ) and “ ≤ ” on Lamp( L ) are the reflexive AMPS IN SLIM RECTANGULAR LATTICES 17 transitive closures of σ and ρ alg , respectively, the desired (2.25) follows from (2.27)and (2.29).Next, we are going to call a certain map a quasi-coloring . (The exact meaningof a quasi-coloring was introduced in Cz´edli [1] and [2] but we do not need it here.)Following Definition 4.3(iv) of [2], we define a map ξ from the set Edge( L ) of edgesof L to Traj( L ) by the rule p ∈ ξ ( p ) for every p ∈ Edge( L ). That is, ξ maps an edgeto the unique trajectory containing it. By Theorem 4.4 of [2], ξ is a quasi-coloring.It follows from Lemma 4.1 of [2] thatthe posets (cid:104) J( L ); ≤(cid:105) and (cid:104) Traj( L ) / Θ ; τ / Θ (cid:105) are isomorphic . (2.31)By (2.25) and its particular case, (2.26), the structures (cid:104) Traj( L ) / Θ , τ / Θ (cid:105) and (cid:104) Lamp( L ); ≤(cid:105) are also isomorphic . (2.32)We have already mentioned around (2.24) and we also know from (2.31) that (cid:104) Traj( L ) / Θ , τ / Θ (cid:105) is a poset. Thus, (cid:104) Lamp( L ); ≤(cid:105) is also a poset by (2.32), provingpart (ii) of Lemma 2.11.Combining (2.31) and (2.32), it follows that (cid:104) J( L ); ≤(cid:105) ∼ = (cid:104) Lamp( L ); ≤(cid:105) , whichis the first half of part (iii) of Lemma 2.11. It is straightforward to extract from(2.6)–(2.8) of Cz´edli [1], or from the proof of (2.31) or that of Theorem 7.3(i) of [2]that the map ψ : (cid:104) Traj( L ) / Θ ; τ / Θ (cid:105) → (cid:104) J(Con L ); ≤(cid:105) definedby u/ Θ (cid:55)→ con(TopE( u )) is a poset isomorphism. (2.33)Observe that every lamp I is of the form Lmp( u ) for some u ∈ Traj( L ); indeed, wecan choose u as ξ ( p ) for some (in fact, any) neon tube p of I . By (2.25) and (2.26),the map ψ : (cid:104) Lamp( L ); ≤(cid:105) → (cid:104) Traj( L ) / Θ ; τ / Θ (cid:105) , definedby Lmp( u ) (cid:55)→ u/ Θ , is also a poset isomorphism. (2.34)Combining (2.21), (2.33), and (2.34), we obtain that ϕ = ψ · ψ . Hence, ϕ is alsoa poset isomorphism, proving part (iii) of Lemma 2.11.Finally, if a partial ordering ≤ is the reflexive transitive closure of a relation ρ ,then the covering relation ≺ with respect to ≤ is obviously a subset of ρ . Thisyields part (iv) and completes the proof of Lemma 2.11. (cid:3) Some consequences of Lemma 2.11 and some properties of lamps
Some statements of this section are explicitly devoted to congruence lattices ofslim planar semimodular lattices; they are called corollaries since they are derivedfrom Lemma 2.11. The rest of the statements of the section deal with lamps.Convention 2.4 raises the (easy) question whether lamps are determined by theirfoots and how. For an element u ∈ L \ { } , let u + denote the join of all covers of u , that is, u + := (cid:95) { y ∈ L : u ≺ y } , provided u (cid:54) = 1 . (3.1)Note that 1 + is undefined. Note also that (cid:87) in (3.1) applies actually to one or twojoinands since each u ∈ L \ { } has either a single cover, or it has exactly two coversby Gr¨atzer and Knapp [21, Lemma 8]. Let x ∈ L and define the element lifted( x )by induction on the number of elements of the principal filter ↑ x as follows.lifted( x ) = , if x = 1; x + , if ∃ y ∈ M( L ) such that y ≺ x + ;lifted( x + ) , otherwise . (3.2) Lemma 3.1. If L is a slim rectangular lattice, then Peak( I ) = lifted(Foot( I )) holdsfor every I ∈ Lamp( L ) .Proof. The proof is trivial by Figure 1 and (2.10). (cid:3)
Lemma 3.2 (Maximal lamps are boundary lamps) . If L is a slim rectangularlattice, then the maximal elements of Lamp( L ) are exactly the boundary lamps.Proof. We know from Gr¨atzer and Knapp [22] that each slim planar semimodularlattice L with at least three elements has a so-called congruence-preserving rectan-gular extension L (cid:48) ; see also Cz´edli [5] and Gr¨atzer and Schmidt [24] for stronger ver-sions of this result. Among other properties of this L (cid:48) , we have that Con L (cid:48) ∼ = Con L .Hence, to simplify the notation,we can assume in the proof that L is a slimrectangular lattice with a fixed C -diagram. (3.3)Note that the same assumption will be made in many other proofs when we knowthat | L | ≥
3. Armed with (3.3), Lemma 3.2 follows trivially from (2.10) andLemma 2.11. (cid:3)
The set of maximal elements of a poset P will be denoted by Max( P ). Corollary 3.3 (P2 property from Gr¨atzer [20]) . If L is a slim planar semimod-ular lattice with at least three elements, then Con L has at least two coatoms or,equivalently, J(Con L ) has at least two maximal elements.Proof. Assume (3.3). The well-known representation theorem of finite distributivelattices (see, for example, Gr¨atzer [15, Theorem 107]) easily implies that Con L has at least two coatoms if an only if J(Con L ) has at least two maximal elements.Hence, Lemma 3.2 applies. (cid:3) The covering relation in a poset P is denoted by ≺ or, if confusion threatens,by ≺ P . For sets A , A , and A , the notation A = A ˙ ∪ A will stand for theconjunction of A ∩ A = ∅ and A = A ∪ A . Corollary 3.4 (Bipartite maximal elements property) . Let L be a slim planarsemimodular lattice with at least three elements and let D := Con L . Then thereexist nonempty sets LeftMax(J( D )) and RightMax(J( D )) such that Max(J( D )) = LeftMax(J( D )) ˙ ∪ RightMax(J( D )) and for each x ∈ J( D ) and y, z ∈ Max(J( D )) , if x ≺ J( D ) y , x ≺ J( D ) z , and y (cid:54) = z , then neither { y, z } ⊆ LeftMax(J( D )) , nor { y, z } ⊆ RightMax(J( D )) . Fur-thermore, when J( D ) = J(Con L ) is represented in the form Lamp( L ) according toLemma 2.11 (iii) , then LeftMax(J( D )) can be chosen so that its members correspondto the boundary lamps on the top left boundary chain of L while the members of RightMax(J( D )) = max J( D ) \ LeftMax(J( D )) correspond to the boundary lampson the top right boundary chain.Proof. Armed with (3.3) again, we know from Lemma 3.2 that the maximal lampsare on the upper boundary. Let LeftMax(Lamp( L )) and RightMax(Lamp( L )) de-note the set of boundary lamps on the top left boundary chain ↑ lc( L ) and those onthe top right boundary chain ↑ rc( L ), respectively. Since L is rectangular, none of AMPS IN SLIM RECTANGULAR LATTICES 19 these two sets is empty. So these two sets form a partition of max Lamp( L ). Letus say that, for I (cid:48) , J (cid:48) ∈ Lamp( L ),Lit( I (cid:48) ) and Lit( J (cid:48) ) are sufficiently disjoint if for ev-ery line segment S of positive length in the plane, if S ⊆ Lit( I (cid:48) ) ∩ Lit( J (cid:48) ), then S is of a normal slope. (3.4)Clearly, if Lit( I (cid:48) ) and Lit( J (cid:48) ) are sufficiently disjoint, then no nonempty open setof R is a subset of Lit( I (cid:48) ) ∩ Lit( J (cid:48) ).Let I, J ∈ LeftMax(Lamp( L )) such that I (cid:54) = J . There are two easy ways tosee that Lit( I ) and Lit( J ) are sufficiently disjoint: either we apply (2.12), or weuse (2.13) with (cid:104) i, j (cid:105) = (cid:104) , k (cid:105) . Suppose, for a contradiction, that K ≺ Lamp( L ) I and K ≺ Lamp( L ) J . Then, by parts (i) and (iv) of Lemma 2.11, (cid:104) K, I (cid:105) ∈ ρ Body .Similarly, (cid:104)
K, J (cid:105) ∈ ρ Body , and so we have that Body( K ) ⊆ Lit( I ) ∩ Lit( J ). Since K is an internal lamp, it contains a precipitous neon tube S , which contradicts thesufficient disjointness of Lit( I ) and Lit( J ). By Lemma 2.11 and left-right symmetry,we conclude Corollary 3.4. (cid:3) Corollary 3.5 (Dioecious maximal elements property) . If L is a slim planar semi-modular lattice, D := Con L , x ∈ J( D ) , y ∈ Max(J( D )) , and x ≺ J( D ) y , then thereexists an element z ∈ J( D ) such that z (cid:54) = y and x ≺ J( D ) z . The adjective “dioecious” above is explained by the idea of interpreting x ≺ y as “ x is a child of y ”. Proof of Corollary 3.5. If | L | <
3, then | J( L ) | ≤ | L | ≥ L is rectangular; see (3.3). Assumethat I ∈ Lamp( L ), J ∈ Max(Lamp( L )), and I ≺ J in Lamp( L ). We know fromLemma 3.2 that I is an internal lamp while J is a boundary lamp. For the sakeof contradiction, suppose that J is the only cover of I in Lamp( L ). By left-rightsymmetry and Lemma 3.2, we can assume that (the only neon tube of) J is on thetop left boundary chain of L . With reference to (2.9), the illuminated sets of thelamps of L , which are boundary lamps, divide the full geometric rectangle of L into pairwise sufficiently disjoint (topologically closed) rectangles T , . . . , T m . Thatis, T , . . . , T m are the squares (that is, the 4-cells) of the initial grid L . By (2.13),the illuminated sets of the boundary lamps of L (rather than L ) divide the fullgeometric rectangle of L into the same rectangles, and the same holds for all L i , i ∈ { , , . . . , k } . We know from (2.10) that, for some i ∈ { , . . . , k } , each of thefour sides of the rectangle CircR( I ) is an edge in L i . Since no two edges of L i crosseach other by planarity (see also Kelly and Rival [25, Lemma 1.2]), it follows from(2.11) that the sides (in fact, edges) of CircR( I ) in L i do not cross the sides of T , . . . , T m . Hence, still in L i , CircR( I ) is fully included in one of the T , . . . , T m .This also holds in L since CircR( I ) is the same in L as in L i by (2.10). Hence,there is lamp K on the top right boundary chain of L such that CircR( I ) ⊆ Lit( K ).Hence, (cid:104) I, K (cid:105) ∈ ρ CircR , whence parts (i) and (ii) of Lemma 2.11 give that
I < K in Lamp( L ). By finiteness, we can pick a lamp K (cid:48) such that I ≺ K (cid:48) ≤ K inLamp( L ). We have assumed that J is the only cover of I , whereby K (cid:48) = J andso J = K (cid:48) ≤ K . The inequality here cannot be strict since both J and K belongto Max(Lamp( L )). Hence, J = K , but this is a contradiction since J is on thetop left boundary chain of L while K is on the top right boundary chain. SinceJ( D ) ∼ = Lamp( L ) by Lemma 2.11, we have proved Corollary 3.5. (cid:3) Corollary 3.6 (Two-cover Theorem from Gr¨atzer [19]) . If L is a slim planarsemimodular lattice and D := Con L , then for every x ∈ J( D ) , the set { y ∈ J( D ) : x ≺ J( D ) y } of covers of x with respect to ≺ J( D ) consists of at most two elements.Proof. Since the case | L | < L ) rather than in J( D ). For lamps I and J of our slimrectangular lattice L , we define I ≺ left J def ⇐⇒ CircR( I ) ⊆ LeftLit( L ) and, similarly, I ≺ right J def ⇐⇒ CircR( I ) ⊆ RightLit( L ) . (3.5)By parts (i) and (iv) of Lemma 2.11, for any I, J ∈ Lamp( L ),if I ≺ J , then I ≺ left J or I ≺ right J ; (3.6)note that I ≺ left J and I ≺ right J can simultaneously hold. Based on (2.10), astraightforward induction on i occurring in (2.9) yields thatfor each I ∈ Lamp( L ), there is at most one J inLamp( L ) such that I ≺ left J . Similarly, I ≺ right K holds for at most one K ∈ Lamp( L ). (3.7)Finally, (3.6) and (3.7) imply Corollary 3.6. (cid:3) Definition 3.7.
For non-horizontal parallel geometric lines T and T , we say that T is to the left of T if T i is of the form {(cid:104) a i , (cid:105) + t · (cid:104) v x , v y (cid:105) : t ∈ R } for i ∈ { , } such that a < a . Here the vector (cid:104) v x , v y (cid:105) is the common direction of T and T while ( a i ,
0) is the intersection point of T i and the x -axis. We denote by T λ T that T is left to T . For parallel line segments S and S of positive lengths, wesay that S is to the left of S , in notation, S λ S , if the line containing S isto the left of the line containing S . Let us emphasize that if S or S is of zerolength, then S λ S fails! Next, let L be a slim rectangular lattice, and let J and J be distinct lamps of L . With reference to Definition 2.7, we say that J and J are left separatory lamps if there is a (unique) i ∈ { , } such thatLRoof( J i ) λ LRoof( J − i ) λ LFloor( J i ) λ LFloor( J − i ) . (3.8)Replacing the left roofs and left floors in (3.8) by right roofs and right floors,respectively, we obtain the concept of right separatory lamps. We say that J and J are separatory lamps if J and J are left separatory or right separatory. Finally,if the line segments LFloor( J ) and LFloor( J ) lie on the same line, then J and J are left floor-aligned . If RFloor( J ) and RFloor( J ) are segments of the sameline, then J and J are right floor-aligned . They are floor-aligned if they are leftfloor-aligned or right floor-aligned. Lemma 3.8. If I and J are distinct lamps of a slim rectangular lattice, then thesetwo lamps are neither separatory, nor floor-aligned.Proof. To prove this by contradiction, suppose that the lemma fails. Let J := I and J := J . Using (2.11), (2.14), and that Foot( J j ) is in the (topological) interiorof CircR( J j ) for j ∈ { , } , we can find an i ∈ { , } such that LRoof( J i ) orLFloor( J i ) crosses the upper right edge of CircR( J − i ) or left-right symmetrically.This contradicts planarity and completes the proof. Alternatively, we can use aninduction on i occurring in (2.9). (cid:3) AMPS IN SLIM RECTANGULAR LATTICES 21
Since this section is intended to be a “toolkit”, we formulate the following lemmahere; not only its proof but also its complete formulation are left to the next section.
Lemma 3.9. If L is a slim rectangular lattice, then Lamp( L ) = Lamp( L k ) satisfies (4.3) . Figure 9.
Two-pendant four-crown4.
Two new properties of congruence lattices of slim planarsemimodular lattices
For posets X and Y , we say that X is a cover-preserving subposet of Y if X ⊆ Y and, for all u, v ∈ X , u ≤ X v ⇐⇒ u ≤ Y v and u ≺ X v ⇐⇒ u ≺ Y v . The poset R given in Figure 9 will be called the two-pendant four-crown ; it is a four-crowndecorated with two “pendants”, z and w . This section is devoted to the followingtwo properties. Definition 4.1 (Two-pendant four-crown property) . We say that a finite distribu-tive lattice D satisfies the two-pendant four-crown property if R given in Figure 9is not a cover-preserving subposet of J( D ) such that the maximal elements of R aremaximal in J( D ). Definition 4.2 (Forbidden marriage property) . We say that a finite distributivelattice D satisfies the forbidden marriage property if for every x, y ∈ J( D ) and z ∈ Max(J( D )), if x (cid:54) = y , x ≺ J( D ) z , and y ≺ J( D ) z , then there is no p ∈ J( D ) suchthat p ≺ J( D ) x and p ≺ J( D ) y .Now we are in the position to formulate the main theorem of the paper. Theorem 4.3 (Main Theorem) . If L is a slim planar semimodular lattice, then (i) Con L satisfies the the forbidden marriage property, and (ii) Con L satisfies the two-pendant four-crown property;see Definitions 4.1 and 4.2. Remark 4.4.
The smallest distributive lattice D that fails to satisfy the forbiddenmarriage property is the eight-element D given in Cz´edli [4]. Now the result of [4],stating that D cannot be represented as the congruence lattice of a slim planarsemimodular lattice, becomes an immediate consequence of Theorem 4.3. In fact,part (i) of Theorem 4.3 is a generalization of [4]. Remark 4.5.
By the well-known representation theorem of finite distributive lat-tices, see, for example, Gr¨atzer [15, Theorem 107], there is a unique finite distribu-tive lattice D R such that J( D R ) ∼ = R . By Theorem 4.3(ii), there is no slim planarsemimodular lattice L such that Con L ∼ = D R . Since D R satisfies the propertiesmentioned in Corollaries 3.3–3.6, so all the previously known properties, and eventhe forbidden marriage property, it follows that part (ii) of Theorem 4.3 is really anew result. Remark 4.6.
A straightforward calculation shows that D R mentioned in Re-mark 4.5 consists of 56 elements. Furthermore, we are going to prove that ev-ery finite distributive lattice with less than 56 elements satisfies the two-pendantfour-crown property. Figure 10.
Illustrating the proof of (4.3)
Proof of Theorem 4.3.
As usual, the case | L | < D := Con L . By Lemma 2.11, we can also assume thatJ( D ) = Lamp( L ). For J , J ∈ Lamp( L ), we say thatthe lamps J and J are independent if there is a(unique) i ∈ { , } such that Peak( J i ) ≤ Foot( J − i ). (4.1)First, we deal with part (i), that is, with the forbidden marriage property. InFigure 10, which is either L or only a part (in fact, an interval) of L , there are fourinternal lamps, S , U , V , and W ; their foots are s , u , v , and w , respectively. Inthis figure, for example, U and V are independent but S and W are not. Actually, { S, W } is the only two-element subset of { S, U, V, W } whose two members are notindependent. It follows from (2.6) and Corollary 6.1 of Cz´edli [5] that, using theterminology of (3.4),if J and J are independent lamps, thenLit( J ) and Lit( J ) are sufficiently disjoint. (4.2)Now assume that Z is a boundary lamp. By left-right symmetry, we can assumethat it is on the top left boundary chain; see Figure 10 where z = Foot( Z ). Theintersection of Lit( Z ) = RightLit( Z ) with the right boundary chain C right ( L ) willbe denoted by E ( Z ); it is the (topologically closed) line segment with endpoints a and b in the Figure. Also, we define the following set of geometric points F ( Z ) := { q ∈ E ( Z ) : ( ∃ U ∈ Lamp( L )) ( (cid:104) U, Z (cid:105) ∈ ρ CircR and q ∈ Lit( U )) } . AMPS IN SLIM RECTANGULAR LATTICES 23
For example, F ( Z ) in Figure 10 is the line segment with endpoints b and c . If F ( Z ) = ∅ or F ( Z ) is a line segment with its upper endpoint being the same as thatof E ( Z ), then we say that there is no gap in F ( Z ). For example, there is no gapin F ( Z ) in Figure 10. With reference to (2.9) and (2.10), we claim thatfor i = 0 , , . . . , k , the lower covers of Z in Lamp( L i ) are pairwise independentand, still in L i , there is no gap in F ( Z ); (4.3)note that our boundary lamp Z is in L and so (4.3) makes sense. Note also that(4.3) implies Lemma 3.9, because L = L k .We prove (4.3) by induction on i . The case i = 0 is trivial since Z has no lowercover in Lamp( L ). In Figure 10, Z has three lower covers: U , V and W . (Since S < W < Z , S is not a lower cover.) Assume that this figure is the relevant part(that is, Lit( Z )) of L i for some i ∈ { , , . . . , k − } . Assume also that in L i +1 , Z obtains a new lower cover, T . We know from (i) and (iv) of Lemma 2.11 that (cid:104) T, Z (cid:105) ∈ ρ CircR . Due to (2.9) and (2.10), CircR( T ) is a distributive 4-cell in thefigure. For every lamp G ∈ Lamp( L i ) such that G < Z (in particular, if (cid:104)
G, Z (cid:105) ∈ ρ CircR ), CircR( T ) cannot be a 4-cell of Lit( G ) since otherwise (cid:104) T, G (cid:105) ∈ ρ CircR wouldlead to
T < G < Z , contradicting T ≺ Z . Also, there can be no G ∈ Lamp( L i ) suchthat (cid:104) G, Z (cid:105) ∈ ρ CircR and G is to the “south-east” of T , because otherwise CircR( T )would not be distributive. It follows that T is one of the dark-grey cells in the figure,whereby even in L i +1 , the lower covers of Z remain pairwise independent and thereis no gap in F ( Z ). Since the figure clearly represents generality, we are done withthe induction step from i to i + 1. This proves (4.3).Finally, Lemmas 2.11 and 3.2 translate part (i) of Theorem 4.3 to the followingstatement on Lamp( L ):if Z is a boundary lamp, X ≺ Z , Y ≺ Z , and X (cid:54) = Y , thenthere exists no P ∈ Lamp( L ) such that P ≺ X and P ≺ Y . (4.4)To see this, assume the premise. For the sake of contradiction, suppose that theredoes exist a P described in (4.4). By (i) and (iv) of Lemma 2.11(iv), (cid:104) P, X (cid:105) ∈ ρ CircR and (cid:104)
P, Y (cid:105) ∈ ρ CircR . Hence, CircR( P ) ⊆ Lit( X ) ∩ Lit( Y ), whereby Lit( X ) andLit( Y ) are not sufficiently disjoint. On the other hand, we obtain from L = L k and(4.3) that Lit( X ) and Lit( Y ) are independent, whence they are sufficiently disjointby (4.2). This is a contradiction proving (4.4) and part (i) of Theorem 4.3.Next, we deal with part (ii). For the sake of contradiction, suppose that D =Con L fails to satisfy the two-pendant four-crown property. This assumption andLemma 2.11 yield that R is a cover-preserving subposet of Lamp( L ) = J( D ) suchthat the four maximal elements of R are also maximal in Lamp( L ); see Defini-tion 4.1. For a, b, · · · ∈ R , the corresponding lamp will be denoted by A , B , . . . ,that is, by the capitalized version of the notation used in Figure 9. By Lemma 3.2, A, B, C, D are boundary lamps. Each of these four lamps is on the top left boundarychain or on the top right boundary chain.By Corollary 3.4, any two consecutive members of the sequence
A, B, C, D, A belong to different top boundary chains since they have a common lower cover. Byleft-right symmetry, we can assume that A and C are on the top left boundarychain while B and D on the top right one. We can assume that C is above A inthe sense that Foot( A ) < Foot( C ) since otherwise we can relabel R according to the “rotational” automorphism that restricts to { a, b, c, d } as (cid:18) a b c dc d a b (cid:19) . Also, we can assume that D is above B since otherwise we can extend (cid:18) a b c da d c b (cid:19) . to a “reflection” automorphism of R and relabel R accordingly. Note that A and C are not necessarily neighboring boundary lamps, that is, we have Peak( A ) ≤ Foot( C ) but Peak( A ) < Foot( C ) need not hold. Similarly, we only have thatPeak( B ) ≤ Foot( D ). The situation is outlined in Figure 11. The foots of lamps inthe figure are black-filled and any of the two distances marked by curly bracketscan be zero. The illuminated sets Lit( X ) and Lit( Y ) are dark-grey while Lit( A )and Lit( C ) are (dark and light) grey. Since X ≺ A and X ≺ B , we know from(i) and (iv) of Lemma 2.11 that (cid:104) X, A (cid:105) ∈ ρ Body and (cid:104)
X, B (cid:105) ∈ ρ Body . Hence,Body( X ) ⊆ Lit( A ) ∩ Lit( B ), in accordance with the figure. Similarly, Y ≺ C and Y ≺ D lead to Body( Y ) ⊆ Lit( C ) ∩ Lit( D ), as it is indicated in Figure 11. Note thatdue to (2.6), the figure is satisfactorily correct in this aspect. Hence, the ∧∧∧∧∧∧∧∧∧ -shapedLit( Y ) is above the ∧∧∧∧∧∧∧∧∧ -shaped Lit( X ). Thus, using (2.6), we obtain thatLit( X ) and Lit( Y ) are sufficiently disjoint; (4.5)see (3.4) for this concept. On the other hand, Z ≺ X and Z ≺ Y together with(i) and (iv) of Lemma 2.11 gives that (cid:104) Z, X (cid:105) ∈ ρ Body and (cid:104)
Z, Y (cid:105) ∈ ρ Body . Hence,Body( Z ) ⊆ Lit( X ) ∩ Lit( Y ). Thus, since Z is an internal lamp by Lemma 3.2,Lit( X ) ∩ Lit( Y ) contains a precipitous neon tube. This contradicts (4.5). Note thatthere is another way to get a contradiction: since (cid:104) Z, X (cid:105) ∈ ρ CircR and (cid:104)
Z, Y (cid:105) ∈ ρ CircR , we have that CircR( Z ) ⊆ Lit( X ) ∩ Lit( Y ), which contradicts the fact thatCircR( Z ) is of positive area (two-dimensional measure) while Lit( X ) ∩ Lit( Y ) is ofarea 0. Any of the two contradictions in itself implies part (ii) and completes theproof of Theorem 4.3. (cid:3) Proof of Remark 4.6.
For the sake of contradiction, suppose that there exists adistributive lattice D such that D fails to satisfy the two-pendant four-crown prop-erty but | D | <
56. Let Q := J( D ). By our assumption, R is a subposet ofthe poset Q . Denote by DnSt( R ) and DnSt( Q ) the lattice of down sets (thatis, order ideals and the emptyset) of R and Q , respectively. For X ⊆ Q , let ↓ Q X := { y ∈ Q : y ≤ x for some x ∈ X } . The map ϕ : DnSt( Q ) → DnSt( R ),defined by X (cid:55)→ X ∩ R is surjective since, for each Y ∈ DnSt( R ), we have that ↓ Q Y ∈ DnSt( Q ) and ϕ ( ↓ Q Y ) = Y . Hence, using the structure theorem mentionedin Remark 4.5, | D | = | DnSt( Q ) | ≥ | DnSt( R ) | = | D R | = 56, which is a contradictionproving Remark 4.6. (cid:3) We conclude the paper with a last remark.
Remark 4.7.
Lamps have several geometric properties. Many of these propertieshave already been mentioned, and there are some other properties of technicalnature, too. These properties would allow us to represent the congruence latticesof slim planar semimodular lattices in a purely geometric (but quite technical)
AMPS IN SLIM RECTANGULAR LATTICES 25
Figure 11.
Illustrating the proof of Theorem 4.3(ii)way. However, this does not seem to be more useful than our technique based on(2.9)–(2.10) and the tools presented in the paper.
Added on February 28, 2021.
Since January 8, 2021, when the first version ofthe present paper was uploaded to arXiv, the tools developed here have successfullybeen used in Cz´edli [6] and Cz´edli and Gr¨atzer [8].
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