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Sublattices of lattices of order-convex sets, III. The case of totally ordered sets
Marina Semenova, Friedrich Wehrung
To cite this version:
Marina Semenova, Friedrich Wehrung. Sublattices of lattices of order-convex sets, III. The case of totally ordered sets. International Journal of Algebra and Computation, World Scientific Publishing, 2004, 14 (3), pp.357-387. �10.1142/S021819670400175X�. �hal-00003977�
ccsd-00003977, version 1 - 21 Jan 2005
THE CASE OF TOTALLY ORDERED SETS
MARINA SEMENOVA AND FRIEDRICH WEHRUNG
Abstract. For a partially ordered setP, letCo(P) denote the lattice of all order-convex subsets ofP. For a positive integern, we denote bySUB(LO) (resp.,SUB(n)) the class of all lattices that can be embedded into a lattice of
the form Y
i∈I
Co(Ti),
wherehTi|i∈Iiis a family ofchains(resp., chains with at mostnelements).
We prove the following results:
(1) Both classes SUB(LO) and SUB(n), for any positive integer n, are locally finite, finitely based varieties of lattices, and we find finite equa- tional bases of these varieties.
(2) The varietySUB(LO) is the quasivariety join of all the varietiesSUB(n), for 1≤n < ω, and it has only countably many subvarieties. We clas- sify these varieties, together with all the finite subdirectly irreducible members ofSUB(LO).
(3) Every finite subdirectly irreducible member ofSUB(LO) is projective withinSUB(LO), and every subquasivariety ofSUB(LO) is a variety.
1. Introduction
For a partially ordered set (from now onposet) (P,E), a subsetX ofP isorder- convex, if x E z E y and {x, y} ⊆ X implies that z ∈ X, for all x, y, z ∈ P. The lattices of the form Co(P) have been characterized by G. Birkhoff and M. K.
Bennett in [2]. In M. Semenova and F. Wehrung [12], the authors solve a problem stated in K. V. Adaricheva, V. A. Gorbunov, and V. I. Tumanov [1], by proving the following result.
Theorem 1. The classSUBof all lattices that can be embedded into some lattice of the form Co(P)forms a variety, defined by three identities, (S),(U), and (B).
In M. Semenova and F. Wehrung [13], this result is extended to special classes of posetsP:
Theorem 2. For a positive integer n, the class SUBn of all lattices that can be embedded into some lattice of the form Co(P), where P is a poset of length at
Date: January 21, 2005.
2000Mathematics Subject Classification. Primary: 06B05, 06B20, 06B15, 06A05, 08C15. Sec- ondary: 05B25.
Key words and phrases. Lattice, embedding, poset, chain, order-convex, variety, join-irreduc- ible, join-seed.
The first author was partially supported by INTAS grant no. YSF: 2001/1-65. The au- thors were partially supported by GA CR grant no. 201/00/0766 and by institutional grant MSM:J13/98:1132000007a.
1
most n, is a variety, defined by the identities (S), (U), (B), together with new identities (Hn) and (Hk,n+1−k), for 1≤k≤n.
In the present paper, we extend these results to sublattices of products of lattices of convex subsets ofchains(i.e., totally ordered sets), thus solving a problem of [12].
More specifically, we denote bySUB(LO) (resp.,SUB(n)) the class of all lattices that can be embedded into a lattice of the form
Y
i∈I
Co(Ti),
wherehTi|i∈Iiis a family of chains (resp., chains with at mostnelements). We prove the following results:
(1) Both classesSUB(LO) andSUB(n) are finitely based varieties of lattices, for any positive integer n. Moreover,SUB(n+ 1) =SUB(LO)∩SUBn
(Theorems 8.2 and 9.4).
(2) By using a result of V. Slav´ık [14], we prove that the varietySUB(LO) is locally finite (Theorem 9.5).
(3) The varietySUB(LO) is the quasivariety join of all the varietiesSUB(n), for 1≤n < ω (Corollary 9.7), and every proper subvariety ofSUB(LO) is finitely generated (Corollary 11.7).
(4) The only proper subvarieties ofSUB(LO) are those betweenSUB(n) and SUB(n+ 1) for some natural number n(Theorem 11.5).
(5) We classify all finite subdirectly irreducible members ofSUB(LO), and we describe exactly the lattice of all subvarieties ofSUB(LO) (Theorem 11.5 to Corollary 11.9).
(6) All finite subdirectly irreducible members of SUB(LO) are projective withinSUB(LO) (Theorem 12.4), and every subquasivariety ofSUB(LO) is a variety (Theorem 12.5).
The main technical result towards the proof thatSUB(LO) is a variety is that the reflexive closure of the join-dependency relationDistransitive, in any member ofSUB(LO) with ‘enough’ join-irreducible elements (Corollary 6.2). This may be viewed as an analogue, for certain join-semidistributive lattices, of the transitivity of perspectivity proved by von Neumann in continuous geometries, see [11].
We refer the reader to our papers [12, 13] for unexplained notation and termi- nology. In particular, the identities (S), (U), and (B), together with their join-ir- reducible translations (Sj), (Uj), and (Bj), and tools such as Stirlitz tracks or the Udav-Bond partition, are defined in [12]. The identities (Hn) and (Hm,n), their join-irreducible translations, and bi-Stirlitz tracks are defined in [13]. We shall of- ten use the trivial fact thatCo(P,E) =Co(P,D), for any poset (P,E), whereD denotes the converse order ofE.
The join-dependency relation on a lattice L, see R. Freese, J. Jeˇzek, and J. B.
Nation [5], is defined on the set J(L) of all join-irreducible elements of L, and it is written DL, orD ifLis understood from the context. Fora∈J(L), we write, as in [12, 13],
[a]D={x∈J(L)|a D x}.
2. Join-seeds and more minimal covers We recall from [13] the following definition:
Definition 2.1. A subset Σ of a latticeLis ajoin-seed, if the following statements hold:
(i) Σ⊆J(L);
(ii) every element ofLis a join of elements of Σ;
(iii) for allp∈Σ and all a,b ∈Lsuch that p≤a∨b andpa, b, there are x≤aandy≤bboth in Σ such thatp≤x∨y is minimal inxandy.
Two important examples of join-seeds are provided by the following lemma, see [13].
Lemma 2.2. Any of the following assumptions implies that the subset Σis a join- seed of the latticeL:
(i) L=Co(P)andΣ ={{p} |p∈P}, for some poset P.
(ii) L is a dually 2-distributive, complete, lower continuous, finitely spatial lattice, andΣ = J(L).
Lemma 2.3. LetLbe a lattice satisfying (B), let Σbe a join-seed ofL, let p∈Σ, letx,y∈[p]D. If the inequalityp≤x∨y holds, then it is minimal in bothxandy.
Proof. From the assumption that x, y ∈ [p]D, it follows that p x, y. Since p≤x∨y and Σ is a join-seed ofL, there areu≤xandv ≤y in Σ such that the inequality p≤u∨v holds and is minimal in both uand v. Furthermore, by the definition of the D relation and since Σ is a join-seed of L, there are x′, y′ ∈ Σ such that both inequalitiesp≤x∨x′ andp≤y∨y′ hold and are minimal inx,x′, y,y′. By applying (Bj) to the inequalitiesp≤x∨x′, u∨v and by observing that px, v, we obtain that p≤x′∨u. Since u≤xand the inequality p≤x∨x′ is minimal inx, we obtain thatu=x. Similarly,v=y.
Lemma 2.4. Let L be a lattice satisfying (B), let Σ be a join-seed of L. Then [p]D∩Σis an antichain ofL, for anyp∈Σ.
Proof. Letx,y ∈[p]D. Since Σ is a join-seed ofL, there arex′,y′ ∈Σ such that both inequalities p ≤ x∨x′ and p ≤ y∨y′ are minimal nontrivial join-covers.
Observe that px, x′, y, y′. If x≤ y, then, since p y =x∨y and L satisfies (Bj), the inequalityp≤x∨y′ holds. Sincex≤y and the inequalityp≤y∨y′ is
minimal iny, we obtain thatx=y.
3. The identity(E)
Let (E) be the following identity in the variablesx, a,b0,b1,b2:
x∧^
i<3
(a∨bi) = _
i<3
x∧bi∧^
j6=i
(a∨bj)
∨ _
σ∈S3
x∧(a∨b∗0,σ)∧(a∨b∗1,σ)∧(a∨bσ(2)) ,
where we denote byS3the group of all permutations of{0,1,2}and we put
b∗0,σ=bσ(0)∧(x∨bσ(1)), (3.1)
b∗1,σ=bσ(1)∧(x∨bσ(2))∧(bσ(0)∨bσ(2)), (3.2) for allσ∈S3.
We now introduce a lattice-theoretical axiom, thejoin-irreducible interpretation of (E), that we will denote by (EΣ).
Definition 3.1. For a lattice L and a subset Σ of J(L), we say that L satisfies (EΣ), if for all elements x,a, b0, b1, and b2 of Σ, if the inequalityx≤a∨bi is a minimal nontrivial join-cover, for every i <3, then there exists σ∈S3 such that bσ(0)≤x∨bσ(1) ≤x∨bσ(2) andbσ(1)≤bσ(0)∨bσ(2).
The geometrical meaning of (EΣ) is illustrated on Figure 1. The lines of that figure represent the ordering of the either the posetP or its dual (and not the or- dering ofL) in caseL=Co(P,E). For example, the left half of Figure 1 represents (up to dualization ofE) the relationsaExEbi, for i <3, so that the inequality {x} ≤ {a} ∨ {bi}holds inL. Similar conventions hold for Figures 2 and 3.
a a
b0 b1 b2
x
x bσ(0)
bσ(1)
bσ(2)
Figure 1. Illustrating (EΣ)
Lemma 3.2. Let L be a lattice, let Σ be a subset of J(L). Then the following statements hold:
(i) If L satisfies (E), thenLsatisfies (EΣ).
(ii) If Σis a join-seed of L andL satisfies both (B) and (EΣ), thenL satis- fies (E).
Proof. (i) Suppose thatx, a,b0,b1,b2∈Σ satisfy the premise of (EΣ). Sincexis join-irreducible and xbi, for alli < 3, we obtain, by applying the identity (E) and using the notation introduced in (3.1) and (3.2), that there existsσ∈S3such that both inequalities x≤a∨b∗0,σ, a∨b∗1,σ hold. Sinceb∗i,σ≤bσ(i), it follows from the minimality ofbσ(i)in the inequalityx≤a∨bσ(i)thatb∗i,σ=bσ(i), for alli <2.
Therefore,bσ(0)≤x∨bσ(1) ≤x∨bσ(2) andbσ(1)≤bσ(0)∨bσ(2).
(ii) Let c (resp., d) denote the left hand side (resp., right hand side) of the identity (E). Sinced≤c holds in any lattice, it suffices to prove that c ≤d. Let p∈Σ withp≤c, we prove thatp≤d. Ifp≤a, thenp≤x∧a≤d. Ifp≤bi, for somei <3, thenp≤x∧bi∧V
j6=i(a∨bj)≤d.
Suppose from now on thatpaand pbi, for all i <3. Sincep≤a∨bi and Σ is a join-seed of L, there are ui ≤a and vi ≤bi in Σ such that the inequality p ≤ ui ∨vi is a minimal nontrivial join-cover, for all i < 3. In particular, ui, vi ∈ [p]D. Put u = u0, and let i < 3. By applying (Bj) to the inequalities p≤u∨v0, ui∨viand observing thatpa(thuspu∨ui), we obtain the inequality p≤u∨vi. Furthermore, by Lemma 2.3, this inequality is minimal in bothuand vi. Hence, by (EΣ), there existsσ∈S3such thatvσ(0)≤p∨vσ(1) ≤p∨vσ(2) and
vσ(1) ≤vσ(0)∨vσ(2). Therefore, by putting v∗0,σ=vσ(0)∧(p∨vσ(1)),
v∗1,σ=vσ(1)∧(p∨vσ(2))∧(vσ(0)∨vσ(2)),
we obtain the equalitiesv0,σ∗ =vσ(0) andv∗1,σ=vσ(1), and the inequalities p≤x∧(u∨v∗0,σ)∧(u∨v∗1,σ)∧(u∨vσ(2))≤d.
Since every element ofLis a join of elements of Σ, the inequalityc≤dfollows.
Corollary 3.3. The lattice Co(T)satisfies the identity (E), for any chain(T,E).
Proof. We apply Lemma 3.2 to L = Co(T) together with the join-seed Σ = {{p} |p∈T}. Let x, a, b0, b1, b2 ∈T such that the inequality{x} ≤ {a} ∨ {bi} is a minimal nontrivial join-cover, for all i < 3. Since Co(T,E) = Co(T,D), we may assume without loss of generality thata⊳x⊳b0, thusx⊳bi, for alli <3.
SinceT is a chain, there existsσ∈S3 such thatbσ(0)Ebσ(1) Ebσ(2), whence {bσ(0)} ≤ {x} ∨ {bσ(1)} ≤ {x} ∨ {bσ(2)} and{bσ(1)} ≤ {bσ(0)} ∨ {bσ(2)}.
HenceCo(T) satisfies (EΣ). SinceCo(T) satisfies (B) (see [12]) and Σ is a join-seed ofCo(T), it follows from Lemma 3.2 thatCo(T) satisfies (E).
Lemma 3.4. LetLbe a join-semidistributive lattice satisfying the identity (E), let a, x ∈ J(L) and b0, b1, b2 ∈ J(L) be distinct such that x ≤a∨bi is a minimal nontrivial join-cover, for all i < 3. Then a∨b0 ≤ a∨b1 ≤ a∨b2 implies that a∨b0< a∨b1< a∨b2 andb1≤b0∨b2.
Proof. Leti,j be distinct in{0,1,2}. Ifa∨bi=a∨bj, then, by the join-semidis- tributivity ofL,x≤a∨bi=a∨(bi∧bj); it follows from the minimality assumption on bi that bi ≤bj. Similarly, bj ≤ bi, whence bi =bj, a contradiction. Thus we have obtained the inequalities
a∨b0< a∨b1< a∨b2. (3.3) On the other hand, it follows from Lemma 3.2 that there exists σ∈S3 such that the inequalities
x∨bσ(0) ≤x∨bσ(1) ≤x∨bσ(2), (3.4)
bσ(1)≤bσ(0)∨bσ(2) (3.5)
hold. From (3.4) it follows thata∨bσ(0)≤a∨bσ(1)≤a∨bσ(2), thus, by (3.3),σis the identity. The conclusion follows from (3.4) and (3.5).
4. The identity(P)
Let (P) be the following identity in the variablesa,b,c,d,b0,b1: a∧(b′∨c)∧(c∨d) = a∧b′∧(c∨d)
∨ a∧d∧(b′∨c)
∨
a∧
b′∧(a∨d)
∨c
∧(c∨d)
∨_
i<2
a∧(bi∨c)∧
b′∧(a∨bi)∧(bi∨d)
∨c
∧(c∨d)
,
where we putb′=b∧(b0∨b1).
We now introduce a lattice-theoretical axiom, thejoin-irreducible interpretation of (P), that we will denote by (PΣ).
Definition 4.1. For a lattice L and a subset Σ of J(L), we say that L satisfies (PΣ), if for all elementsa,b,c,d,b0,b1 in Σ, if both inequalitiesa≤b∨c, c∨dare minimal nontrivial join-covers andb≤b0∨b1, then either b≤a∨dor there exists i <2 such thata≤bi∨c andb≤a∨bi, bi∨d.
The geometrical meaning of (PΣ) is illustrated on Figure 2. Horizontal lines are meant to suggest that “no side is chosen yet”. For example, the non-horizontal lines in the left half of Figure 2 represent various inequalities such as c Ea Ed and c E a E b (in case L = Co(P,E)), while the horizontal line represents the inequalitiesb1−iEbEbi, for somei <2. A similar convention applies to Figure 3.
a a
a
b b
b
c c
c d
d
d
b0 b0
bi
Case where
Case where
b≤a∨d
b1−i
b1
b1
b≤a∨bi
b≤bi∨d a≤bi∨c
Figure 2. Illustrating (PΣ)
Lemma 4.2. Let L be a lattice, let Σ be a subset of J(L). Then the following statements hold:
(i) If L satisfies (P), thenLsatisfies (PΣ).
(ii) If Σis a join-seed of L andL satisfies both (B) and (PΣ), thenL satis- fies (P).
Proof. (i) Let a, b, c, d, b0, b1 ∈ Σ satisfy the premise of (PΣ). Observe that b∧(b0∨b1) =b, thus the left hand side of the identity (P) computed with these parameters equalsa. Sinceab, dandais join-irreducible, eithera≤ b∧(a∨d)
∨c ora≤bi∨c anda≤ b∧(a∨bi)∧(bi∨d)
∨c, for somei <2. In the first case, from the fact that the cover a≤b∨c is minimal in b it follows thatb≤a∨din the first case, andb≤a∨bi, bi∨din the second case.
(ii) Let e (resp., f) denote the left hand side (resp., right hand side) of the identity (P). Letp∈ Σ such that p≤e, we prove that p≤f. If eitherp≤ c or p≤b′ orp≤dthis is obvious, so suppose, from now on, thatpc, b′, d. Since Σ is a join-seed of L, there areu ≤b′ together withv, v′ ≤c andw ≤d in Σ such that both inequalities
p≤u∨v, (4.1)
p≤v′∨w (4.2)
are minimal nontrivial join-covers. In particular,u,v,v′, w∈[p]D. Furthermore, by applying (Bj) to the inequalities (4.1) and (4.2) and observing that pv∨v′ (becausepc), we obtain the inequality
p≤v∨w. (4.3)
Furthermore, it follows from Lemma 2.3 that (4.3) is a minimal nontrivial join- cover. Since Σ is a join-seed ofL, there areui≤bi in Σ∪ {0}, fori <2, such that u≤u0∨u1. Suppose first thatu0, u1∈Σ. SinceLsatisfies (PΣ), either
u≤p∨w (4.4)
or
p≤ui∨vandu≤p∨ui, ui∨w, for somei <2. (4.5) The conclusion (4.5) also holds ifuj = 0, for somej <2, becauseu≤u1−j.
If (4.4) holds, then p≤a∧
u∧(p∨w)
∨v
∧(v∨w)≤f.
If (4.5) holds, then
p≤a∧(ui∨v)∧
u∧(p∨ui)∧(ui∨w)
∨v
∧(v∨w)≤f.
Since every element of L is a join of elements of Σ, the inequality e ≤f follows.
Sincef ≤eholds in any lattice, we obtain thate=f. Corollary 4.3. The lattice Co(T) satisfies (P), for every chain (T,E).
Proof. We apply Lemma 4.2 to L = Co(T) together with the join-seed Σ = {{p} |p∈T}. Let a, b, c, d, b0, b1 ∈ T such that both inequalities {a} ≤ {b} ∨ {c},{c} ∨ {d}are minimal nontrivial join-covers and{b} ≤ {b0} ∨ {b1}. Since Co(T,E) =Co(T,D), we may assume without loss of generality thatc⊳a⊳b, d.
Furthermore, from {b} ≤ {b0} ∨ {b1} it follows that there exists i < 2 such that bEbi. SinceT is a chain, either bEdordEb. In the first case,{b} ≤ {a} ∨ {d}.
In the second case,{a} ≤ {bi} ∨ {c} and{b} ≤ {a} ∨ {bi},{bi} ∨ {d}.
HenceCo(T) satisfies (PΣ). By Lemma 4.2,Co(T) satisfies (P).
5. The identity (HS)
Let (HS) be the following identity in the variablesa,b,c, b0,b1: a∧(b′∨c) =(a∧b′)∨_
i<2
ha∧ (b∧bi)∨ci
∨_
i<2
a∧
b′∧(a∨bi)
∨c
∧(bi∨c)∧(b∨b1−i)
∨_
i<2
a∧
b′∧(a∨bi)
∨c
∧(b0∨c)∧(b1∨c)
,
where we put b′ = b∧(b0∨b1). Since the right hand side of (HS) lies obviously below the right hand side of the identity (S) while the left hand sides are the same, we obtain immediately the following result.
Lemma 5.1. The identity (HS)implies the Stirlitz identity (S).
As observed in [12], (S) implies both join-semidistributivity and dual 2-distrib- utivity. Therefore, we obtain the following consequence.
Lemma 5.2. The identity (HS)implies both join-semidistributivity and dual2-dis- tributivity.
We now introduce a lattice-theoretical axiom, thejoin-irreducible interpretation of (HS), that we will denote by (HSΣ).
Definition 5.3. For a lattice L and a subset Σ of J(L), we say that L satisfies (HSΣ), if for all elementsa, b, c, b0, b1 in Σ, ifa6= b, the inequalitya ≤b∨c is minimal inb, andb≤b0∨b1is a nontrivial join-cover, then there existsi <2 such thatb≤a∨bi and eithera≤bi∨c, b∨b1−i ora≤b0∨c, b1∨c.
The geometrical meaning of (HSΣ) is illustrated on Figure 3.
a a
a
b0 b b1 b
b
Case where
c
c c
Case where
bi bi
b1−i
a≤bi∨c a≤b∨b1−i
b≤a∨bi
a≤b0∨c a≤b1∨c b≤a∨bi
b1−i
Figure 3. Illustrating (HSΣ)
Lemma 5.4. Let L be a lattice, let Σ be a subset of J(L). Then the following statements hold:
(i) If L satisfies (HS), thenL satisfies (HSΣ).
(ii) If Σis a join-seed of LandL satisfies (HSΣ), then Lsatisfies (HS).
Proof. (i) Let a, b, c, b0, b1 ∈ Σ satisfy the premise of (HSΣ). Observe that b′ =b∧(b0∨b1) = b and a∧(b′∨c) = a. Since a ≤ b∨c is minimal inb and b∧bi < b, it follows from the join-irreducibility ofa that there existsi < 2 such that one of the following inequalities holds:
a≤
b∧(a∨bi)
∨c
∧(bi∨c)∧(b∨b1−i), a≤
b∧(a∨bi)
∨c
∧(b0∨c)∧(b1∨c).
From the minimality ofbina≤b∨cit follows thatb≤a∨bi. Furthermore, in the first casea≤bi∨c, b∨b1−i while in the second casea≤b0∨c, b1∨c.
(ii) Let d (resp., e) denote the left hand side (resp., right hand side) of the identity (HS). Let p ∈ Σ such that p ≤ d, we prove that p ≤ e. Ifp ≤ b′ then p≤d∧b′=a∧b′, ifp≤cthenp≤a∧c, in both casesp≤e. Suppose from now on that pb′, c. Since Σ is a join-seed ofL, there areu≤b′ andv≤c in Σ such
that p≤u∨v is a minimal nontrivial join-cover. If u≤bi, for some i <2, then u≤b∧bi, whence
p≤a∧(u∨v)≤a∧ (b∧bi)∨c
≤e.
Suppose from now on thatub0, b1. Since Σ is a join-seed ofL, there areu0≤b0
and u1 ≤ b1 in Σ such that u ≤ u0∨u1 is a minimal nontrivial join-cover. By (HSΣ), there existsi < 2 such that u≤p∨ui and either p≤ui∨v, u∨u1−i or p≤u0∨v, u1∨v. In the first case,
p≤a∧(ui∨v)∧(u∨u1−i)∧
u∧(p∨ui)
∨v
≤e.
In the second case,
p≤a∧(u0∨v)∧(u1∨v)∧
u∧(p∨ui)
∨v
≤e.
Since every element ofL is a join of elements of Σ, we obtain that d≤e. Since
e≤dholds in any lattice, we obtain thatd=e.
Corollary 5.5. The lattice Co(T) satisfies (HS), for every chain(T,E).
Proof. We apply Lemma 5.4 to L = Co(T) together with the join-seed Σ = {{p} |p∈T}. Leta,b,c,b0,b1∈T such thata6=b, the inequality{a} ≤ {b} ∨ {c}
is minimal in b (thusa6=c), and{b} ≤ {b0} ∨ {b1}. Since Co(T,E) =Co(T,D), we may assume without loss of generality thatc⊳a⊳b. Furthermore, there exists i <2 such thatbEbi, whence{b} ≤ {a} ∨ {bi}. SinceT is a chain, either b1−iEa or aEb1−i. In the first case,{a} ≤ {bi} ∨ {c},{b} ∨ {b1−i}. In the second case, {a} ≤ {b0} ∨ {c},{b1} ∨ {c}.
HenceCo(T) satisfies (HSΣ). By Lemma 5.4,Co(T) satisfies (HS).
6. The Transitivity Lemma
The main purpose of the present section is to prove the following technical lemma, which provides a large supply of minimal coverings.
Lemma 6.1(The Transitivity Lemma). LetL be a lattice satisfying the identities (HS),(U),(B),(E), and (P), letΣ be a join-seed ofL, and let a,b,c,b0,b1∈Σ such that botha≤b∨c andb≤b0∨b1 are minimal nontrivial join-covers. Then there existsi <2 such the following statements hold:
(i) the inequalityb≤a∨biholds, and both inequalitiesb≤c∨bianda≤c∨bi
are minimal nontrivial join-covers;
(ii) one of the following two statements holds:
(ii.1) a≤bi∨c, b1−i∨b and, ifa6=b1−i, then the inequalitya≤b0∨b1 is a minimal nontrivial join-cover;
(ii.2) a≤b0∨c, b1∨c and, ifa6=b1−i, then the inequalitya≤b1−i∨c is a minimal nontrivial join-cover.
The situation may be partly viewed on Figure 3.
Proof. It follows from Lemma 5.4 that there existsi <2 such that
b≤a∨biand either a≤bi∨c, b1−i∨bor a≤b0∨c, b1∨c. (6.1) Sinceb≤bi∨cis a nontrivial join-cover and Σ is a join-seed ofL, there arex≤bi
andc′≤cin Σ such thatb≤x∨c′ is a minimal nontrivial join-cover. By applying (Bj) to the inequalitiesb≤bi∨b1−i, x∨c′ and observing thatbbi=bi∨x, we
