Avoidable structures, II: finite distributive lattices and nicely ...

30 W. DZIOBIAK, J. JEŽEK, AND R. MCKENZIE
Proof. (1): Each of the five lattices is avoidable in 〈D,≤ 2 〉, since the first four
of them constitute a 0,1-anti-chain together with the lattices J 2 n (n ≥ 0), and
the last constitutes a 0,1-anti-chain together with D n (n ≥ 1). Thus every finite
distributive lattice containing a 0,1-sublattice isomorphic to one of the five lattices
is avoidable in 〈D,≤ 2 〉. Let L be a finite distributive lattice containing no 0,1-
sublattice isomorphic to one of the five lattices. Since L 0,1-avoids the first four
lattices, L is of the form 1+ o K+ o 1 for a finite lattice K (or L is the two-element
chain). Since it 0,1-avoids I 0 , the lattice K avoids B 2,3 and hence L is of the form
C 0 + ′ o ...+ ′ o C k−1 where k ≥ 1, C 0 and C k−1 are two-element lattices and every
C i is either D 1 or the two-element lattice. Clearly, all such lattices L are 0,1-
embeddable into all sufficiently large lattices D n , into all sufficiently large lattices
J 0 n, into all sufficiently large lattices J 1 n and into all sufficiently large lattices J 2 n.
Thus it follows from Theorem 7.1(1) that L is unavoidable in 〈D,≤ 2 〉.
(2): Thethreelattices areavoidable in〈D,≤ 2 〉, sincethefirstofthem constitutes
a 1-anti-chain (i.e., anti-chain with respect to ≤ 1 ) together with the lattices B 2,3 + ′ o
D n and the last two constitute a 1-anti-chain together with the lattices J n + o 1. Let
L be a finite distributive lattice containing no 1-sublattice isomorphic to one of the
three lattices. Since D 1 ≰ 1 L, L is of the form K+ o 1 for a finite lattice K (or L is
trivial). Since K avoids both 1+ o B 2,3 and I 3 , L is of the form C 0 + ′ o...+ ′ oC k−1
wherek ≥ 1, C k−1 isthetwo-elementlattice, C 0 isasublatticeofB 2,3 andeveryC i
with i > 0 is either D 1 or the two-element lattice. (To see that C 0 is a sublattice of
B 2,3 , usethefactsthat1+ o B 2,3 ≤ B 2,4 andI 3 ≰ K.) Clearly, allsuchlatticesLare
1-embeddable into all sufficiently large lattices B 2,3 + ′ o D n and into all sufficiently
large lattices J n + o 1. Thus it follows from Theorem 7.1(2) that L is unavoidable
in 〈D,≤ 1 〉.
(3): This statement is dual to (2). •
It follows that the quasi-ordered sets 〈D,≤ 2 〉, 〈D,≤ 1 〉 and 〈D,≤ 0 〉 contain no
least complete infinite anti-chain. The quasi-ordered set 〈D,≤〉 is thus rather exceptional.
References
[1] W. Dziobiak, J. Ježek and R. McKenzie, Avoidable structures, I: Finite ordered sets,
semilattices and lattices (manuscript).
[2] J. B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, J.
Combinatorial Theory Ser. A 13 (1972), 297–305.
Department of Mathematics, University of Puerto Rico, Mayagüez Campus, Mayagüez
PR 00681–9018, USA
E-mail address: w.dziobiak@gmail.com
Department of Mathematics, Charles University, Prague, Czech Republic
E-mail address: jezek@karlin.mff.cuni.cz
Department of Mathematics, Vanderbilt University, Nashville, TN 37235, USA
E-mail address: ralph.mckenzie@vanderbilt.edu