On maximal sublattices of finite lattices (original) (raw)

Maximal sublattices of finite distributive lattices

Algebra Universalis, 1996

Algebraic properties of lattices of quotients of nite posets are considered. Using the known duality between the category of all nite posets together with all order-preserving maps and the category of all nite distributive (0; 1)-lattices together with all (0; 1)-lattice homomorphisms, algebraic and arithmetic properties of maximal proper sublattices and, in particular, Frattini sublattices of nite distributive (0; 1)-lattices are thereby obtained.

Maximal sublattices and Frattini sublattices of bounded lattices

Journal of the Australian Mathematical Society, 1997

We investigate the number and size of the maximal sublattices of a finite lattice. For any positive integerk, there is a finite lattice L with more that ]L]ksublattices. On the other hand, there are arbitrary large finite lattices which contain a maximal sublattice with only 14 elements. It is shown that every bounded lattice is isomorphic to the Frattini sublattice (the intersection of all maximal sublattices) of a finite bounded lattice.

Finite sublattices of a free lattice

Transactions of the American Mathematical Society, 1982

Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice. Introduction. The aim of this paper is to show that a finite semidistributive lattice satisfying Whitman's condition can be embedded in a free lattice. This confirms a conjecture of Bjarni Jónsson, and indeed our proof will follow the line of approach originally suggested by him in unpubhshed notes around 1960. This approach was later described in Jónsson and Nation [15], to which the reader is referred for a more complete discussion of the background material and related work than will be given here. Let us recall some relevant definitions and results. A finite sublattice of a free lattice satisfies Whitman's condition [23] (W) ab < c + d iff a < c + d or b < c + d or ab < c or ab < d and the semidistributive laws introduced by Jónsson [12] (SDV) u = a + b = a + c implies u = a + be, (SDA) u = ab = ac implies u = a(b + c). As in [15], we shall refer to a finite lattice satisfying these three conditions as an S-lattice. We will often use the following (equivalent) form of the semidistributive laws [14]. (SDV) u = 2 a,,-2 bj implies u = 2,-2, a,bp (SDA) u = n a,: = LI bj implies u = II, II, (a,. + bj). Let J(L) denote the set of nonzero join-irreducible elements in a finite lattice L. Every element p G J(L) has a unique lower cover, which we will denote by p^. If />" G J(L), letpt<1 = (pf)+. Dually, M(L) denotes the set of nonunit meet-irreducible elements of L, and for y G M(L), y* >y. In a finite semidistributive lattice there is a bijection between J(L) and M(L), p <-» k(p) = 2 {x G L: x > pt andx £p}. (In fact, A. Day has shown that this characterizes finite semidistributive lattices [4].) Now px = p, iff x > p" and x %p, and, by (SDA), p/c(p) = p"; thus k(p) is the largest element in L with this property. Repeatedly we will use the following observations.

Lattices That Are the Join of Two Proper Sublattices

Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in Sub L? That is, when do there exist proper sublattices A, B such that L = A ∨ B? In particular, could it be that every nontrivial lattice has this property, in which case every element of Sub L would be finitely join reducible? The authors would like to thank David Wasserman and M. E. Adams for bringing this problem to our attention, along with some elementary observations and helpful discussion. Let us mention a related problem. Recall the following result of Tom Whaley [4].

Chapter VI: Finite Distributive Lattices

1983

We continue our study of the finite ideals of 2 in this chapter by showing that every finite distributive lattice is isomorphic to an ideal of Q>. This result is proved using techniques extending those introduced in Chap. V. Different trees are used, and we introduce tables which provide reduction procedures from the top degree of the ideal; these tables are obtained from representations of distributive lattices. As an application, we show that the set of minimal degrees forms an automorphism base for 2. Many of the applications which we obtain in later chapters from the complete characterization of the countable ideals of 2 can be obtained from the fact that all countable distributive lattices are isomorphic to ideals of 2. We use Exercise 4.17 of this chapter to indicate how to obtain the characterization of distributive ideals of 2. This exercise allows the reader to proceed directly to Chap. VIII.2 from the end of this chapter. The results of Appendix B.I are needed for this chapter.

The fundamental theorem of finite semidistributive lattices

Selecta Mathematica, 2021

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set X with some additional structure, such that L is isomorphic to the admissible subsets of X ordered by inclusion; in this case, X and its additional structure are uniquely determined by L." The additional structure on X is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices.

The lattice of subsemilattices of a semilattice

Algebra Universalis, 1994

This note makes two observations about lattices of subsemilattices. First, we establish relationship between direct decompositions of such lattices and ordinal sum decompositions of semilattices. Then we give a characterization of the subsemilattice-lattices.

The structure of congruence lattices of finite semilattices

Algebra and Logic, 1996

G. ghitomirskii gave a description of congruence lattices of semilattices in the second-order language. Here we describe finite lattices belonging to this class in terms of properties shared by coatoms (minimal nonidentity elements of a lattice}. As a consequence, a characterization of finite semiIattices with isomorphic congruence lattices is obtained. LEMMA [2]. The congruence lattice of a finite lower semilattice P is dually isomorphic to Sub+ (P). Note that if a semilattice P has a greatest element 1p, then the lattice Sub+ (P) coincides with Subv (P), the lattice of upper subsemiJattices of P with Op. Lattices of the form Subv (P) were described in , where it was shown that the property of having no special sequences of atoms, called cycles, is equivalent to being lower bounded (see also ). It was noted in [8] that lattices Sub+(P) satisfy a similar condition, and i = O, n -1. For a biatomic lattice, this definition coincides with the well-known notion of a C-cycle (see [10, ll D. For our further reasoning, we need the following: LEMMA 1.2 (see ). Let L be a finite biatomic lattice which satisfies (D2) and has no cycles and let ]P be its socle. Then the lattice S(]?) of subsystems of 1? is isomorphic to L. Specifically, if the socles 171 and ~2 of two such lattices L1 and L2 are isomorphic, then LI ~ L2.