Maximal sublattices and Frattini sublattices of bounded lattices (original) (raw)
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On maximal sublattices of finite lattices
Discrete Mathematics, 1999
We discuss the possible structures for and mutual relationships between a finite distributive lattice L, a maximal sublattice M of L and the corresponding 'remainder' R = L\M with the aid of Birkhoff duality, and contrast the results with the analogous situations for a general finite lattice L.
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.
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].
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.
A note on subortogonal lattices
ArXiv, 2016
It is shown that, given any kkk-dimensional lattice Lambda\LambdaLambda, there is a lattice sequence Lambdaw\Lambda_wLambdaw, winmathbbZw\in \mathbb ZwinmathbbZ, with suborthogonal lattice LambdaosubsetLambda\Lambda_o \subset \LambdaLambdaosubsetLambda, converging to Lambda\LambdaLambda (unless equivalence), also we discuss the conditions for faster convergence.
Lattices with large minimal extensions
Algebra Universalis - ALGEBRA UNIV, 2001
This paper characterizes those finite lattices which are a maximal sublattice of an infinite lattice. There are 145 minimal lattices with this property, and a finite lattice has an infinite minimal extension if and only if it contains one of these 145 as a sublattice.
1983
We completely characterize the finite ideals of <2) in this chapter as the set of all finite lattices. It is not known whether all finite lattices have finite homogeneous lattice tables, so we replace these tables with weakly homogeneous sequential lattice tables which are possessed by all finite lattices. We extend the methods of Chap. VI, using such tables to embed finite lattices as ideals of 9>. This embedding theorem is used to locate decidable fragments of Th(^); the V 2-theory of Q) is decidable, but the V 3theory of 3) is undecidable. Results from Appendices A.2 and B.2 are used in this chapter.