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L. Next one checks that it is subdirectly irreducible: the monolith of L collapses each of the sets fa s : s 2 Zg;:::;f n s :s2Z gto a single point.
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Context 1
... L denote the lattice in Figure 1, as labeled. This lattice is most naturally pictured on the surface of a sphere. ...
Context 2
... same technique has been used for all the gures in this paper. Thus L= is given in Figure 2, and is easily seen to be a subdirect product of two copies of the lattice F in Figure 3. Lemma 2. Let L be the lattice in Figure 1. Then every nitely generated, subdirectly irreducible lattice i n V L is in HSL. ...
Citations
... Our first examples of inherently nonfinitely based lattices were inspired by the lattices used in Nation [32] to refute the Finite Height Conjecture, while the general method we use is obtained from that found in Baker, McNulty, and Werner [4] modified by a variant of the doubling construction of Alan Day [11]. ...
We give a general method for constructing lattices L whose equational theoriesare inherently nonfinitely based. This means that the equational class (that is, thePreprint submitted to Elsevier Preprint 21 March 2002## ### #Page 2 of 42Go BackFull ScreenPrintClosevariety) generated by L is locally finite and that L belongs to no locally finitefinitely axiomatizable equational class. We also provide an example of a latticewhich fails to be inherently nonfinitely based but...
A variety is primitive if every subquasivariety is equational, i.e. a subvariety. In this paper, we explore the connection between primitive lattice varieties and Whitman’s condition [Formula: see text]. For example, if every finite subdirectly irreducible lattice in a locally finite variety [Formula: see text] satisfies Whitman’s condition [Formula: see text], then [Formula: see text] is primitive. This allows us to construct infinitely many sequences of primitive lattice varieties, and to show that there are [Formula: see text] such varieties. Some lattices that fail [Formula: see text] also generate primitive varieties. But if [Formula: see text] is a [Formula: see text]-failure interval in a finite subdirectly irreducible lattice [Formula: see text], and [Formula: see text] denotes the lattice with [Formula: see text] doubled, then [Formula: see text] is never primitive.
This book started with Lattice Theory, First Concepts, in 1971. Then came General Lattice Theory, First Edition, in 1978, and the Second Edition twenty years later. Since the publication of the first edition in 1978, General Lattice Theory has become the authoritative introduction to lattice theory for graduate students and the standard reference for researchers. The First Edition set out to introduce and survey lattice theory. Some 12,000 papers have been published in the field since then; so Lattice Theory: Foundation focuses on introducing the field, laying the foundation for special topics and applications. Lattice Theory: Foundation, based on the previous three books, covers the fundamental concepts and results. The main topics are distributivity, congruences, constructions, modularity and semimodularity, varieties, and free products. The chapter on constructions is new, all the other chapters are revised and expanded versions from the earlier volumes. Over 40 "diamond sections", many written by leading specialists in these fields, provide a brief glimpse into special topics beyond the basics. Lattice theory has come a long way... For those who appreciate lattice theory, or who are curious about its techniques and intriguing internal problems, Professor Grätzer's lucid new book provides a most valuable guide to many recent developments. Even a cursory reading should provide those few who may still believe that lattice theory is superficial or naive, with convincing evidence of its technical depth and sophistication." Garrett Birkhoff (Bulletin of the American Mathematical Society) Grätzer's book General Lattice Theory has become the lattice theorist's bible." (Mathematical Reviews)
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.
In this paper it is shown that under certain general conditions, if the amalgamation class of a lattice variety contains
a member which does not have a 2-congruence, then the amalgamation class is not closed under ultrapowers or direct products.
In particular, the amalgamation class is not first order axiomatizable.
It is shown that the variety generated by a finite modular lattice has only finitely many locally finite covers.
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. In [12], I. Rival showed that if L is a maximal sublattice of a distributive lattice K with |K| > 2, then |K|#(3/2)|L|. In [1] the authors took up the question for more general varieties. In particular a 14 element lattice was constructed which is a maximal sublattice of an infinite lattice. The question of how small such a lattice could be was raised. In this paper we will use the term big lattice to mean a finite lattice which is a maximal sublattice in an infinite lattice, and small lattice for a finite lattice which is not big. Our main result provides an algorithm for determining whether a given finite lattice is big. This allows us to give many interesting examples of both big and small lattices. We will show that M 3 is big but no smaller lattice is, answering the question mentioned above. On the other hand, N 5 is small. Using this algorithm, we produce a complete list of all 145 minimal big lattices. The minimal big lattices (up to dual isomorphism) are denoted by G i for 1 # i # 81, and are drawn in the Figures 30--38 in Section 21. A finite lattice is big if and only if it contains some G i or G d i as a sublattice. 1. The Plan The outline of the paper, after the preliminaries, goes as follows. Section 3 proves that a superlattice of a big lattice is big. Section 4 proves that a linear sum of small lattices is small. Section 5 gives some examples of small lattices. Section 6 gives the main construction which will be used to show that a lattice is big. This construction involves glu...
