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Subdirectly Irreducible Members of Products of Lattice Varieties
... This result has many applications. For instance, we apply it in[7] to show that DoD and MoD are varieties; we use it in [8] to show that every lattice in VoW can be embedded into a subdirectly irreducible member. This paper starts our investigation of products of lattice varieties. ...
I think it is hopeless to try to give a concise account of the over 300 publications of George Grätzer and E. Tamás Schmidt.
There is a danger of getting lost in too many details (for instance, mentioning Grätzer [G35], where the concept of Mal’cev
condition was introduced and named). Hence, I will be guided by the following five restrictions. (1) I will discuss only lattice
theoretic results. (2) I will try to concentrate on their deepest results, (3) on results with the largest impact, and (4)
on series of papers. (5) I will give preference to joint or at least partially joint works. Even then, I will be able to cover
only a small part of their results satisfying these five criteria.
We show that every lattice variety can be generated by a special type of subdirectly irreducible lattice. As an appli-cation we prove that there are only tr !vial solutions of the equation X. Y = X v Y for lattice varieties X and Y. The special case X = Y was earlier considered by A. Day.
New appendices by the author with B.A. Davey, R. Freese, B. Ganter, M. Greferath, P. Jipsen, H.A. Priestley, H. Rose, E. T. Schmidt, S. E. Schmidt, F. Wehrung, and R. Wille.
In [5], Mal'cev generalized the group theoretical results of H. Neumann (see [6] Chapter 2) to produce the notion of the product, of two subclasses of a given variety of algebras, Following the group theorectic example, members of were called extensions of algebras in by algebras in
Let V be a variety (equational class) of lattices. We denote by FV(to) the free _V lattice on ~o generators. For elements a < b in FV(to), the interval [a, b] is a sublattice of FV(to), and the variety _W generated by this interval is a subvariety of V. We first investigate the possible varieties _W that can arise in this way from a given _V. Of particular interest are those varieties _V for which the variety generated by every nontrivial interval in FV(to) is again V. We say such varieties have generic intervals. We then discuss the amalgamation property and some of its weaker relatives. The main result of this section is that if a variety _V has the amalgamation property or any of these weakened versions, then _V must have generic intervals. We conclude the paper with a number of examples which flesh out these definitions and results. This includes solutions to some problems posed by B. J6nsson in [16] and provides an example of a variety of lattices without the amalgamation property which has the two element chain in its amalgamation class (cf. Problem V.9 of [6]).
Multiplication of classes of algebraic systems, Siberian Math 5 4 ~ 267; English transl., The metamathematics of algebraic systems
- A I Mal
A. I. Mal'cev, Multiplication of classes of algebraic systems, Siberian Math. J. 8 (1967), 2 5 4 ~ 267; English transl., The metamathematics of algebraic systems, North-Holland, Amsterdam, 1971, pp. 422-446
General lattice theory (Russian translation
- G Gratzer
G. Gratzer, General lattice theory, Pure and Appl. Math. Series, Academic Press, New York;
Mathematische Reihe, Band 52, Birkhauser Verlag, Basel; Akademie Verlag, Berlin, 1978.
(Russian translation: "MIR", Moscow, 1982.)
Algebras whose congruence lattices are distributive
B. J6nsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-
121.