Congruence lattices of locally finite algebras

It is shown that there exist algebraic lattices that cannot be represented as the congruence lattice of a locally finite algebra. Received August 11, 2004; accepted in final form January 6, 2005.     

Congruence lattices of finite algebras and factorizations of groups

Making use of deep results from group theory we prove that if a finite algebra has permutable congruences and its congruence lattice is M_n, then n-1 is a prime power.     

The order dimension of relatively free lattices

If V is a nontrivial lattice variety and is an infinite cardinal, then the order dimension of FV() is the smallest cardinal such that 2.This research was supported in part by NSF Grants DMS 87-03540 (Nation) and DMS 86-01576 (Schmerl).     

Elementary equivalence and relatively free products of lattices

It is shown that if is a non-trivial variety of lattices, then there existA, B, C ∈ such thatB≡C but notA*B≡A*C. Except in the case when is the variety of all distributive lattices,A can be taken to consist of just one element. For the varietyD of all distributive lattices, it is shown that for anyB, C and any finiteA, B≡C if and only ifA*B≡A*C. The work of the first author was supported in part by NSF Grant GP-29129, and the work of the second author by a grant from the N.R.C. of Canada. Presented by C. C. Chang.     

Congruence lattices of function lattices

Thefunction lattice L ^P is the lattice of all isotone maps from a posetP into a latticeL.D. Duffus, B. Jónsson, and I. Rival proved in 1978 that for afinite poset P, the congruence lattice ofL ^P is a direct power of the congruence lattice ofL; the exponent is |P|.This result fails for infiniteP. However, utilizing a generalization of theL ^P construction, theL[D] construction (the extension ofL byD, whereD is a bounded distributive lattice), the second author proved in 1979 that ConL[D] is isomorphic to (ConL) [ConD] for afinite lattice L.In this paper we prove that the isomorphism ConL[D](ConL)[ConD] holds for a latticeL and a bounded distributive latticeD iff either ConL orD is finite.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

On independent systems in unitary relatively free algebras

This paper introduces a family of polynomial systems of quite general form (triangular systems) in a unitary relatively free algebra F with associative degrees. These systems generate a subalgebra that is isomorphic to the algebra F. The proof of independency is based on some simple algebro-geometric considerations.     

On the Hilbert series op relatively free algebras

Subvarieties of the variety of algebras genelaetu by the algebra of the upper triangular matrices are studied in this paper. The main result is that the relatively free algebras have rational Hilbert (or Poincarê) series when the base field is infinite. As a corollary, in characteristic 0 the rationality is obtained for varieties not containing the algebra of all 2 × 2 matrices. Some of the results are transfered to Lie and Jordan algebras.     

The GK dimension of relatively free algebras of PI-algebras

We prove a strict relation between the Gelfand–Kirillov (GK) dimension of the relatively free (graded) algebra of a PI-algebra and its (graded) exponent. As a consequence we show a Bahturin–Zaicev type result relating the GK dimension of the relatively free algebra of a graded PI-algebra and the one of its neutral part. We also get that the growth of the relatively free graded algebra of a matrix algebra is maximal when the grading is fine. Finally we compute the graded GK dimension of the matrix algebra with any grading and the graded GK dimension of any verbally prime algebra endowed with an elementary grading.     

Partition theorems for unary algebras

Those unary algebras (A, ?) (A a set, ? a mapping from A into A) are characterized for which an analog of Ramsey's theorem holds.     

Endomorphisms of relatively free algebras with weak exchange properties

The structure of the endomorphism monoid of a stable basis algebra A is described. It is shown to be an abundant monoid; the subsemigroup of endomorphisms of finite rank has a regular semigroup of left quotients.     

Quasiidentities of finite unary algebras

Translated from Algebra i Logika, Vol. 28, No. 5, pp. 493–512, September–October, 1989.