Kalle Kaarli

Polynomial completeness in algebraic systems

ALGEBRAS, LATTICES, AND VARIETIES Algebras, Languages, Clones, Varieties Congruence Properties CHARACTERIZATIONS OF EQUIVALENCE LATTICES Introduction Arithmeticity Compatible Function Lifting PRIMALITY AND GENERALIZATIONS Primality and Functional Completeness Near Unanimity Varieties Arithmetical Varieties Generalizations of Primality Categorical Equivalence AFFINE COMPLETE VARIETIES Introduction and Instructive Examples General properties Varieties with a Finite Residual Bound Locally Finite Affine Complete Varieties POLYNOMIAL COMPLETENESS IN SPECIAL VARIETIES Strictly Locally Affine Complete Algebras Modules Lattices Algebras Based on Distributive Lattices Semilattices Miscellaneous Results

A CHARACTERIZATION OF THE INVERSE MONOID OF BI-CONGRUENCES OF CERTAIN ALGEBRAS

This paper provides an abstract characterization of the inverse monoids that appear as monoids of bi-congruences of finite minimal algebras generating arithmetical varieties. As a tool, a matrix construction is introduced which might be of independent interest in inverse semigroup theory. Using this construction as well as Ramseys theorem, we embed a certain kind of inverse monoid into a factorizable monoid of the same kind. As noticed by M. Lawson, this embedding entails that the embedded finite monoids have finite F-unitary cover.

Endoprimal abelian groups

A group A is said to be endoprimal if its term functions are precisely the functions which permute with all endomorphisms of A . In this paper we describe endoprimal groups in the following three classes of abelian groups: torsion groups, torsionfree groups of rank at most 2, direct sums of a torsion group and a torsionfree group of rank 1.

On varieties generated by functionally complete algebras

In [4] A. F. Pixley stated a problem due to A. L. Foster: is a functionally complete algebra having no non-trivial subalgebras necessarily categorical? In this paper we show that, in general, the answer to this question is negative. However the answer becomes affirmative, if we replace the word “non-trivial” by the word “proper”.

On endoprimality of torsion-free abelian groups of rank 3

This paper gives a complete description of the behaviour of torsion-free abelian groups of rank 3 with respect to endoprimality.

On Minimal Varieties of Near-Rings

It is well known that every minimal variety of associative rings is generated by a finite ring of prime order, in particular it is locally finite. In this paper we focus at locally finite minimal varieties of near-rings. They are exactly the varieties generated by finite strictly simple near-rings. We prove that every finite strictly simple near-ring is either a near-ring with the so-called trivial multiplication on a group of prime order or a finite planar near-ring whose additive group is elementary abelian. We describe the multiplicative subgroups of Galois fields which lead to strictly simple Ferrero near-rings and prove that in this way one obtains all finite strictly simple near-rings satisfying the identity xyz= yxz. In particular, this proves that the finite, strictly simple near-rings with non-prime order are abundant.

Strictly locally order affine complete lattices

We call a latticeL strictly locally order-affine complete if, given a finite subsemilatticeS ofLn, every functionf: S →L which preserves congruences and order, is a polynomial function. The main results are the following: (1) all relatively complemented lattices are strictly locally order-affine complete; (2) a finite modular lattice is strictly locally order-affine complete if and only if it is relatively complemented. These results extend and generalize the earlier results of D. Dorninger [2] and R. Wille [9, 10].

SUBALGEBRAS OF THE SQUARES OF FINITE MINIMAL MAJORITY ALGEBRAS

This paper provides an abstract characterization of the monoids that appear as monoids of subalgebras of the square of finite minimal algebras admitting a majority term. As a tool, a matrix construction similar to the one introduced in [A characterization of the inverse monoid of bi-congruences of certain algebras, Int. J. Algebra Comput.6 (2009) 791–808] is used.

Unary polynomials on a class of semidirect products of finite groups

We describe unary polynomial functions on nite groups G that are semidirect products of an elementary abelian group of exponent p and a cyclic group of prime order q , p ≠ q .

ON RADICAL THEORY OF NON-ZEROSYMMETRIC NEAR-RINGS

Abstract This paper is based on an expository talk given at the International Conference on Radicals and Rings ICOR 97. Our aim is not to give a comprehensive survey of the subject. We rather wish to present some selected results and on their basis to analyze the development of radical theory of near-rings, emphasizing the role of non-zerosymmetric near-rings. Besides we present some new results on Jacobson type radicals of non-zerosymmetric near-rings.