László Márki

Semi-abelian categories

Abstract The notion of semi-abelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abelian-group and module theory. In modern terms, semi-abelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to “old” exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semi-abelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar non-abelian structures.

Ideals and Clots in Pointed Regular Categories

We clarify the relationship between ideals, clots, and normal subobjects in a pointed regular category with finite coproducts.

LIMIT PRESERVATION PROPERTIES OF THE GREATEST SEMILATTICE IMAGE FUNCTOR

We study what kinds of limits are preserved by the greatest semilattice image functor from the category of all semigroups to its subcategory of all semilattices.

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.

Locally semisimple coverings

Abstract We introduce the notion of a locally semisimple covering with respect to a class X of objects in a given exact category, and classify these coverings in terms of internal-category actions inside X . This is similar to the classification of ordinary covering spaces of a “good” topological space in terms of its fundamental-group actions. Locally semisimple coverings are essentially the same as the maps with fibres in X ; examples are e.g. ring and group homomorphisms with semisimple kernels (with respect to a given radical) or continuous maps of compact Hausdorff spaces with totally disconnected fibres.

Matrix representations of finitely generated Grassmann algebras and some consequences

We prove that the m-generated Grassmann algebra can be embedded into a 2m−1×2m−1 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for n × n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.

Kurosh–Amitsur Radicals via a Weakened Galois Connection

Abstract In the present paper we consider a correspondence weaker than Galois connection and prove that this produces Kurosh–Amitsur radicals in a very general setting including all universal classes of Ω-groups. As a framework we introduce a simple combinatorial structure which uses mappings between complete lattices.

Affine complete semilattices

This paper presents a structural characterization of affine complete and locally affine complete semilattices.

COMPLETELY 0-SIMPLE SEMIGROUPS OF LEFT QUOTIENTS OF A SEMIGROUP

In this paper we investigate the class of all completely 0-simple semigroups of left quotients of a given semigroup S. We show that (if the class is non-empty) it has a ‘greatest’ member ∈S which is in a sense the free completely 0-simple semigroup on S, and describe how the other members can be obtained as homomorphic images of .