In this paper we introduce a notion of divisibility and primality on k-ary trees and we nd a relation between indecomposable preex code and prime trees. This relation allows to work on trees instead than directly on preex codes. In this way we can prove a density theorem for indecomposable preex codes and we can give an algorithm to test the indecomposability of a maximal preex code.

First a problem of Ralph McKenzie is answered by proving that in a nitely directly representable variety every directly indecomposable algebra must be nite. Then we show that there is no local proof of the fundamental theorem of Abelian algebras nor of H. P. Gumm's permutability results. This part may also be of interest for those investigating non-modular Abelian algebras. Finally we provide a...

The subject of this paper are general properties of direct sum decompositions of automata. Using certain properties of the lattice Sub(A) of subautomata of an automaton A and its Boolean part, lattices of direct sum congruences and direct sum decompositions of A are characterized. We show that every automaton A can be represented as a direct sum of direct sum indecomposable automata, and that t...

The aim of this paper is double. From one side we survey the knowledge we have acquired these last ten years about the lattice of all λ-theories (= equational extensions of untyped λ-calculus) and the models of lambda calculus via universal algebra. This includes positive or negative answers to several questions raised in these years as well as several independent results, the state of the art ...

A module is called absolutely indecomposable if it is directly indecomposable in every generic extension of the universe. We want to show the existence of large abelian groups that are absolutely indecomposable. This will follow from a more general result about R-modules over a large class of commutative rings R with endomorphism ring R which remains the same when passing to a generic extension...

If A is a primitive matrix, then there is a smallest power of A (its fully indecomposable exponent) that is fully indecomposable, and a smallest power of A (its strict fully indecomposable exponent) starting from which all powers are fully indecomposable. We obtain bounds on these two exponents.

For a given complete lattice L, we investigate whether L can be decomposed as a direct product of directly indecomposable lattices. We prove that this is the case if every element of L is a join of join-irreducible elements and dually, thus extending to non-algebraic lattices a result of L. Libkin. We illustrate this by various examples and counterexamples.

For a given complete lattice L, we investigate whether L can be decomposed as a direct product of directly indecomposable lattices. We prove that this is the case if every element of L is a join of join-irreducible elements and dually, thus extending to non-algebraic lattices a result of L. Libkin. We illustrate this by various examples and counterexamples.

A matching is indecomposable if it does not contain a nontrivial contiguous segment of vertices whose neighbors are entirely contained in the segment. We prove a Ramsey-like result for indecomposable matchings, showing that every sufficiently long indecomposable matching contains a long indecomposable matching of one of three types: interleavings, broken nestings, and proper pin sequences.

In this paper we study the Cartan matrix associated to Ext-projective stratifying system induced by a basic and τ-rigid object M in mod(A) means of g-vectors indecomposable direct summands M. particular show that group module can be calculated directly using these vectors. Moreover characterise systems coming from modules have diagonal matrix.