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X-maximal congruences and relevant sets for algebras

Staruch B., Staruch B.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2016

Vol. 21

93--104

We introduce and investigate X-maximal congruences and relevant sets for a given algebra. We describe interrelations among these concepts and atomicity of the congruence lattice and the number of atoms. We also investigate subdirect decomposition of algebras into subdirectly irreducible factors.

On the directly and subdirectly irreducible many-sorted algebras

Climent-Vidal J., Soliveres-Tur J.

Demonstratio Mathematica

2015

Vol. 48, nr 1

1--12

A theorem of single-sorted universal algebra asserts that every finite algebra can be represented as a product of a finite family of finite directly irreducible algebras. In this article, we show that the many-sorted counterpart of the above theorem is also true, but under the condition of requiring, in the definition of directly reducible many-sorted algebra, that the supports of the factors should be included in the support of the many-sorted algebra. Moreover, we show that the theorem of Birkhoff, according to which every single-sorted algebra is isomorphic to a subdirect product of subdirectly irreducible algebras, is also true in the field of many-sorted algebras.