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On the directly and subdirectly irreducible many-sorted algebras

100%

Climent-Vidal J. , Soliveres-Tur J.

Demonstratio Mathematica

2015

tom 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.

X-maximal congruences and relevant sets for algebras

84%

Staruch B. , Staruch B.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2016

tom 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.