On the directly and subdirectly irreducible many-sorted algebras

• Autorzy: Climent-Vidal J. , Soliveres-Tur J.
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

many-sorted algebras; support of a many-sorted algebra; directly irreducible many-sorted algebra; subdirectly irreducible many-sorted algebra

PL

algebra; twierdzenie Birkhoffa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2015
• Tom: Vol. 48, nr 1
• Strony: 1--12
• Opis fizyczny: Bibliogr. 5 poz.

Twórcy

autor

Climent-Vidal J.

• Universidad De Valencia, Departamento De Lógica Y Filosofía De La Ciencia, E-46010 Valencia, Spain

autor

Soliveres-Tur J.

• Universidad De Valencia, Departamento De Lógica Y Filosofía De La Ciencia, E-46010 Valencia, Spain

Bibliografia

• [1] G. Birkhoff, Subdirect unions in universal algebra, Amer. Math. Soc. 50 (1944), 764–768.
• [2] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Springer-Velag, 1981.
• [3] J. Climent, L. Fernandino, On the relation between many-sorted uniform 2-algebraic closure operators and many-sorted algebras, Collect. Math. 40 (1990), 93–101.
• [4] J. Climent, J. Soliveres, On many-sorted algebraic closure operators, Math. Nachr. 266 (2004), 81–84.
• [5] J. Climent, J. Soliveres, When is the insertion of the generators injective for a sur-reflective subcategory of a category of many-sorted algebras?, Houston J. Math. 35 (2009), 363–372.