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Demonstratio Mathematica

Editor-in-Chief: Vetro, Calogero

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CiteScore 2017: 0.28

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2391-4661
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Volume 48, Issue 1

Issues

On the Directly and Subdirectly Irreducible Many-Sorted Algebras

J. Climent Vidal / J. Soliveres Tur
Published Online: 2015-03-11 | DOI: https://doi.org/10.1515/dema-2015-0001

Abstract

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.

Keywords: many-sorted algebra; support of a many-sorted algebra; directly irreducible many-sorted algebra; subdirectly irreducible many-sorted algebra

References

  • [1] G. Birkhoff, Subdirect unions in universal algebra, Amer. Math. Soc. 50 (1944), 764-768.Google Scholar

  • [2] S. Burris, H. P. Sankappanavar, A Course in Universal Algebra, Springer-Velag, 1981.CrossrefGoogle Scholar

  • [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.Google Scholar

  • [4] J. Climent, J. Soliveres, On many-sorted algebraic closure operators, Math. Nachr. 266 (2004), 81-84.Google Scholar

  • [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. Google Scholar

About the article

Received: 2013-11-04

Revised: 2014-04-08

Published Online: 2015-03-11

Published in Print: 2015-03-01


Citation Information: Demonstratio Mathematica, Volume 48, Issue 1, Pages 1–12, ISSN (Online) 2391-4661, ISSN (Print) 0420-1213, DOI: https://doi.org/10.1515/dema-2015-0001.

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© by J. Climent Vidal. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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[2]
Murat Bekar, Fouzi Hathout, and Yusuf Yayli
Asian-European Journal of Mathematics, 2017, Page 1850008

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