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Algebraic Aspects of Conditional Independence and Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Thomas Kahle, Johannes Rauh, Seth Sullivant
According to Proposition 1.2.2, the corresponding decomposition operation for ideals is to write ideals as the intersection of other ideals. However, for general ideals, the situation is much more complicated than for varieties. The situation simplifies for radical ideals (which are in a one-to-one correspondence with varieties). This case is discussed next. The general case is summarized afterwards.
Banach Algebras
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A maximal ideal of A $ \mathcal{A} $ is a proper ideal that is not contained in a larger proper ideal. Here is an interesting and illuminating example.
Two notions of MV-algebraic semisimplicity relative to fixed MV-chains
Published in Journal of Applied Non-Classical Logics, 2022
Celestin Lele, Jean B. Nganou, Jean M. Wagoum
Note that every -maximal ideal is prime. Indeed, if M is -maximal, then for some . It follows that A/M is isomorphic to a sub-MV-algebra of . Since is an MV-chain, then A/M is an MV-chain, which implies that M is prime. In addition it is clear from the definition that a proper ideal M of A is -maximal if and only if A/M is isomorphic to a sub-MV-algebra of .
A set-theoretic proof of the representation of MV-algebras by sheaves
Published in Journal of Applied Non-Classical Logics, 2022
Alejandro Estrada, Yuri A. Poveda
In order to prove injectivity, it suffices to show that Ker. Let Ker, then , that is, for all . This implies that , i.e. for all . By Proposition 2.2, we know that the intersection of all the prime ideals is the trivial ideal. So a = 0 and is injective.