Congruences and ideals in pseudoeffect algebras |
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Authors: | Hai-Yang Li Sheng-Gang Li |
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Affiliation: | (1) Department of Mathematics, Shaanxi Normal University, 710062 Xi’an, People’s Republic of China;(2) School of Science, Xi’an Polytechnic University, 710048 Xi’an, People’s Republic of China |
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Abstract: | This paper is devoted to congruences and ideals in pseudoeffect algebras. Let I be a normal ideal in a pseudoeffect algebra E. We show that: (1) the relation ~ I induced by I is a congruence if and only if for every a∈E, I∩ [0,a] is upper directed; (2) the relation ~ I induced by I is a strong congruence if and only if I is a normal weak Riesz ideal in a pseudoeffect algebra E. Moreover, we introduce a stronger concept of congruence—namely Riesz strong congruence—and we prove that, if I is a normal weak Riesz ideal in a pseudoeffect algebra E, then ~ I is a Riesz strong congruence and, conversely, if ~ is a Riesz strong congruence, then I = [0]~ is a normal weak Riesz ideal, and ~ I = ~. This work was supported by the National Natural Science Foundation of China (Grant No. 10271069). |
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Keywords: | Pseudoeffect algebras Ideals Riesz ideals Normal weak Riesz ideals Congruences Strong congruences Riesz strong congruences |
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