Riesz spaces for which every ideal is a projection band
✍ Scribed by C.B Huijsmans
- Publisher
- Elsevier Science
- Year
- 1976
- Weight
- 317 KB
- Volume
- 79
- Category
- Article
- ISSN
- 1385-7258
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✦ Synopsis
It was proved by the present author (see [2], theorem 3) that a Riesz space L is hyper-archimedean (i.e., L/A is Archimedean for every ideal A in L) if and only if the distributive lattice &p (L) of all principal ideals in L, partially ordered by inclusion, is a Boolean ring (and hence a Boolean algebra if L has, in addition, a strong unit). Actually, the following theorem holds (for the terminology and the notations used we refer to [4] ; all Riesz spaces considered in this paper are non-trivial). THEOREM 1. In a Riesz space L the following conditions are equivalent. (i) L is hyper-archimedean. (ii) ,G?~ (L) is a Boolean ring. (iii) Every principal ideal in L is a projection band, i.e., L= A, @ AUd for all u E L-f-, where A, denotes the principal ideal generated by u. (iv) The base sets {P) U ( u E L+) of the hull-kernel topology in the collection 9 of all proper prime iad in L are open and closed.