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A note on the number of subquantales. (English) Zbl 1283.06030
The paper continues the study of S. Han and B. Zhao [Algebra Univers. 61, No. 1, 97–114 (2009; Zbl 1183.06010)] on quantic conuclei, which are in one-to-one correspondence with subquantales of quantales [K. I. Rosenthal, Quantales and their applications. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. (1990; Zbl 0703.06007)]. The main result of the paper (Proposition 2.8 on page 197) shows that every finite quantale which has at least five elements, has at least five subquantales. The proof of this claim is based in Lemmas 2.5, 2.6, 2.7 on pages 193–197, which take almost the whole paper and which are rather technical in nature (and whose claims, moreover, somewhat intersect). In continuation, the authors provide a sequence of quantales with growing cardinality, all of which have exactly five subquantales (Example 2.10 on page 198 and its particular case Example 2.9 on page 197). In the last section of the paper, the authors turn their attention to ideals of quantales, which are in one-to-one correspondence with (their previously introduced) ideal conuclei (Definition 3.3 on page 199). Based in the results of D. Kruml and J. Paseka [in: Handbook of Algebra 5, 323–362 (2008; Zbl 1219.06016)] and J. Paseka [in: Proceedings of the 8th Prague topological symposium, Prague, Czech Republic, 1996. North Bay, ON: Topology Atlas. 314–328 (1997; Zbl 0920.06007)] on the quantale \(\mathcal{Q}(S)\) of all sup-lattice endomorphisms \(f:S\rightarrow S\), they show that, given a completely distributive lattice \(S\), the quantale \(\mathcal{Q}(S)\) has only two ideals (Proposition 3.6 on page 199).
The paper is relatively well written (just a couple of typos), self-contained, and all of its (however, sometimes quite technical) results are easy to follow.
MSC:
06F07 Quantales
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