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Simple quantales. (English) Zbl 0920.06007
Simon, Petr (ed.), Proceedings of the 8th Prague topological symposium, Prague, Czech Republic, August 18–24, 1996. North Bay, ON: Topology Atlas, 314-328 (1997).
A quantale is a complete lattice in which an associative multiplication is defined that distributes over arbitrary suprema; it is called simple if it contains no nontrivial quotients. An element \(x\) of a quantale \(Q\) is right-sided (left-sided) if \(x\cdot 1\leq x\) \((1\cdot x\leq x)\); it is two-sided if it is both right-sided and left-sided. A quantale \(Q\) is a factor if 0 and 1 are the only two-sided elements; it is \(m\)-trivial if \(x\cdot y=0\) for all \(x,y\in Q\). \(Q\) is called faithful if \(x\cdot a=y\cdot a\) and \(b\cdot x=b\cdot y\) for any right-sided element \(a\) and any left-sided element \(b\) implies \(x=y\); it is discrete if \((a\to_l 0)\to_r 0=a\) and \((b\to_r0)\to_l 0=b\) for any right-sided element \(a\) and any left-sided element \(b\); here \(a \to_r\) – \((b\to_l-)\) denotes the right adjoint to \(a\cdot-\) \((-\cdot b)\). A quantale is semi-idempotent if both the set of its right-sided elements and the set of its left-sided elements are idempotent quantales (i.e., the multiplication \(\cdot\) is idempotent).
The author proves the following characterization theorem: A quantale is simple if and only if it is a discrete faithful factor or an \(m\)-trivial factor. Also, factorial (spatial, resp.) quantales are studied, i.e. quantales which have enough strong homomorphisms (homomorphisms preserving 1) into semi-idempotent factor (simple, resp.) quantales to separate elements of \(Q\).
For the entire collection see [Zbl 0907.00034].
Reviewer: V.Novák (Brno)

06F05 Ordered semigroups and monoids
54A05 Topological spaces and generalizations (closure spaces, etc.)
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