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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)

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