##
**Quantales and their applications.**
*(English)*
Zbl 0703.06007

Pitman Research Notes in Mathematics Series, 234. Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc. 165 p. (1990).

What is a quantale? Read half a dozen different papers with the word “quantale” in their titles, and you will encounter half a dozen different definitions. Quantale theory, it seems, is a perfect example of an area of mathematics which is driven by an attractive terminology rather than a unifying concept. And yet there is a unifying concept there: at its simplest, it is that of a complete lattice equipped with a binay multiplication which distributes (on either side) over arbitrary joins. Of course, such objects have been studied since long before C. J. Mulvey first committed the word “quantale” to print in 1986: their first appearance was probably in the late 1930s, in a series of papers by M. Ward and R. P. Dilworth (who called them residuated lattices - residuation being their name, derived from the ideal theory of rings, for the right adjoints to multiplication by a fixed element). And there is no doubt that the concept is a unifying one: this is testified to by the number of times it has been re-invented, under different names, by workers in widely separated areas, as well as by the list of 20 diverse examples which follows the basic definition on pages 17-22 of this book.

The problems begin when you ask what additional axioms the multiplication should satisfy. Most people (though not all) are happy to assume that it should be associative (a condition satisfied in all the significant examples); but should it be commutative? Should it be idempotent? Should it have a (one-sided or two-sided) unit, and if so should the unit be the top element of the lattice? Again, should the lattice be an arbitrary complete lattice, or should it satisfy some further condition such as modularity, distributivity, compactness of the top element, meet- continuity or continuity? Obviously, the more conditions you impose, the more theorems you will be able to prove; and consideration of the various examples has led different authors to radically different conclusions about the right level of generality at which to work. In the present volume, the author sensibly assumes none of the above conditions (apart from associativity) at the outset, but does not shrink from imposing them as and when they become needed for particular developments of the theory.

If you impose all the above conditions on the multiplication (but none, apart from what is forced by this, on the underlying lattice), then the multiplication is forced to coincide with binary meet, and you arrive at the concept of a frame, which has been extensively studied in connection with the lattice-theoretic approach to topology now generally known as locale theory (for a general account, see the reviewer’s book “Stone spaces” (1982; Zbl 0499.54001). Mulvey coined the name “quantale” as a fusion of “quantum logic” and “locale”, the idea being that the multiplication plays the role of a “non-commutative conjunction” familiar in quantum theory, but that in other respect a quantale is a generalized locale. (In passing, it is unfortunate that, whereas in locale theory we have two words “frame” and “locale” to describe the algebraic and topological aspects of what are extensionally the same objects, the word “quantale” is having to do both jobs simultaneously in this more general context. The reviewer has for some time been trying to propagate the word “skewframe” as a synonym for “quantale” in its algebraic aspects, but so far without success). To emphasize this idea, Mulvey used an ampersand (&) to denote the multiplication, and it is perhaps the attractiveness of this notation as much as the inherent rightness of the name “quantale” that has brought the latter so quickly to its present dominant position.

The book under review is the first to be entirely devoted to quantales, though they made an earlier appearance in the final chapter of “Topology via logic” by S. Vickers (1989; Zbl 0668.54001). It begins with a brief review of partially ordered sets and frames (the latter relying heavily on the reviewer’s book, op. cit.), and then devotes chapters 2 and 3 to the basic algebra of quantales and their quotients. The next two chapters deal with applications of quantales, first to the ideal theory of rings (largely following work of H. Simmons, and of S. B. Niefield and the author) and then to the representation theory of non-commutative \(C^*\)-algebras (the goal for which Mulvey originally introduced quantales, and which has since been pursued by F. Borceux, G. van den Bossche, J. Rosický and others). In the latter, the author introduces a notion of sheaf on a quantale, but does not develop it very far; in general, this book tends to concentrate on the algebraic rather than the topological aspects of quantale theory. The final brief chapter deals with “Girard quantales”, which were recently introduced by D. Yetter as a means of modelling the propositional part of J. Y. Girard’s linear logic; as the author admits, their theory is relatively undeveloped as yet.

It will be seen that the book contains little that is not already in print elsewhere (and the author is scrupulous about giving credit to the authors of the original papers whose work he reproduces), but it is certainly valuable to have it all presented between a single pair of covers, and in a uniform notation and terminology. The author’s style is a little abrupt, and there are one or two misprints which might confuse the beginner (for example, in the definition of dualizing element in Definition 6.1.1, and of the ordering on \(\tilde Q\) in Theorem 6.1.3), but otherwise it is easy to read. The book ends with a useful bibliography containing most of the papers so far published on quantales (though a few appear unhelpfully as “Preprint”, rather than any more specific reference).

The problems begin when you ask what additional axioms the multiplication should satisfy. Most people (though not all) are happy to assume that it should be associative (a condition satisfied in all the significant examples); but should it be commutative? Should it be idempotent? Should it have a (one-sided or two-sided) unit, and if so should the unit be the top element of the lattice? Again, should the lattice be an arbitrary complete lattice, or should it satisfy some further condition such as modularity, distributivity, compactness of the top element, meet- continuity or continuity? Obviously, the more conditions you impose, the more theorems you will be able to prove; and consideration of the various examples has led different authors to radically different conclusions about the right level of generality at which to work. In the present volume, the author sensibly assumes none of the above conditions (apart from associativity) at the outset, but does not shrink from imposing them as and when they become needed for particular developments of the theory.

If you impose all the above conditions on the multiplication (but none, apart from what is forced by this, on the underlying lattice), then the multiplication is forced to coincide with binary meet, and you arrive at the concept of a frame, which has been extensively studied in connection with the lattice-theoretic approach to topology now generally known as locale theory (for a general account, see the reviewer’s book “Stone spaces” (1982; Zbl 0499.54001). Mulvey coined the name “quantale” as a fusion of “quantum logic” and “locale”, the idea being that the multiplication plays the role of a “non-commutative conjunction” familiar in quantum theory, but that in other respect a quantale is a generalized locale. (In passing, it is unfortunate that, whereas in locale theory we have two words “frame” and “locale” to describe the algebraic and topological aspects of what are extensionally the same objects, the word “quantale” is having to do both jobs simultaneously in this more general context. The reviewer has for some time been trying to propagate the word “skewframe” as a synonym for “quantale” in its algebraic aspects, but so far without success). To emphasize this idea, Mulvey used an ampersand (&) to denote the multiplication, and it is perhaps the attractiveness of this notation as much as the inherent rightness of the name “quantale” that has brought the latter so quickly to its present dominant position.

The book under review is the first to be entirely devoted to quantales, though they made an earlier appearance in the final chapter of “Topology via logic” by S. Vickers (1989; Zbl 0668.54001). It begins with a brief review of partially ordered sets and frames (the latter relying heavily on the reviewer’s book, op. cit.), and then devotes chapters 2 and 3 to the basic algebra of quantales and their quotients. The next two chapters deal with applications of quantales, first to the ideal theory of rings (largely following work of H. Simmons, and of S. B. Niefield and the author) and then to the representation theory of non-commutative \(C^*\)-algebras (the goal for which Mulvey originally introduced quantales, and which has since been pursued by F. Borceux, G. van den Bossche, J. Rosický and others). In the latter, the author introduces a notion of sheaf on a quantale, but does not develop it very far; in general, this book tends to concentrate on the algebraic rather than the topological aspects of quantale theory. The final brief chapter deals with “Girard quantales”, which were recently introduced by D. Yetter as a means of modelling the propositional part of J. Y. Girard’s linear logic; as the author admits, their theory is relatively undeveloped as yet.

It will be seen that the book contains little that is not already in print elsewhere (and the author is scrupulous about giving credit to the authors of the original papers whose work he reproduces), but it is certainly valuable to have it all presented between a single pair of covers, and in a uniform notation and terminology. The author’s style is a little abrupt, and there are one or two misprints which might confuse the beginner (for example, in the definition of dualizing element in Definition 6.1.1, and of the ordering on \(\tilde Q\) in Theorem 6.1.3), but otherwise it is easy to read. The book ends with a useful bibliography containing most of the papers so far published on quantales (though a few appear unhelpfully as “Preprint”, rather than any more specific reference).

Reviewer: P.T.Johnstone

### MSC:

06F05 | Ordered semigroups and monoids |

18B35 | Preorders, orders, domains and lattices (viewed as categories) |

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

03G25 | Other algebras related to logic |

06D20 | Heyting algebras (lattice-theoretic aspects) |

13A15 | Ideals and multiplicative ideal theory in commutative rings |

16D99 | Modules, bimodules and ideals in associative algebras |

54A05 | Topological spaces and generalizations (closure spaces, etc.) |

46L05 | General theory of \(C^*\)-algebras |

### Keywords:

quantale; residuated lattices; non-commutative conjunction; generalized locale; frame; skewframe; ideal theory of rings; representation theory of non-commutative \(C^ *\)-algebras; sheaf on a quantale; Girard quantales; linear logic
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\textit{K. I. Rosenthal}, Quantales and their applications. Harlow: Longman Scientific \& Technical; New York: John Wiley \& Sons, Inc. (1990; Zbl 0703.06007)

### Online Encyclopedia of Integer Sequences:

Number of quantales on n elements, up to isomorphism.Number of semi-unital quantales on n elements, up to isomorphism.

Number of unital quantales on n elements, up to isomorphism.

Number of left-sided quantales on n elements, up to isomorphism. Also number of right-sided quantales on n elements, up to isomorphism.

Number of strictly left-sided quantales on n elements, up to isomorphism. Also number of strictly right-sided quantales on n elements, up to isomorphism.

Number of two-sided quantales on n elements, up to isomorphism.

Number of integral quantales on n elements, up to isomorphism.

Number of balanced quantales on n elements, up to isomorphism.