Orthomodular lattices. Algebraic approach.

*(English)*Zbl 0558.06008The lattice-properties of the set of closed subspaces of a Hilbert space initiated the definition of the orthomodular law by Husimi 1937. It was not until the years of 1960 that the algebraic structure of lattices with an orthomodular orthocomplementation was developed. The author presents in his book elementary results of the new theory. Chapter I contains, for the reader not familiar with universal algebra and lattice theory, a brief outline of its basic concepts which are used in the book. Chapter II contains too little of the elementary theory of orthomodular lattices L, for instance the definition of the ”center” of L is postponed until VII. 6. Also, the characterizations given in II. 1.3 and II. 5.2 for ortho- and orthomodular lattices are not very helpful, - there are more useful equivalent forms of their axioms. The concept of commutativity is not well presented, some of the important theorems using this concept are missing. In II. 4.3, for instance, the properties that \(a\in L\) or its complement is comparable with \(b\in L\) or its complement are given as implying commutativity of a and b whereas, more generally, commutativity is the smallest binary relation on L containing \(\leq\) and which satisfies that the elements of L commuting with a fixed element \(a\in L\) form a subalgebra of L.

The skew operations on an ortholattice introduced in chapter III are discussed in the literature under the name of implication; for instance the ”skew join” a\({\dot \vee}b\) coincides with the implication \(b'\to_ 3a\) [see G. Kalmbach, ”Orthomodular lattices” (1983; Zbl 0512.06011), p. 239]. There is a long description of the two-generated free orthomodular lattice in III. 2. and an example which shows that the three-generated one is not finite. The sections III. 3. and III. 4. are devoted to the Hilbert space examples in the theory of orthomodular lattices. The author’s usage of ”amalgam” in chapter IV for Greechie’s ”past job of orthomodular lattices from Boolean algebras” may mislead, at first glance, some oldtimers in the theory. There are also too few Greechie diagrams given in this section and the important result of M. Dichtl on Greechie diagrams is missing. In chapter V, congruences and its connection with p-ideals in a generalized orthomodular lattice G are studied and Marsden’s characterization of distributive quotients of G via commutators is given. In chapter VI the group-theoretical term ”solvability” of G has no bearing for the orthomodular theory since in VI. 3.7 it is shown that the commutator sublattice G’ coincides with G”. Some important basic facts on orthomodular lattices can be found in chapter VII. For the ”physics” related discussion in VIII. 1. one obtains a better information from the book of E. G. Beltrametti and G. Cassinelli [”The logic of quantum mechanics” (1981; Zbl 0504.03026)]. There are a few more results in chapter VIII on compatibility and some remarks about the extensive literatur on orthologics and manuals introduced by Foulis-Randall. In the dimension theory VIII. 3. the proof of 3.16 could easily be stated on three more pages, but the reader is referred to the literature. Solutions to the exercises are given at the end of the book.

In general, the polynomial identities-computations are overstressed in this book on orthomodular lattices and most statements and proofs of the deeper results of the theory are not found in it.

The skew operations on an ortholattice introduced in chapter III are discussed in the literature under the name of implication; for instance the ”skew join” a\({\dot \vee}b\) coincides with the implication \(b'\to_ 3a\) [see G. Kalmbach, ”Orthomodular lattices” (1983; Zbl 0512.06011), p. 239]. There is a long description of the two-generated free orthomodular lattice in III. 2. and an example which shows that the three-generated one is not finite. The sections III. 3. and III. 4. are devoted to the Hilbert space examples in the theory of orthomodular lattices. The author’s usage of ”amalgam” in chapter IV for Greechie’s ”past job of orthomodular lattices from Boolean algebras” may mislead, at first glance, some oldtimers in the theory. There are also too few Greechie diagrams given in this section and the important result of M. Dichtl on Greechie diagrams is missing. In chapter V, congruences and its connection with p-ideals in a generalized orthomodular lattice G are studied and Marsden’s characterization of distributive quotients of G via commutators is given. In chapter VI the group-theoretical term ”solvability” of G has no bearing for the orthomodular theory since in VI. 3.7 it is shown that the commutator sublattice G’ coincides with G”. Some important basic facts on orthomodular lattices can be found in chapter VII. For the ”physics” related discussion in VIII. 1. one obtains a better information from the book of E. G. Beltrametti and G. Cassinelli [”The logic of quantum mechanics” (1981; Zbl 0504.03026)]. There are a few more results in chapter VIII on compatibility and some remarks about the extensive literatur on orthologics and manuals introduced by Foulis-Randall. In the dimension theory VIII. 3. the proof of 3.16 could easily be stated on three more pages, but the reader is referred to the literature. Solutions to the exercises are given at the end of the book.

In general, the polynomial identities-computations are overstressed in this book on orthomodular lattices and most statements and proofs of the deeper results of the theory are not found in it.

Reviewer: G.Kalmbach

##### MSC:

06C15 | Complemented lattices, orthocomplemented lattices and posets |

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

06C20 | Complemented modular lattices, continuous geometries |

46C05 | Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) |

81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |

03G12 | Quantum logic |

81P05 | General and philosophical questions in quantum theory |