##
**Semimodular lattices. Theory and applications.**
*(English)*
Zbl 0957.06008

Encyclopedia of Mathematics and Its Applications. 73. Cambridge: Cambridge University Press. xiv, 370 p. (1999).

For a lattice \((L,\vee,\wedge)\) all of whose elements \(a,b\) fulfil the implication
\[
\text{If }a\wedge b\text{ is a lower cover of \(a\), then \(b\) is a lower cover of }a\vee b. \tag{Sm}
\]
Wilcox coined the term semimodular. For lattices of finite length semimodularity represents “one half” of modularity, because the latter is implied by the conjunction of (Sm) and its reverse.

Although in the absence of finite length the concept of semimodularity “ramifies in a most unpleasant way” (G. Birkhoff 1967; cf. this book, page x) the author succeeded in writing a most pleasant book which among others presents the different facets of the concept of semimodularity in lattices of infinite length as they have been revealed by Wilcox, Mac Lane, and others.

The book under review is an extended version of the author’s earlier monograph [Semimodular lattices. Dedicated to Garrett Birkhoff, Teubner-Texte zur Mathematik. 125. Stuttgart etc.: B. G. Teubner Verlagsgesellschaft (1991; Zbl 0743.06008)]. To the profit of the reader the book proceeds in a historico-genetical manner. The text as well as the corresponding notes contain an impressing wealth of material on semimodularity. The style of writing spans the gap between a research monograph reviewing results only and a detailed textbook – giving those proofs which hitherto have not been provided by other textbooks. Many exercises are given and open problems are mentioned. Hence the book is useful for students as well as researchers interested in the subject of semimodularity.

The nine chapters of the book are entitled as follows: 1) From Boolean algebras to semimodular lattices; 2) M-symmetric lattices; 3) Conditions related to semimodularity, 0-conditions, and disjointness properties; 4) Supersolvable and admissible lattices; consistent and strong lattices; 5) The covering graph; 6) Semimodular lattices of finite length; 7) Local distributivity; 8) Local modularity; 9) Congruence semimodularity.

Chapters 2 and 3 analyze so-called conditions related to semimodularity, i.e. conditions which are equivalent to semimodularity for lattices of finite length. Chapter 2 examines Wilcox’ concept of M-symmetry referring also to partly unpublished work of Wilcox which was fed from the relationship between projective and affine geometry. The main reason for Wilcox’ objection against (Sm) was that (Sm) refers to the covering relation such that it holds trivially in any lattice with only continuous chains. Orthomodular lattices and M-symmetry are revisited in 2.6, whence here quantum logics are touched upon.

Chapter 3 takes a look at Mac Lane’s covering-free approach to semimodularity: A lattice is said to satisfy Mac Lane’s condition if for any \(x,y,z\in L\) with \(y\wedge z< x< z< y\vee x\) there exists \(t\in L\) such that \(y\wedge z< t< y\) and \(x= (x\vee t)\wedge z\), which means that inside \(L\) any (non-modular) pentagon extends to a (semimodular) centered hexagon. The author shows in 3.1 how Mac Lane’s condition (Mac), Birkhoff’s condition (Bi), semimodularity (Sm), M-symmetry (Ms) are related in general, and that in particular they are equivalent in upper continuous strongly atomic lattices. Here (Bi) stands for a weakening of (Sm), namely: If \(a\wedge b\) is a lower cover of \(a\) and \(b\), then \(a\) and \(b\) are lower covers of \(a\vee b\). Chapter 3.2 looks at (Mac), (Bi), (Sm) and (Ms) as conditions on the ideal lattice or the lattice of dual ideals of \(L\) and shows how such requirements relate to properties of \(L\) itself. 3.3 studies the simplifications which are gained by assuming chain conditions (ACC or DCC). In 3.4 “local” properties are studied which arise when semimodularity, modularity and distributivity are required only “at 0” (for instance setting \(a=0\) in (Sm)). Disjointness properties are also examined. Chapter 3.5 gives many results for lattices which have complementation properties, in particular with respect to the question of when “local” properties become global ones.

Chapter 4 starts with the Möbius function of a poset (of a finite semimodular lattice, in particular). The author then studies supersolvable lattices [cf. R. Stanley, Algebra Univers. 2, 197-217 (1972; Zbl 0256.06002)], which may or may not be semimodular, and he shows how finite semimodular lattices and supersolvable lattices fit into the framework of admissible lattices and of Cohen-Macaulay posets [cf. R. Stanley, Algebra Univers. 4, 361-371 (1974; Zbl 0303.06006); A. Björner, Trans. Am. Math. Soc. 260, 159-183 (1980; Zbl 0441.06002)]. Consistency [cf. J. P. S. Kung, Math. Proc. Camb. Philos. Soc. 101, 221-231 (1987; Zbl 0626.06008)], and strongness [cf. U. Faigle, J. Comb. Theory, Ser. B 28, 26-51 (1980; Zbl 0416.05029)] are examined in detail, for instance they are shown to be independent properties in general, which coincide for finite semimodular lattices.

In chapter 5 the covering graph is a principal tool. The author among others describe covering graph isomorphisms of graded balanced lattices by direct product decompositions, thereby generalizing results of J. Jakubík [Czech. Math. J. 4, 131-142 (1954; Zbl 0059.02602)], and D. Duffus and I. Rival [Discrete Math. 19, 139-158 (1977; Zbl 0372.06005)]. Finite semimodular lattices with centrally symmetric covering graph are finite Boolean lattices; this statement is shown to hold even if semimodularity is replaced by strongness of the dual lattice.

Chapter 6 gives embedding results for finite semimodular lattices and many other classes of lattices, it considers a description of semimodular lattices by closure operators, or by selectors.

Chapter 7 gives characterizations of local distributivity for finite lattices mainly following S. P. Avann [Math. Ann. 154, 420-426 (1964; Zbl 0202.31703), and Math. Ann. 142, 345-354 (1961; Zbl 0094.01603)].

Chapter 8 contains for instance a characterization of the Kurosh-Ore replacement property for strongly atomic algebraic lattices, and applications to lattices of subnormal subgroups.

Chapter 9 refers to algebras whose congruence lattice is semimodular. The semigroup case is given special attention.

Each of the nine chapters is divided into subchapters each of which is supplemented by notes and references.

For convenience of the reader a table of notations, a detailed subject index, and a master reference list containing all citations are given. All this underlines the overall impression of a very well organized book it is a pleasure to read in. It will certainly become a standard source.

Although in the absence of finite length the concept of semimodularity “ramifies in a most unpleasant way” (G. Birkhoff 1967; cf. this book, page x) the author succeeded in writing a most pleasant book which among others presents the different facets of the concept of semimodularity in lattices of infinite length as they have been revealed by Wilcox, Mac Lane, and others.

The book under review is an extended version of the author’s earlier monograph [Semimodular lattices. Dedicated to Garrett Birkhoff, Teubner-Texte zur Mathematik. 125. Stuttgart etc.: B. G. Teubner Verlagsgesellschaft (1991; Zbl 0743.06008)]. To the profit of the reader the book proceeds in a historico-genetical manner. The text as well as the corresponding notes contain an impressing wealth of material on semimodularity. The style of writing spans the gap between a research monograph reviewing results only and a detailed textbook – giving those proofs which hitherto have not been provided by other textbooks. Many exercises are given and open problems are mentioned. Hence the book is useful for students as well as researchers interested in the subject of semimodularity.

The nine chapters of the book are entitled as follows: 1) From Boolean algebras to semimodular lattices; 2) M-symmetric lattices; 3) Conditions related to semimodularity, 0-conditions, and disjointness properties; 4) Supersolvable and admissible lattices; consistent and strong lattices; 5) The covering graph; 6) Semimodular lattices of finite length; 7) Local distributivity; 8) Local modularity; 9) Congruence semimodularity.

Chapters 2 and 3 analyze so-called conditions related to semimodularity, i.e. conditions which are equivalent to semimodularity for lattices of finite length. Chapter 2 examines Wilcox’ concept of M-symmetry referring also to partly unpublished work of Wilcox which was fed from the relationship between projective and affine geometry. The main reason for Wilcox’ objection against (Sm) was that (Sm) refers to the covering relation such that it holds trivially in any lattice with only continuous chains. Orthomodular lattices and M-symmetry are revisited in 2.6, whence here quantum logics are touched upon.

Chapter 3 takes a look at Mac Lane’s covering-free approach to semimodularity: A lattice is said to satisfy Mac Lane’s condition if for any \(x,y,z\in L\) with \(y\wedge z< x< z< y\vee x\) there exists \(t\in L\) such that \(y\wedge z< t< y\) and \(x= (x\vee t)\wedge z\), which means that inside \(L\) any (non-modular) pentagon extends to a (semimodular) centered hexagon. The author shows in 3.1 how Mac Lane’s condition (Mac), Birkhoff’s condition (Bi), semimodularity (Sm), M-symmetry (Ms) are related in general, and that in particular they are equivalent in upper continuous strongly atomic lattices. Here (Bi) stands for a weakening of (Sm), namely: If \(a\wedge b\) is a lower cover of \(a\) and \(b\), then \(a\) and \(b\) are lower covers of \(a\vee b\). Chapter 3.2 looks at (Mac), (Bi), (Sm) and (Ms) as conditions on the ideal lattice or the lattice of dual ideals of \(L\) and shows how such requirements relate to properties of \(L\) itself. 3.3 studies the simplifications which are gained by assuming chain conditions (ACC or DCC). In 3.4 “local” properties are studied which arise when semimodularity, modularity and distributivity are required only “at 0” (for instance setting \(a=0\) in (Sm)). Disjointness properties are also examined. Chapter 3.5 gives many results for lattices which have complementation properties, in particular with respect to the question of when “local” properties become global ones.

Chapter 4 starts with the Möbius function of a poset (of a finite semimodular lattice, in particular). The author then studies supersolvable lattices [cf. R. Stanley, Algebra Univers. 2, 197-217 (1972; Zbl 0256.06002)], which may or may not be semimodular, and he shows how finite semimodular lattices and supersolvable lattices fit into the framework of admissible lattices and of Cohen-Macaulay posets [cf. R. Stanley, Algebra Univers. 4, 361-371 (1974; Zbl 0303.06006); A. Björner, Trans. Am. Math. Soc. 260, 159-183 (1980; Zbl 0441.06002)]. Consistency [cf. J. P. S. Kung, Math. Proc. Camb. Philos. Soc. 101, 221-231 (1987; Zbl 0626.06008)], and strongness [cf. U. Faigle, J. Comb. Theory, Ser. B 28, 26-51 (1980; Zbl 0416.05029)] are examined in detail, for instance they are shown to be independent properties in general, which coincide for finite semimodular lattices.

In chapter 5 the covering graph is a principal tool. The author among others describe covering graph isomorphisms of graded balanced lattices by direct product decompositions, thereby generalizing results of J. Jakubík [Czech. Math. J. 4, 131-142 (1954; Zbl 0059.02602)], and D. Duffus and I. Rival [Discrete Math. 19, 139-158 (1977; Zbl 0372.06005)]. Finite semimodular lattices with centrally symmetric covering graph are finite Boolean lattices; this statement is shown to hold even if semimodularity is replaced by strongness of the dual lattice.

Chapter 6 gives embedding results for finite semimodular lattices and many other classes of lattices, it considers a description of semimodular lattices by closure operators, or by selectors.

Chapter 7 gives characterizations of local distributivity for finite lattices mainly following S. P. Avann [Math. Ann. 154, 420-426 (1964; Zbl 0202.31703), and Math. Ann. 142, 345-354 (1961; Zbl 0094.01603)].

Chapter 8 contains for instance a characterization of the Kurosh-Ore replacement property for strongly atomic algebraic lattices, and applications to lattices of subnormal subgroups.

Chapter 9 refers to algebras whose congruence lattice is semimodular. The semigroup case is given special attention.

Each of the nine chapters is divided into subchapters each of which is supplemented by notes and references.

For convenience of the reader a table of notations, a detailed subject index, and a master reference list containing all citations are given. All this underlines the overall impression of a very well organized book it is a pleasure to read in. It will certainly become a standard source.

Reviewer: Horst Szambien (Garbsen)

### MSC:

06C10 | Semimodular lattices, geometric lattices |

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

06A07 | Combinatorics of partially ordered sets |