##
**Ockham algebras.**
*(English)*
Zbl 0835.06011

Oxford: Oxford University Press. 241 p. (1994).

Ockham algebra theory is a newly discovered very small corner of mathematics. All published work on Ockham algebra theory has appeared within the last fifteen years. No more than thirty people have even worked on it and no more than sixty papers dealing explicitly with it have even appeared (at least half of them are due to the authors of this book). The monograph under review is the first attempt to give a systematic and self-contained account of the research in this area.

An algebra \(L = (L; \vee, \wedge, f,0,1)\) of type \((2,2,1, 0,0)\) in which \((L; \vee, \wedge, 0,1)\) is a bounded distributive lattice and the operation \(f\) satisfies the following identities: \(f(0) = 1\), \(f(1) = 0\), \(f(x \vee y) = f(x) \wedge f(y)\) and \(f(x \wedge y) = f(x ) \vee f(y)\), i.e. \(f\) is a dual lattice endomorphism of \(L\), is called an Ockham algebra. The concept of an Ockham algebra arose from attempts to generalize the notion of a Boolean algebra. Important steps in this history are De Morgan algebras and Stone algebras.

The class of Ockham algebras is equational, i.e., it is a variety. Very important subvarieties, the Berman classes \(K_{p,q}\), were introduced by J. Berman [Aequationes Math. 16, 165-171 (1977; Zbl 0395.06007)] defined by adjoining the identity \(f^{2p + q} = f^q\) for \(p \geq 1\) and \(q \geq 0\). (Note that \(f^0 = \text{id}\) and \(f^n (x) = (f(f^{n - 1} (x))\).) The classes \(K_{p,q}\) are related as follows: \(K_{p,q} \leq K_{r,s}\) if and only if \(p |r\) and \(r \leq s\). Now, \(K_{1,0}\) is in fact the variety of all De Morgan algebras, and the subvariety of Boolean algebras can be characterized by the additional identity \(x \wedge f(x) = 0\). Similarly, \(x \wedge f(x) = 0\) defines the Stone algebras as a subvariety of \(K_{1,1}\). The class of De Morgan- Stone algebras, or MS-algebras, is defined as a subvariety of \(K_{1,1}\) satisfying the identity \(x \wedge f^2(x) = x\).

The book contains sixteen chapters. In the first four ones, after the basic ideas, standard constructions, and important examples, the authors study congruence relations and subdirectly irreducible Ockham algebras. Duality theory is treated in the next chapter. Since Ockham algebras are special bounded distributive lattices, they are dually equivalent to a suitable Priestley space endowed with a continuous order-preserving mapping \(g\), the Ockham space. Combining both algebraic and topological approaches, the authors obtained insight into the lattice of subvarieties of Ockham algebras. Recall that an element \(a\) of an Ockham algebra \((L;f)\) is called a fixed point of \(f\), if \(f(a) = a\). Dually, an element \(x\) of an Ockham space \((X;g)\) is said to be a fixed point of \(g\) if \(g(x) = x\). It can be shown that the subset \(\text{Fix} L\) is an antichain (possibly empty) and that a fixed point of \(f\) corresponds to a special downset ( = order ideal) of the dual space \(X\). Conversely, a fixed point of \(g\) corresponds to a particular prime ideal of \(L\). This and similar results can be found in chapter 6. By a fixed point separating congruence on \(L\) the authors mean a congruence relation that separates every pair of fixed points of \(L\). Chapter 7 covers in some detail the topic of fixed point separating congruences. In chapters 8 and 9 the authors concentrate on the class \(K_{1,1}\), its subvarieties, subdirectly irreducibles, and congruences.

If \(L\) is a finite distributive lattice and \(V\) a subvariety of Ockham algebras, then \(L\) is said to be a relative \(V\)-algebra if every interval \([a,b]\) of \(L\) can be given the structure of an Ockham algebra belonging to \(V\). Chapter 11 considers the relative Ockham algebras.

The last four chapters are devoted to the study of double MS-algebras. Recall that an algebra \((L; \vee, \wedge,^\circ, ^+, 0,1)\) of type \((2,2,1,1,0,0)\) is called a double MS-algebra if (i) \((L; \vee, \wedge,^\circ, 0,1) \) is an MS-algebra, (ii) \((L^{\text{op}}; \wedge, \vee,^+, 0,1)\) is an MS-algebra, (iii) \(a^{+ \circ} = a^{++}\) and (iv) \(A^{\circ +} = a^{\circ \circ}\). Dual spaces to double MS-algebras, subdirectly irreducibles and congruences on double MS-algebras are investigated in chapters 12-14. In the last chapter, the connection between equational theories of the MS-algebra \((L;^\circ)\) and the MS-algebra \((L^{\text{op}};^+)\) of a double MS-algebra \((L;^\circ,^+)\) is investigated.

The text is well and clearly written, and it contains complete references to the literature. Many examples illustrate and sharpen the theory. This book could form a stimulating introduction for a student contemplating research in this area. In order to meet this expectation, it needed a number of good open questions and problems, which is, unfortunately, not the case.

An algebra \(L = (L; \vee, \wedge, f,0,1)\) of type \((2,2,1, 0,0)\) in which \((L; \vee, \wedge, 0,1)\) is a bounded distributive lattice and the operation \(f\) satisfies the following identities: \(f(0) = 1\), \(f(1) = 0\), \(f(x \vee y) = f(x) \wedge f(y)\) and \(f(x \wedge y) = f(x ) \vee f(y)\), i.e. \(f\) is a dual lattice endomorphism of \(L\), is called an Ockham algebra. The concept of an Ockham algebra arose from attempts to generalize the notion of a Boolean algebra. Important steps in this history are De Morgan algebras and Stone algebras.

The class of Ockham algebras is equational, i.e., it is a variety. Very important subvarieties, the Berman classes \(K_{p,q}\), were introduced by J. Berman [Aequationes Math. 16, 165-171 (1977; Zbl 0395.06007)] defined by adjoining the identity \(f^{2p + q} = f^q\) for \(p \geq 1\) and \(q \geq 0\). (Note that \(f^0 = \text{id}\) and \(f^n (x) = (f(f^{n - 1} (x))\).) The classes \(K_{p,q}\) are related as follows: \(K_{p,q} \leq K_{r,s}\) if and only if \(p |r\) and \(r \leq s\). Now, \(K_{1,0}\) is in fact the variety of all De Morgan algebras, and the subvariety of Boolean algebras can be characterized by the additional identity \(x \wedge f(x) = 0\). Similarly, \(x \wedge f(x) = 0\) defines the Stone algebras as a subvariety of \(K_{1,1}\). The class of De Morgan- Stone algebras, or MS-algebras, is defined as a subvariety of \(K_{1,1}\) satisfying the identity \(x \wedge f^2(x) = x\).

The book contains sixteen chapters. In the first four ones, after the basic ideas, standard constructions, and important examples, the authors study congruence relations and subdirectly irreducible Ockham algebras. Duality theory is treated in the next chapter. Since Ockham algebras are special bounded distributive lattices, they are dually equivalent to a suitable Priestley space endowed with a continuous order-preserving mapping \(g\), the Ockham space. Combining both algebraic and topological approaches, the authors obtained insight into the lattice of subvarieties of Ockham algebras. Recall that an element \(a\) of an Ockham algebra \((L;f)\) is called a fixed point of \(f\), if \(f(a) = a\). Dually, an element \(x\) of an Ockham space \((X;g)\) is said to be a fixed point of \(g\) if \(g(x) = x\). It can be shown that the subset \(\text{Fix} L\) is an antichain (possibly empty) and that a fixed point of \(f\) corresponds to a special downset ( = order ideal) of the dual space \(X\). Conversely, a fixed point of \(g\) corresponds to a particular prime ideal of \(L\). This and similar results can be found in chapter 6. By a fixed point separating congruence on \(L\) the authors mean a congruence relation that separates every pair of fixed points of \(L\). Chapter 7 covers in some detail the topic of fixed point separating congruences. In chapters 8 and 9 the authors concentrate on the class \(K_{1,1}\), its subvarieties, subdirectly irreducibles, and congruences.

If \(L\) is a finite distributive lattice and \(V\) a subvariety of Ockham algebras, then \(L\) is said to be a relative \(V\)-algebra if every interval \([a,b]\) of \(L\) can be given the structure of an Ockham algebra belonging to \(V\). Chapter 11 considers the relative Ockham algebras.

The last four chapters are devoted to the study of double MS-algebras. Recall that an algebra \((L; \vee, \wedge,^\circ, ^+, 0,1)\) of type \((2,2,1,1,0,0)\) is called a double MS-algebra if (i) \((L; \vee, \wedge,^\circ, 0,1) \) is an MS-algebra, (ii) \((L^{\text{op}}; \wedge, \vee,^+, 0,1)\) is an MS-algebra, (iii) \(a^{+ \circ} = a^{++}\) and (iv) \(A^{\circ +} = a^{\circ \circ}\). Dual spaces to double MS-algebras, subdirectly irreducibles and congruences on double MS-algebras are investigated in chapters 12-14. In the last chapter, the connection between equational theories of the MS-algebra \((L;^\circ)\) and the MS-algebra \((L^{\text{op}};^+)\) of a double MS-algebra \((L;^\circ,^+)\) is investigated.

The text is well and clearly written, and it contains complete references to the literature. Many examples illustrate and sharpen the theory. This book could form a stimulating introduction for a student contemplating research in this area. In order to meet this expectation, it needed a number of good open questions and problems, which is, unfortunately, not the case.

Reviewer: T.Katriňák (Bratislava)

### MSC:

06D30 | De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects) |

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