Modal theory. An algebraic approach to order, geometry, and convexity.

*(English)*Zbl 0553.08001
Research and Exposition in Mathematics, 9. Berlin: Heldermann Verlag. XII, 158 p. DM 38.00 (1985).

Beside the classical structures of algebra, like groups, rings, lattices etc., in the last years several new structures have been introduced and studied in view of applications to other branches of mathematics or outside of mathematics. The present authors suggest such a structure which they call mode and which is intended as a unifying tool in the study of semilattices, affine geometry, convex sets and possibly other fields. A mode is a finitary algebra (A,\(\Omega)\) such that every operation \(\omega\in \Omega\) is idempotent, i.e. \(\omega (x,...,x)=x\), and entropic, \(i.e.\)

\(\omega\) (\(\omega\) ’(x\({}_{11},...,x_{1p}),...,\omega '(x_{n1},...,x_{np}))=\omega '(\omega (x_{11},...,x_{n1}),...,\omega (x_{1p},...,x_{np}))\)

for every \(\omega\) ’\(\in \Omega\). Semilattices (S,\(\vee)\) and more generally, the semigroups known as normal bands are examples of modes. Also, given a subset S of a unitary commutative ring R, every R-module E yields a mode under the operations \(\omega_ p(x,y)=x(1-p)+yp\), for \(p\in S\); these modes provide a description of affine geometry. In the particular case when R is the field of reals and S is the open unit interval \(I^ 0\), the subalgebras of \((E,I^ 0)\) are precisely the convex subsets of E. The class of barycentric algebras is defined as the equational class generated by the convex subsets of E viewed as modes; it also comprises semilattices, simplicial complexes and the modes of subspaces of real affine spaces.

A second basic definition is that of a modal. By this term is meant an algebra (A,\(\Omega\),\(\vee)\) where (A,\(\Omega)\) is a mode, (A,\(\vee)\) is a semilattice and each \(\omega\in \Omega\) distributes over \(\vee\). Various sets of submodes form modals under joins and complex products. Another important example of a modal is the algebra \(({\mathbb{R}},I^ 0,\max).\)

The book develops the algebraic theory of modes and modals, including homomorphisms and congruences, free algebras, representation theorems, varieties etc. The authors also discuss lines of further research to support their belief that modal theory may become a useful and illuminating tool in affine geometry, convex sets, harmonic analysis, computer science and possibly other fields.

\(\omega\) (\(\omega\) ’(x\({}_{11},...,x_{1p}),...,\omega '(x_{n1},...,x_{np}))=\omega '(\omega (x_{11},...,x_{n1}),...,\omega (x_{1p},...,x_{np}))\)

for every \(\omega\) ’\(\in \Omega\). Semilattices (S,\(\vee)\) and more generally, the semigroups known as normal bands are examples of modes. Also, given a subset S of a unitary commutative ring R, every R-module E yields a mode under the operations \(\omega_ p(x,y)=x(1-p)+yp\), for \(p\in S\); these modes provide a description of affine geometry. In the particular case when R is the field of reals and S is the open unit interval \(I^ 0\), the subalgebras of \((E,I^ 0)\) are precisely the convex subsets of E. The class of barycentric algebras is defined as the equational class generated by the convex subsets of E viewed as modes; it also comprises semilattices, simplicial complexes and the modes of subspaces of real affine spaces.

A second basic definition is that of a modal. By this term is meant an algebra (A,\(\Omega\),\(\vee)\) where (A,\(\Omega)\) is a mode, (A,\(\vee)\) is a semilattice and each \(\omega\in \Omega\) distributes over \(\vee\). Various sets of submodes form modals under joins and complex products. Another important example of a modal is the algebra \(({\mathbb{R}},I^ 0,\max).\)

The book develops the algebraic theory of modes and modals, including homomorphisms and congruences, free algebras, representation theorems, varieties etc. The authors also discuss lines of further research to support their belief that modal theory may become a useful and illuminating tool in affine geometry, convex sets, harmonic analysis, computer science and possibly other fields.

Reviewer: S.Rudeanu

##### MSC:

08-02 | Research exposition (monographs, survey articles) pertaining to general algebraic systems |

08A30 | Subalgebras, congruence relations |

06A12 | Semilattices |

52A01 | Axiomatic and generalized convexity |

08A05 | Structure theory of algebraic structures |

20M07 | Varieties and pseudovarieties of semigroups |

51N10 | Affine analytic geometry |

52A10 | Convex sets in \(2\) dimensions (including convex curves) |

53C99 | Global differential geometry |

57N99 | Topological manifolds |

43A99 | Abstract harmonic analysis |