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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.
Reviewer: S.Rudeanu

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