An introduction to the theory of modes and modals.(English)Zbl 0776.08003

Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 3, 241-262 (1992).
[For the entire collection see Zbl 0745.00034.]
This work is a short survey of the theory of modes and modals. A mode is an idempotent algebra $$(A,\Omega)$$ which is entropic in the sense that the following equality $w(w'(x_{11},\dots,x_{n1}),\dots, w'(x_{1m},\dots,x_{nm}))= w'(w(x_{11},\dots, x_{ 1m}),\dots, w(x_{n1},\dots, x_{nm})$ holds for all fundamental operations $$w,w'\in\Omega$$ and all $$x_{ij}\in A$$. An algebra $$(A,+,\Omega)$$, where $$(A,+)$$ is a semilattice, $$(A,\Omega)$$ is a mode with distributive laws for all $$w\in\Omega$$, is called a modal (compare with the notion of a module). Natural examples of modes are semilattices, affine spaces over prime fields, the algebras $$(A,(0,1))$$, where $$A$$ is a convex subset of $$\mathbb{R}^ n$$ and $$p\in (0,1)$$ is a binary operation defined by $$p(x,y)=px+(1-p)y$$, idempotent medial groupoids of Kepka and Ježek. Modes (and modals) appear in many branches of pure mathematics and also in applied mathematics, e.g., in game theory and mathematical economy. There are 86 papers referred to.

MSC:

 08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems 06-02 Research exposition (monographs, survey articles) pertaining to ordered structures 51-02 Research exposition (monographs, survey articles) pertaining to geometry 52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry

Zbl 0745.00034