×

Antimorphic action. Categories of algebraic structures with involutions or anti-endomorphisms. (English) Zbl 0609.18003

Research and Exposition in Mathematics. 12. Berlin: Heldermann Verlag. xix, 189 p. DM 48.00 (1986).
The central concept of this work is that of a category \(M-\mathfrak X\) whose objects are those of a category \(\mathfrak X\) on which a given monoid \(M\) acts, in such a way that the members of a prescribed submonoid \(M^+\) give rise to endomorphisms of each \(M-\mathfrak X\)-object while the members of \(M^- = M\setminus M^+\) give rise to anti-endomorphisms. Other properties of \(M\) that are required are \((M^-)^2\le M^+\) and \(M^- M^+\cup M^+M^-\le M^-\). Such a monoid is called a \(\pm\)-monoid, and \(M^-\) is allowed to be empty. Two interesting \(\pm\)-monoids are
\(J=\{1,\kappa\); \(\kappa^ 2=1\}\) with \(J^+=\{1\}\) and \(J^-=\{\kappa \}\), and \(P=\{1,\alpha,\alpha^ 2\); \(\alpha^ 3=\alpha \}\) with \(P^+=\{1,\alpha^ 2\}\) and \(P^-=\{\alpha \}.\)
The scope of application of such a notion can be adequately illustrated by taking \(\mathfrak X\) to be the variety of semigroups, or of bounded distributive lattices, or of rings (with a \(1)\), in which case \(J-\mathfrak X\) is, respectively, the variety of semigroups with an involution, or of de Morgan algebras, or of rings with an involution. Likewise, when \(X=\mathfrak{BD}\), the variety of bounded distributive lattices, \(P-\mathfrak X\) contains the coreflective subvariety of Stone algebras [those in which \(x\wedge \alpha(x)=0]\) and the coreflective subvariety of polar algebras [those in which \(x\le \alpha^2(x)]\), also known as MS-algebras. Very many others are similarly obtained.
This scholarly ‘essay’ is hugely categorical in nature and for that reason the notation gets somewhat complicated at times. The main investigations are rather too complicated to expound here and are concerned with the situation where the original variety is congruence- distributive and the monoid finite, with the deepest results obtained for the variety of bounded distributive lattices.

MSC:

18C10 Theories (e.g., algebraic theories), structure, and semantics
18-02 Research exposition (monographs, survey articles) pertaining to category theory
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
18C05 Equational categories
06B20 Varieties of lattices
08C05 Categories of algebras
20M07 Varieties and pseudovarieties of semigroups
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
08B10 Congruence modularity, congruence distributivity
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)