×

Abstract clone theory. (English) Zbl 0792.08005

Rosenberg, Ivo (ed.) et al., Algebras and orders. Proceedings of the NATO Advanced Study Institute and Séminaire de mathématiques supérieures, Montréal, Canada, July 29 - August 9, 1991. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 389, 507-530 (1993).
This interesting expository paper gives a useful survey of abstract clone theory. Unfortunately, many of the definitions and results are too technical to be stated easily here.
A concrete clone is a family of finitary operations on a set, closed under composition and containing all projections. When we wish to define an abstract clone we can use either the language of category theory or that of partial algebras; the two definitions are equivalent. Morphisms of clones can be defined in either manifestation.
A representation of a clone \({\mathbf C}\) is a clone homomorphism from \({\mathbf C}\) to \(\text{Clo}(A)\), the clone of all operations on a set \(A\). The concept of representations can be used to define a variety \(\text{Var}({\mathbf C})\) associated with \({\mathbf C}\). Every variety \(\mathcal V\) has associated with it a clone \({\mathbf C}({\mathcal V})\), the clone of all term operations on the free algebra of rank \(\omega\) in \(\mathcal V\) and we have that \(\text{Var}({\mathbf C}({\mathcal V}))\equiv {\mathcal V}\), where two varieties are equivalent if each has an interpretation in the other. We also have that \({\mathbf C}\cong {\mathbf C}(\text{Var}({\mathbf C}))\). It follows that equivalence classes of varieties correspond to isomorphism classes of clones. Surprisingly, it is still possible to distinguish between varieties in an equivalence class in terms of clones, using the concept of clone presentations.
Other topics covered in the paper are tensor products of varieties and applications of the theory of clones to Mal’tsev conditions.
For the entire collection see [Zbl 0778.00036].

MSC:

08B99 Varieties
08A55 Partial algebras
08C05 Categories of algebras
08A40 Operations and polynomials in algebraic structures, primal algebras
08B10 Congruence modularity, congruence distributivity
PDFBibTeX XMLCite