Tame congruence theory (Q2782447)

The development of tame congruence theory in the 1980s by Hobby and McKenzie has revolutionized the study of finite algebraic structures and locally finite varieties (a tame is a pair \((\alpha,\beta)\) of congruences satisfying some conditions: uniformity, incompressibility, separation, connectedness, etc.). Numerous enhancements and applications of the theory have been found by various mathematicians since then.NEWLINENEWLINENEWLINEThese notes (Chapter 1, ``The structure of finite algebras'', and Chapter 2, ``Varieties'') are derived from lectures given by the second author while in residence at the Fields Institute during the programme on algebraic model theory. They contain a presentation of the material and ideas at the core of tame congruence theory, as well as an application to the study of residually small varieties.NEWLINENEWLINENEWLINEReaders interested in learning more about the subject are encouraged to read the origial work on tame congruence theory by \textit{D. Hobby} and \textit{R. McKenzie} [The structure of finite algebra, Contemporary Mathematics 76, Providence, RI: American Mathematical Society (1988; Zbl 0721.08001)].NEWLINENEWLINENEWLINEThe results on residual smallness primarily come from work of \textit{K. A. Kernes}, \textit{E. W. Kiss} and \textit{M. A. Valeriote} and can be found in Ann. Pure Appl. Logic 99, 137-169 (1999; Zbl 0930.08003).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00023].