Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 1, pp. 181-204 (1998)

Clones Closed with Respect to Permutation Groups or Transformation Semigroups

V. V. Gorlov; R. Pöschel

Dept. of Mathematics, Byelorussian State University
pr-t F. Skoriny 4, 220050 Minsk, Belarus

Technische Universität Dresden, Institut für Algebra
01062 Dresden, Germany

Abstract: This paper presents a new approach to the classification of finite algebras with respect to actions of permutation groups or transformation semigroups on the clone of term operations. In particular it is a contribution to the structural investigation of the (uncountable) lattice ${\cal L}_A$ of clones on a finite set $A$, which can be divided into (sometimes finitely many) intervals represented by so-called $G$-clones. A Galois theory for $G$-clones is developed. Concrete results are presented for clones - and, more generally, for closed classes - on base sets $A$ with two or three elements.

Keywords: finite algebras; permutation groups; clone; Galois theory

Classification (MSC91): 20B25; 08A40

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