Algebraic Structure of TCI-Groupoids: Classes, Clones, Constructions

Abstract

Some aspects of the algebraic structure of TCI-groupoids (totally commutative idempotent groupoids) are presented. Among other things, a full description of the corresponding binary clones is given. To investigate the algebraic structure of TCI-groupoids and their clones the families of two-generated subgroupoids are used. Next the algebraic structure of these families is studied in detail. We present a special construction which allows us to build each finite TCI-groupoid using the two-element semilattice as a kind of elementary "brick". Furthermore it is proved that the class of all regular TCI-groupoids satisfying binary nonbalanced identities is a nonaxiomatizable proper subclass of the class of all regular TCI-groupoids.