We show that the well-known equivalences between logic, circuits and algebra,
have their analogies for TC0 resp. Maj-Logic.
For the algebraic part we characterize the languages in TC0 as inverse morphic
images of certain groups. Necessarily these are infinite, since
nonregular sets are concerned. To limit the power of these infinite algebraic
objects, we equip them with a finite type set and introduce the notion of a
finitely typed (infinite) monoid.
For this algebraic characterisations we can show that there is a tight
correspondences between the algebraic subvareties to subclasses of TC0 and
subfamilies of logic formulas.