  We consider ideals and Boolean combinations of ideals. For the regular
languages within these classes we give expressively complete automaton models.
In addition, we consider general properties of regular ideals and their Boolean
combinations. These properties include effective algebraic characterizations
and lattice identities.
  In the main part of this paper we consider the following deterministic
one-way automaton models: unions of flip automata, weak automata, and
Staiger-Wagner automata. We show that each of these models is expressively
complete for regular Boolean combination of right ideals. Right ideals over
finite words resemble the open sets in the Cantor topology over infinite words.
An omega-regular language is a Boolean combination of open sets if and only if
it is recognizable by a deterministic Staiger-Wagner automaton; and our result
can be seen as a finitary version of this classical theorem. In addition, we
also consider the canonical automaton models for right ideals, prefix-closed
languages, and factorial languages.
  In the last section, we consider a two-way automaton model which is known to
be expressively complete for two-variable first-order logic. We show that the
above concepts can be adapted to these two-way automata such that the resulting
languages are the right ideals (resp. prefix-closed languages, resp. Boolean
combinations of right ideals) definable in two-variable first-order logic.
