
4. PARTITION PAIRS AND PAIR ALGEBRA
Def. A partition pair (,') on the automaton is an ordered pair of partitions on S such that implies .
Def. Given an automaton
A, we say that state s goes into state t under input x iff
For a subset B of S we define
and we say that the subset B goes into set B' under input x iff
.
Thus (,') is a partition pair (p.p.) on A iff the blocks of are mapped into the blocks of ' by A. That is, for every x in I and in , there exists a in such that
.
4.4. Lemma: If (,') and (,') are partition pairs on A, then
(i) (×
, '×')
is a p.p. on A and
(ii) (+
, '+')
is a p.p. on A.
Note that (,I) and (0,) are trivially p.p.
Def.
If is a partition
on S of A, let
is a p.p. on A} and
is a p.p. on A}.
Def. Let L_{1} and L_{2 }be finite lattices. Then a subset of is a pair algebra on if and only if the two following postulates hold:
4.8. Another definition for operator m
Def. Let be a pair algebra on . For x in L_{1 }we define in D}.
.
NB! Pair algebra associated with an automaton:
Def. For a automaton , define:
Last update: 3 August, 2004