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By Julius T. Tou (Eds.)

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Suppose, in fact, that αΓΛά = 0 and aKJ ä = 1 aC\b = 0 aKJb = 1. Thenb = 6 Π 1 = 6 Π (a\J a) = ( ί ) Π ά ) υ ( δ Π α ) = ( ί > Π α ) υ θ = (br\ä) = äC\b = OU (ar\b) = ( ά Π α ) υ («Π6) = α Π (oU5) = ä Γ\ 1 = â. 28 Theorem. A distributive, complemented lattice is a Boolean algebra. The proof involves only checking that all the laws of Boolean algebra hold, which is routine. 29 The set of switching functions of n variables provides the most familiar practical example of the results of Sects. 28. 30 The partitions of the state set S of an automaton are particularly important in reduction and decomposition theory.

HISTORICAL REMARKS The discipline of regular expressions was originally introduced to describe automata in Kleene's paper [2], which was written way back in 1951 and published in 1956. For many years since, this discipline was interesting to switching theorists. Recently, however, there as been considerable interest in this language among people who do advanced programming. It is likely that their interest in the regular-expression language is due to many different reasons, but one important reason is that the regular operators turn up frequently in the advanced study of mechanical language.

25 The importance of the last two theorems is that studying the set of homomorphic images of O is equivalent to studying the set of congruence relations on D. This fact is much used in algebraic automata theory. 26 Our previous study of partitions now suggests these definitions for combining congruences R and S on an algebra ©. (a, b) Ç R Γ\ S <=> a and b £ the same congruence class (block) of R and to the same congruence class of S. (a, b) G R U S <=> 3 a finite sequence a = s0, Si, s2, · · ·, s^-i, s* = b of elements of 4 9 s,· and s»+i G the same congruence class of R or to the same congruence class of S.

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