http://www.xrce.xerox.com/research/mltt/fst/articles/fsc-91/fsc91.html

In general, the intersection of two regular relations is not regular. 
For example, let R1 be the relation 
<X^n:A^n B*>,
and R2 the relation 
<X^n:A*B^n>. 
R1 corresponds to a finite-state transducer that maps any 
number of X's to the same number of A's 
followed by any number of trailing B's, 
R2 matches the number of X's and B's allowing any number of 
preceding A's. The intersection R1[intersection]R2, 
<X^n:A^n B^n>, cannot be generated by a finite-state automaton because it maps a regular set to a context-free language.

A regular relation is a finite-state language in which the expressions are composed of symbol pairs rather than single symbols, a mapping of one regular set to another one. Simple finite-state languages (regular sets) are in one-to-one correspondence with ordinary finite-state automata; a regular relation corresponds to a finite-state transducer. 

A finite-state transducer is in other respects like an ordinary finite-state automaton except that it considers two strings rather than one. It can be conceived as a network consisting of states and labeled arcs. Final states are represented as double circles, non-final states by single circles. In a simple automaton, the arc labels are single symbols, in a transducer the labels are symbol pairs, x:y. 

Kaplan and Kay made another important observation that Johnson also had noted in passing: regular relations are closed under serial composition. If we can model two rules applying in sequence by constructing a pair of transducers so that the output of the first transducer is the input side of the second one, we can also produce a single equivalent transducer by composition. The composed machine maps the input of the first transducer to the output of the second without generating any intermediate result. There is no corresponding composition operation for rewrite rules.

In general a composite transducer is larger than its components. In the worst case, the number of states in the composite machine is the product of the sizes of the original transducers.

http://www.ics.uci.edu/~eppstein/161/960215.html#trav
http://www.ics.uci.edu/~eppstein/161/glossary.html