The automata that define representations of monomial algebras
Keywords monomial algebras, path algebras, string algebras, automata, regular languages, representations
Status Algebras and Representation Theory 11 (2008) 207--214.
It is well known that the sets of strings that define all representations of
string algebras and many representations of other
quotients of path algebras form a regular set, and hence are defined by
finite state automata.
This short article aims to explain this connection between representation
theory and automata theory in elementary terms; no technical background in
either representation theory or automata theory is assumed.
The article describes the structure of the set of strings of a
monomial algebra as a locally testable and hence regular
set, and describes explicitly the construction of the
automaton, illustrating the construction with an elementary example.
it explains how the sets of strings and bands of a monomial algebra correspond
to the sets of paths and closed (non-powered) circuits in a finite
graph, and how the growth rate of the set of bands is immediately
visible from that graph.
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