Synchronizing non-uniform maps
نویسندگان
چکیده
Let Ω be a set with n elements. A transformation monoid on Ω is a set of maps from Ω to itself which is closed under composition and contains the identity. A transformation monoid M is synchronizing if it contains a map with rank 1 (that is, whose image has only one element). The notion comes from automata theory. A (finite deterministic) automaton consists of a set Ω of states and a set S of transitions or maps on Ω. It is said to be synchronizing if there is a word in the transitions which evaluates to a map of rank 1. (Such a word is called a reset word.) So (Ω, S) is synchronizing if and only if the monoid M = 〈S〉 generated by S is synchronizing. It is easy to tell whether an automaton is synchronizing, but hard to say what the length of the shortest reset word is.
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