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(k, l)-kernels, (k, l)-semikernels, k-Grundy functions and duality for state splittings

Summary: (k, l)-kernels, (k, l)-semikernels, k-Grundy
functions and duality for state splittings
Hortensia Galeana-S´anchez and Ricardo G´omez
Instituto de Matem´aticas de la Universidad Nacional Aut´onoma de M´exico
Circuito Exterior, Ciudad Universitaria C.P. 04510, M´exico D.F. M´exico
Line digraphs can be obtained by sequences of state splittings, a partic-
ular kind of operation widely used in symbolic dynamics [12]. Properties
of line digraphs inherited from the source have been studied, for instance
in [7] Harminc showed that the cardinalities of the sets of kernels and so-
lutions (kernel's dual definition) of a digraph and its line digraph coincide.
We extend this for (k, l)-kernels in the context of state splittings and also
look at (k, l)-semikernels, k-Grundy functions and their duals.
Keywords: state splitting; line digraph; kernel; Grundy function; duality
Mathematics Subject Classification: 05C20
1 Introduction
State splitting is a fundamental operation in symbolic dynamics (see [12] or [8]).
A shift of finite type is a dynamical system (homeomorphic to a Cantor set)
consisting of all possible doubly infinite paths in a digraph and correspond to
doubly infinite sequences of symbols (the vertices). Performing state splittings


Source: Aíza, Ricardo Gómez - Instituto de Matemáticas, Universidad Nacional Autónoma de México


Collections: Mathematics