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A Study of Pseudo-Periodic and Pseudo-Bordered Words for Functions Beyond Identity and Involution
Periodicity, primitivity and borderedness are some of the fundamental notions in combinatorics on words. Motivated by the Watson-Crick complementarity of DNA strands wherein a word (strand) over the DNA alphabet \{A, G, C, T\} and its Watson-Crick complement are informationally ``identical , these notions have been extended to consider pseudo-periodicity and pseudo-borderedness obtained by replacing the ``identity function with ``pseudo-identity functions (antimorphic involution in case of Watson-Crick complementarity). For a given alphabet , an antimorphic involution is an antimorphism, i.e., for all and an involution, i.e., for all . In this thesis, we continue the study of pseudo-periodic and pseudo-bordered words for pseudo-identity functions including involutions.
To start with, we propose a binary word operation, -catenation, that generates -powers (pseudo-powers) of a word for any morphic or antimorphic involution . We investigate various properties of this operation including closure properties of various classes of languages under it, and its connection with the previously defined notion of -primitive words.
A non-empty word is said to be -bordered if there exists a non-empty word which is a prefix of while is a suffix of . We investigate the properties of -bordered (pseudo-bordered) and -unbordered (pseudo-unbordered) words for pseudo-identity functions with the property that is either a morphism or an antimorphism with , for a given , or is a literal morphism or an antimorphism.
Lastly, we initiate a new line of study by exploring the disjunctivity properties of sets of pseudo-bordered and pseudo-unbordered words and some other related languages for various pseudo-identity functions. In particular, we consider such properties for morphic involutions and prove that, for any , the set of all words with exactly -borders is disjunctive (under certain conditions)