Compatibility and Conjugacy on Partial Arrays.

Area of Science:

• Combinatorics on words
• Computational biology
• Genomics

Background:

• Partial words, introduced by Berstel and Boasson, are used in gene comparison.
• Gene alignment can be modeled as compatible partial words.
• Fundamental properties of partial words are foundational in this field.

Purpose of the Study:

• To investigate the applicability of partial word properties to partial arrays.
• To examine the concepts of compatibility and conjugacy in the context of partial arrays.
• To introduce and analyze relaxed versions of compatibility and conjugacy: k-compatibility and k-conjugacy.

Main Methods:

• Extending the definitions of compatibility and conjugacy to partial arrays.
• Developing theoretical frameworks for analyzing k-compatibility and k-conjugacy.
• Comparative analysis of properties between partial words and partial arrays.

Main Results:

• Demonstrates that fundamental properties of partial words, such as compatibility and conjugacy, can be extended to partial arrays.
• Introduces and defines k-compatibility as a relaxation of the compatibility relation for partial arrays.
• Introduces and defines k-conjugacy for partial arrays, offering a new perspective on array relationships.

Conclusions:

• Partial arrays share significant structural similarities with partial words, allowing for the extension of established combinatorial concepts.
• The introduction of k-compatibility and k-conjugacy provides novel tools for analyzing relationships within and between partial arrays.
• This research bridges combinatorics on words and computational biology by offering new analytical frameworks for gene comparison and related sequence analyses.