Core Concepts

The free product of two self-similar or automaton semigroups is self-similar (an automaton semigroup) if there exists a homomorphism from one of the base semigroups to the other.

Abstract

The paper investigates the closure properties of self-similar and automaton semigroups under free products. The main result shows that the free product of two self-similar or automaton semigroups S and T is self-similar (an automaton semigroup) if there exists a homomorphism from one of the base semigroups to the other.
The key steps are:
The authors construct an automaton C that generates the free product S ⋆ T, where S and T are self-similar semigroups with a homomorphism between them. The construction uses a combination of marked and unmarked letters to distinguish the elements of the free product.
They prove that the action of S ⋆ T on the language generated by C is faithful, showing that C indeed generates S ⋆ T as a self-similar semigroup.
The authors also show that the constructed automaton C is finite (and complete) if the original automata generating S and T were finite (and complete), respectively. This implies that the free product of two automaton semigroups is also an automaton semigroup under the same homomorphism condition.
The existence of a homomorphism between the base semigroups is a very weak requirement, satisfied for example if one of the semigroups contains an idempotent. The authors explore the limits of this condition and show that no simple or 0-simple idempotent-free semigroup is a finitely generated self-similar or automaton semigroup.
As a byproduct, the authors show that a new free generator can be adjoined to any self-similar or automaton semigroup without losing the property of self-similarity or being an automaton semigroup.

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Deeper Inquiries

The main result of preserving self-similarity in free products of semigroups has several potential applications in various areas of computer science. In formal language theory, self-similar presentations of semigroups play a crucial role in understanding the structure and behavior of formal languages. By preserving self-similarity in free products, researchers can explore new ways to model and analyze language hierarchies, automata, and grammars. This can lead to advancements in language recognition, parsing algorithms, and pattern matching techniques.
In computational complexity, the ability to maintain self-similarity in free products can impact the design and analysis of algorithms. Self-similar structures often exhibit certain regularities that can be exploited to optimize algorithmic processes, reduce time complexity, and improve overall efficiency. By leveraging the properties of self-similar semigroups in free products, researchers can potentially develop novel algorithmic approaches for solving complex computational problems.
Furthermore, in other fields of computer science such as machine learning and artificial intelligence, self-similarity can be utilized to enhance data representation, feature extraction, and pattern recognition tasks. By applying the results of preserving self-similarity in free products, researchers can potentially improve the performance of machine learning models, clustering algorithms, and data analysis techniques by leveraging the inherent structure and properties of self-similar semigroups.

The homomorphism condition in the main result can potentially be further relaxed while still preserving the self-similarity or automaton semigroup property of the free product. One possible direction for relaxation could involve exploring weaker forms of homomorphisms, such as quasi-homomorphisms or approximate homomorphisms. By allowing for more flexibility in the mapping between the base semigroups, researchers may discover new conditions under which the self-similarity property is maintained in the free product.
Additionally, investigating the impact of partial homomorphisms or homomorphisms with certain constraints on the structure of the semigroups could provide insights into the minimal requirements for preserving self-similarity in free products. By exploring different types of mappings and their effects on the resulting automaton semigroups, researchers can potentially uncover new relationships between the homomorphism condition and the self-similarity property.
Overall, by relaxing the homomorphism condition in various ways and analyzing its implications on the self-similarity of free products, researchers can expand the understanding of the interplay between algebraic structures and automaton properties in semigroup theory.

The properties of self-similarity and automaton semigroups in the context of semigroup theory have connections to other algebraic and topological properties of semigroups, such as residual finiteness and amenability.
Residual finiteness, which characterizes the ability to approximate elements of a semigroup by finite subsets, can be related to self-similarity in the sense that self-similar semigroups often exhibit certain regularities that can be approximated effectively. Understanding the interplay between residual finiteness and self-similarity in semigroups can provide insights into the structural properties of these algebraic systems and their computational implications.
Similarly, the concept of amenability in semigroup theory, which relates to the existence of invariant means on the semigroup, can be linked to the properties of automaton semigroups. The self-similarity and regularity inherent in automaton semigroups can potentially influence the existence and behavior of invariant means, leading to a deeper understanding of the amenability of these semigroups.
Exploring the connections between self-similarity, automaton semigroups, residual finiteness, and amenability can provide a comprehensive view of the structural and computational properties of semigroups, shedding light on the intricate relationships between different algebraic and topological characteristics in semigroup theory.

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